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Search: a047615 -id:a047615
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A047461
Numbers that are congruent to {1, 4} mod 8.
+10
28
1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 44, 49, 52, 57, 60, 65, 68, 73, 76, 81, 84, 89, 92, 97, 100, 105, 108, 113, 116, 121, 124, 129, 132, 137, 140, 145, 148, 153, 156, 161, 164, 169, 172, 177, 180, 185, 188, 193, 196, 201, 204, 209, 212, 217, 220, 225, 228, 233
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
From R. J. Mathar, Oct 29 2008: (Start)
G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(n) + a(n+1) = A004770(n).
a(n+1) - a(n) = A010703(n). (End)
a(n) = 8*floor((n-1)/2) + 4 - 3*(n mod 2). - Reinhard Zumkeller, Dec 15 2008
a(n) = A153125(n,n). - Reinhard Zumkeller, Dec 20 2008
a(n) = 8*n - a(n-1) - 11 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (7 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
a(1)=1, a(2)=4, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jun 18 2013
a(n) = 1 + A004526(n)*3 + A004526(n-1)*5. - Gregory R. Bryant, Apr 16 2014
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 1.
E.g.f.: (8 - exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/4 + sqrt(2)*arccoth(sqrt(2))/8. - Amiram Eldar, Dec 11 2021
MAPLE
seq(coeff(series(factorial(n)*((8-exp(-x)+(8*x-7)*exp(x))/2), x, n+1), x, n), n=1..60); # Muniru A Asiru, Jul 23 2018
MATHEMATICA
Flatten[(#+{1, 4})&/@(8Range[0, 30])] (* or *) LinearRecurrence[ {1, 1, -1}, {1, 4, 9}, 60] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[ Series[(4x^2 + 3x + 1)/((x + 1) (x - 1)^2), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 24 2018 *)
PROG
(Maxima) makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */
(GAP) Filtered([1..250], n->n mod 8=1 or n mod 8 =4); # Muniru A Asiru, Jul 23 2018
(Magma) [4*n-3 - ((n+1) mod 2): n in [1..70]]; // G. C. Greubel, Mar 15 2024
(SageMath) [4*n-3 - ((n+1)%2) for n in range(1, 71)] # G. C. Greubel, Mar 15 2024
(Python)
def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A004526, A004770, A010703, A017077, A017113, A047398, A047452.
Cf. A047470, A047524, A047535, A047615, A047617, A153125, A342819.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047470
Numbers that are congruent to {0, 3} mod 8.
+10
21
0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, 43, 48, 51, 56, 59, 64, 67, 72, 75, 80, 83, 88, 91, 96, 99, 104, 107, 112, 115, 120, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 171, 176, 179, 184, 187, 192, 195, 200, 203, 208, 211, 216, 219, 224, 227, 232
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Maximum number of squares attacked by a queen on an n X n chessboard. - Stewart Gordon, Mar 23 2001
Equivalently, maximum vertex degree in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017
Number of squares attacked by a queen on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 19 2001
List of squared distances between points of diamond 'lattice' with minimal distance sqrt(3). - Arnold Neumaier (Arnold.Neumaier(AT)univie.ac.at), Aug 01 2003
Draw a figure-eight knot diagram on the plane and assign a list of nonnegative numbers at each crossing as follows. Start with 0 and choose a crossing on the knot. Pick a direction and walk around the knot, appending the following nonnegative number everytime a crossing is visited. Two series of sequences are obtained: this sequence, A047535, A047452, A047617 and A047615, A047461, A047452, A047398 (see example). - Franck Maminirina Ramaharo, Jul 22 2018
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Maximum Vertex Degree.
Eric Weisstein's World of Mathematics, Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = a(n-1) + 4 + (-1)^n.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = A042948(n) + A005843(n).
G.f.: (3x+5*x^2)/((1-x)*(1-x^2)).
a(n) = 8*n - a(n-1) - 13 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A171497(k). - Philippe Deléham, Oct 17 2011
a(n) = 4*n -(9 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
E.g.f: (10 - exp(-x) + (8*x - 9)*exp(x))/2. - Franck Maminirina Ramaharo, Jul 22 2018
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/2 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021
EXAMPLE
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
Consider the following equivalent figure-eight knot diagrams:
+---------------------+ +-----------------n
| | | |
| +---------B-----+ | w-----A---e
| | | | | | | |
| n-----C---+ | | | | | |
| | | | | | <=> | +-------B-----s |
| | +---D-----+ | | | | |
| | | | | | | |
w-----A---------e | +---C-------D---------+
| | | |
s---------------------+ +-------+
Uppercases A,B,C,D denote crossings, and lowercases n,s,w,e denote directions. Due to symmetry and ambient isotopy, all possible sequences are obtained by starting from crossing A and choose either direction 'n' or 's'.
Direction 'n':
A: 0, 3, 8, 11, 16, 19, 24, 27, 32, 35, 40, ... (this sequence);
B: 4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, ... A047535;
C: 1, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, ... A047452;
D: 2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, ... A047617.
Direction 's':
A: 0, 5, 8, 13, 16, 21, 24, 29, 32, 37, 40, ... A047615;
B: 1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, ... A047461;
C: 2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, ... A047524;
D: 3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, ... A047398.
(End)
MAPLE
a:=n->add(4+(-1)^j, j=1..n):seq(a(n), n=0..64); # Zerinvary Lajos, Dec 13 2008
MATHEMATICA
With[{c = 8 Range[0, 30]}, Sort[Join[c, c + 3]]] (* Harvey P. Dale, Oct 11 2011 *)
Table[(8 n - 9 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Jun 20 2017 *)
LinearRecurrence[{1, 1, -1}, {0, 3, 8}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(x (3 + 5 x))/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 20 2017 *)
PROG
(PARI) forstep(n=0, 200, [3, 5], print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011
(GAP) a:=[0, 3, 8];; for n in [4..50] do a[n]:=a[n-1]+a[n-2]-a[n-3]; od; a; # Muniru A Asiru, Jul 23 2018
(Python)
def A047470(n): return (n-1<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A042948, A047398, A047461, A047452, A047524, A047535, A047615, A047617.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Vincenzo Librandi, Aug 06 2010
STATUS
approved
A047452
Numbers that are congruent to {1, 6} mod 8.
+10
19
1, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 89, 94, 97, 102, 105, 110, 113, 118, 121, 126, 129, 134, 137, 142, 145, 150, 153, 158, 161, 166, 169, 174, 177, 182, 185, 190
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Except for 1, numbers whose binary reflected Gray code (A014550) ends with 01. - Amiram Eldar, May 17 2021
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
G.f.: x*(1+5*x+2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
E.g.f.: (4 + exp(-x) + (8*x - 5)*exp(x))/2. - Ilya Gutkovskiy, May 25 2016
a(n) = A047615(n) + 1. - Franck Maminirina Ramaharo, Jul 23 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021
MAPLE
seq(coeff(series(factorial(n)*((4+exp(-x)+(8*x-5)*exp(x))/2), x, n+1), x, n), n=1..60); # Muniru A Asiru, Jul 24 2018
MATHEMATICA
Table[(8 n - 5 + (-1)^n)/2, {n, 1, 100}] (* Franck Maminirina Ramaharo, Jul 23 2018 *)
CoefficientList[ Series[(2x^2 + 5x + 1)/((x - 1)^2 (x + 1)), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 6, 9}, 51] (* Robert G. Wilson v, Jul 24 2018 *)
PROG
(Maxima) makelist((8*n - 5 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 23 2018 */
(GAP) Filtered([0..250], n->n mod 8=1 or n mod 8=6); # Muniru A Asiru, Jul 24 2018
(Python)
def A047452(n): return (n<<2)-2-(n&1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A014550, A047398, A047461, A047470, A047524, A047535, A047615, A047617.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047617
Numbers that are congruent to {2, 5} mod 8.
+10
19
2, 5, 10, 13, 18, 21, 26, 29, 34, 37, 42, 45, 50, 53, 58, 61, 66, 69, 74, 77, 82, 85, 90, 93, 98, 101, 106, 109, 114, 117, 122, 125, 130, 133, 138, 141, 146, 149, 154, 157, 162, 165, 170, 173, 178, 181, 186, 189, 194, 197, 202, 205, 210, 213, 218, 221, 226, 229, 234
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers whose binary reflected Gray code (A014550) ends with 11. - Amiram Eldar, May 17 2021
LINKS
Table of n, a(n) for n=1..59.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 8*n - a(n-1) - 9 (with a(1)=2). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (5 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: (2+3*x+3*x^2)/((-1+x)^2*(1+x)). - Harvey P. Dale, Feb 23 2016
a(1)=2, a(2)=5, a(3)=10, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Feb 23 2016
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 2.
E.g.f.: (6 - exp(-x) + (8*x - 5)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 - log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021
MATHEMATICA
Select[Range[300], MemberQ[{2, 5}, Mod[#, 8]]&] (* or *) LinearRecurrence[ {1, 1, -1}, {2, 5, 10}, 80] (* Harvey P. Dale, Feb 23 2016 *)
PROG
(Maxima) makelist(4*n -(5 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */
(Python)
def A047617(n): return (n-1<<2)+1+(n&1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A014550, A047398, A047452, A047461, A047470, A047524, A047535, A047615.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Vincenzo Librandi, Aug 06 2010
STATUS
approved
A047524
Numbers that are congruent to {2, 7} mod 8.
+10
18
2, 7, 10, 15, 18, 23, 26, 31, 34, 39, 42, 47, 50, 55, 58, 63, 66, 71, 74, 79, 82, 87, 90, 95, 98, 103, 106, 111, 114, 119, 122, 127, 130, 135, 138, 143, 146, 151, 154, 159, 162, 167, 170, 175, 178, 183, 186, 191, 194, 199, 202, 207, 210, 215, 218, 223, 226, 231, 234
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A195605 is a subsequence. - Bruno Berselli, Sep 21 2011
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Mar 22 2011: (Start)
a(n) = 4*n - 3/2 + (-1)^n/2.
G.f.: x*(2+5*x+x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = 4*n - (n mod 2) - 1.
a(n) = A047615(n) + 2.
a(2*n) = A004771(n-1).
a(2*n-1) = A017089(n-1).
E.g.f.: ((8*x - 3)*exp(x) + exp(-x) + 2)/2. (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Muniru A Asiru, Aug 06 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 - log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021
MAPLE
seq(coeff(series(x*(2+5*x+x^2)/((1+x)*(1-x)^2), x, n+1), x, n), n=1..60); # Muniru A Asiru, Aug 06 2018
MATHEMATICA
Select[Range[300], MemberQ[{2, 7}, Mod[#, 8]]&] (* or *)
LinearRecurrence[ {1, 1, -1}, {2, 7, 10}, 60] (* Harvey P. Dale, Nov 05 2017 *)
CoefficientList[ Series[(x^2 + 5x + 2)/((x - 1)^2 (x + 1)), {x, 0, 60}], x] (* Robert G. Wilson v, Aug 07 2018 *)
PROG
(Maxima) makelist(4*n - mod(n, 2) - 1, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */
(PARI) is(n) = #setintersect([n%8], [2, 7]) > 0 \\ Felix Fröhlich, Aug 06 2018
(GAP) Filtered([0..250], n->n mod 8=2 or n mod 8=7); # Muniru A Asiru, Aug 06 2018
(Python)
def A047524(n): return (n<<2)-1-(n&1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A047398, A047461, A047452, A047470, A047535, A047615, A047617, A195605.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Vincenzo Librandi, Aug 06 2010
STATUS
approved
A047535
Numbers that are congruent to {4, 7} mod 8.
+10
17
4, 7, 12, 15, 20, 23, 28, 31, 36, 39, 44, 47, 52, 55, 60, 63, 68, 71, 76, 79, 84, 87, 92, 95, 100, 103, 108, 111, 116, 119, 124, 127, 132, 135, 140, 143, 148, 151, 156, 159, 164, 167, 172, 175, 180, 183, 188, 191, 196, 199, 204, 207, 212, 215, 220, 223, 228, 231
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Union of A004771 and A017113.
LINKS
Table of n, a(n) for n=1..58.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 8*n - a(n-1) - 5 (with a(1)=4). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n -(1 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
G.f.: x*(4+3*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 10 2015
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 4.
E.g.f.: (2 - exp(-x) + (8*x - 1)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 11 2021
MAPLE
A047535:=n->4*n - (1 + (-1)^n)/2; seq(A047535(n), n=1..100); # Wesley Ivan Hurt, Feb 24 2014
MATHEMATICA
Table[4n - (1 + (-1)^n)/2, {n, 100}] (* Wesley Ivan Hurt, Feb 24 2014 *)
PROG
(Maxima) makelist(4*n - (1 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */
(Python)
def A047535(n): return (n<<2)-(n&1^1) # Chai Wah Wu, Mar 30 2024
CROSSREFS
Cf. A004771, A017113, A047398, A047461, A047452, A047470, A047524, A047615, A047617.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047398
Numbers that are congruent to {3, 6} mod 8.
+10
16
3, 6, 11, 14, 19, 22, 27, 30, 35, 38, 43, 46, 51, 54, 59, 62, 67, 70, 75, 78, 83, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 147, 150, 155, 158, 163, 166, 171, 174, 179, 182, 187, 190, 195, 198, 203, 206, 211, 214, 219, 222, 227, 230
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..58.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 8*n - a(n-1) - 7, n > 1. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 4*n - (3 + (-1)^n)/2.
G.f.: x*(3+3*x+2*x^2) / ( (1+x)*(x-1)^2 ). (End)
From Franck Maminirina Ramaharo, Aug 06 2018: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 3.
a(n) = 4*n + (n mod 2) - 2.
a(n) = A047470(n) + 3.
a(2*n) = A017137(n-1).
a(2*n-1) = A017101(n-1).
E.g.f.: ((8*x - 3)*exp(x) - exp(-x) + 4)/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/16 + log(2)/8 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021
MAPLE
A047398:=n->4*n-(3+(-1)^n)/2: seq(A047398(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2017
MATHEMATICA
Flatten[# + {3, 6} & /@ (8 Range[0, 28])] (* Arkadiusz Wesolowski, Sep 25 2012 *)
LinearRecurrence[{1, 1, -1}, {3, 6, 11}, 60] (* Harvey P. Dale, Oct 26 2020 *)
PROG
(Maxima) makelist(4*n + mod(n, 2) - 2, n, 1, 100); /* Franck Maminirina Ramaharo, Aug 06 2018 */
(Python)
def A047398(n): return ((n<<2)|(n&1))-2 # Chai Wah Wu, Mar 30 2024
CROSSREFS
Union of A017101 and A017137.
Cf. A047461, A047452, A047470, A047524, A047535, A047615, A047617.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047481
Numbers that are congruent to {0, 2, 5, 7} mod 8.
+10
2
0, 2, 5, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 26, 29, 31, 32, 34, 37, 39, 40, 42, 45, 47, 48, 50, 53, 55, 56, 58, 61, 63, 64, 66, 69, 71, 72, 74, 77, 79, 80, 82, 85, 87, 88, 90, 93, 95, 96, 98, 101, 103, 104, 106, 109, 111, 112, 114, 117, 119, 120, 122, 125
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Complement of A047415.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
FORMULA
From Colin Barker, May 14 2012: (Start)
a(n) = (1/4+i/4)*((-3+3*i)-i*(-i)^n+i^n+(4-4*i)*n) where i=sqrt(-1).
G.f.: x^2*(2+x+x^2)/((1-x)^2*(1+x^2)). (End)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4. - Vincenzo Librandi, May 16 2012
a(n) = (-1*((-1)^((n-1)/2-(-1)^n/4-1/4)))/2+2*(n-1)+1/2.
a(n) = cos(n*Pi/2)-1/2*cos((n-1)*Pi/2)-1/2*cos(n*Pi/2)+2*(n-1)+1/2. - Cédric Christian Bernard Gagneux, Mar 05 2014
a(2k) = A047524, a(2k-1) = A047615(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (2 - sin(x) + cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4 - Pi/16. - Amiram Eldar, Dec 21 2021
MAPLE
A047481:=n->(-1*((-1)^((n-1)/2-(-1)^n/4-1/4)))/2+2*(n-1)+1/2: seq(A047481(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
MATHEMATICA
Select[Range[0, 300], MemberQ[{0, 2, 5, 7}, Mod[#, 8]]&] (* Vincenzo Librandi, May 16 2012 *)
LinearRecurrence[{2, -2, 2, -1}, {0, 2, 5, 7}, 70] (* Harvey P. Dale, May 28 2017 *)
PROG
(Magma) I:=[0, 2, 5, 7, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012
(PARI) a(n)=[-1, 0, 2, 5][n%4]+n\4*8 \\ Charles R Greathouse IV, Mar 05 2014
(PARI) x='x+O('x^100); concat(0, Vec(x^2*(2+x+x^2)/((1-x)^2*(1+x^2)))) \\ Altug Alkan, Dec 24 2015
CROSSREFS
Cf. A047415, A047524, A047615.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047430
Numbers that are congruent to {0, 4, 5, 6} mod 8.
+10
1
0, 4, 5, 6, 8, 12, 13, 14, 16, 20, 21, 22, 24, 28, 29, 30, 32, 36, 37, 38, 40, 44, 45, 46, 48, 52, 53, 54, 56, 60, 61, 62, 64, 68, 69, 70, 72, 76, 77, 78, 80, 84, 85, 86, 88, 92, 93, 94, 96, 100, 101, 102, 104, 108, 109, 110, 112, 116, 117, 118, 120, 124
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..62.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
G.f.: x^2*(4+x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 25 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-5+i^(2*n)-(2+i)*i^(-n)-(2-i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047406(k), a(2k-1) = A047615(k). (End)
E.g.f.: (4 - sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*log(2+sqrt(2))/8 - (2-sqrt(2))*(Pi-log(2))/16. - Amiram Eldar, Dec 23 2021
MAPLE
A047430:=n->(8*n-5+I^(2*n)-(2+I)*I^(-n)-(2-I)*I^n)/4: seq(A047430(n), n=1..100); # Wesley Ivan Hurt, May 25 2016
MATHEMATICA
Table[(8n-5+I^(2n)-(2+I)*I^(-n)-(2-I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 25 2016 *)
Select[Range[0, 124], MemberQ[{0, 4, 5, 6}, Mod[#, 8]] &] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 5, 6, 8}, 100] (* Harvey P. Dale, Aug 05 2023 *)
PROG
(Magma) [n : n in [0..150] | n mod 8 in [0, 4, 5, 6]]; // Wesley Ivan Hurt, May 25 2016
CROSSREFS
Cf. A047406, A047615.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A047439
Numbers that are congruent to {0, 1, 5, 6} mod 8.
+10
1
0, 1, 5, 6, 8, 9, 13, 14, 16, 17, 21, 22, 24, 25, 29, 30, 32, 33, 37, 38, 40, 41, 45, 46, 48, 49, 53, 54, 56, 57, 61, 62, 64, 65, 69, 70, 72, 73, 77, 78, 80, 81, 85, 86, 88, 89, 93, 94, 96, 97, 101, 102, 104, 105, 109, 110, 112, 113, 117, 118, 120, 121, 125
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..63.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=5 and b(k)=2^(k+1) for k>1. - Philippe Deléham, Oct 19 2011
G.f.: x^2*(1+4*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
a(n) = Sum_{i=1..n} gcd(i+2, i-2). - Wesley Ivan Hurt, Jan 23 2014
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = 2n+(1+i)*(4i-4-(1-i)*i^(2n)-i^(1-n)+i^n)/4 where i=sqrt(-1).
a(2n) = A047452, a(2n-1) = A047615(n). (End)
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021
MAPLE
A047439:=n->add(gcd(i+2, i-2), i=1..n); seq(A047439(n), n=0..100); # Wesley Ivan Hurt, Jan 23 2014
MATHEMATICA
Table[Sum[GCD[i + 2, i - 2], {i, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 23 2014 *)
PROG
(Magma) [n : n in [0..150] | n mod 8 in [0, 1, 5, 6]]; // Wesley Ivan Hurt, May 22 2016
CROSSREFS
Cf. A030308, A047452, A047615.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Wesley Ivan Hurt, May 22 2016
STATUS
approved
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