Displaying 1-10 of 19 results found.
1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478
COMMENTS
If A=[ A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[ A010701] 3 (3, 3, 3, ...); X=[ A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010
FORMULA
G.f.: (1 + 8*x)/(1 - x)^2.
MATHEMATICA
LinearRecurrence[{2, -1}, {1, 10}, 60] (* Harvey P. Dale, Dec 27 2014 *)
PROG
(Sage) [i+1 for i in range(480) if gcd(i, 9) == 9] # Zerinvary Lajos, May 20 2009 (Haskell)
a017173 = (+ 1) . (* 9)
CROSSREFS
Cf. A093644 ((9, 1) Pascal, column m=1).
6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 382, 390, 398, 406, 414, 422, 430
FORMULA
G.f.: (6 + 2*x) / (1 - x)^2.
E.g.f.: (6 + 8*x) * exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
EXAMPLE
G.f. = 6 + 14*x + 22*x^2 + 30*x^3 + 38*x^4 + 46*x^5 + 54*x^6 + 62*x^7 + ...
MATHEMATICA
8Range[0, 60]+6 (* or *) LinearRecurrence[{2, -1}, {6, 14}, 60] (* Harvey P. Dale, Nov 14 2021 *)
PROG
(Haskell)
(PARI) Vec((6+2*x)/(1-x)^2 + O(x^100)) \\ Altug Alkan, Oct 23 2015
Numbers with digital root 1, 4, 7 or 9.
+10
20
1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142
COMMENTS
All squares are members (see A070433). May also be defined as: possible sums of digits of squares. - Zak Seidov, Feb 11 2008 First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - Zak Seidov, Feb 11 2008 Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.
a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - Gary Detlefs, Mar 19 2010 The sum of digits of any term belongs to the sequence. Also the products of any terms belong to the sequence.
Positive integers of the forms x^2 + (2*m+1)*x*y + (m^2+m-2)*y^2, for integers m.
This sequence is closed under multiplication. (End)
FORMULA
O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).
a(n) = a(n-4) + 9. (End)
a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - Gary Detlefs, Mar 19 2010 a(n) = 3*n - floor(n/4) - 2*floor((n+3)/4). - Ridouane Oudra, Jan 21 2024 E.g.f.: (cos(x) + (9*x - 1)*cosh(x) - 3*sin(x) + (9*x - 2)*sinh(x))/4. - Stefano Spezia, Feb 21 2024
MAPLE
seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # Gary Detlefs, Mar 19 2010
MATHEMATICA
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 4, 7, 9, 10}, 70] (* Harvey P. Dale, Aug 29 2015 *)
a(n) = 2*Fibonacci(2*n+1) - 1.
+10
12
1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345, 279167724889, 730870592323
COMMENTS
Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.
Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2009
FORMULA
a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller, May 02 2006 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Nov 02 2016
MATHEMATICA
a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
Table[2*Fibonacci[2*n+1]-1, {n, 0, 30}] (* G. C. Greubel, Apr 22 2018 *) LinearRecurrence[{4, -4, 1}, {1, 3, 9}, 30] (* Harvey P. Dale, Sep 22 2020 *)
PROG
(PARI) Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016 (Sage) [2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019 (GAP) List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019
11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739
COMMENTS
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube. Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.
FORMULA
G.f.: (11 + 2*x)/(1 - x)^2.
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
MATHEMATICA
13 Range[0, 60] + 11
Range[11, 800, 13]
LinearRecurrence[{2, -1}, {11, 24}, 60] (* Harvey P. Dale, Jun 14 2023 *)
PROG
(PARI) vector(60, n, n--; 13*n+11)
(Sage) [13*n+11 for n in range(61)]
(Python) [13*n+11 for n in range(61)]
(Maxima) makelist(13*n+11, n, 0, 60);
(Magma) [13*n+11: n in [0..60]];
CROSSREFS
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
Odd composite numbers congruent to 7 modulo 9.
+10
8
25, 115, 133, 169, 187, 205, 259, 295, 385, 403, 475, 493, 511, 529, 565, 583, 637, 655, 745, 763, 781, 799, 817, 835, 871, 889, 925, 943, 961, 979, 1015, 1105, 1141, 1159, 1177, 1195, 1267, 1285, 1339, 1357, 1375, 1393, 1411, 1465, 1501
MATHEMATICA
Select[18Range[100] + 7, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 25 2014 *) Select[Range[1, 1501, 2], CompositeQ[#]&&Mod[#, 9]==7&] (* or *) Select[Range[7, 1501, 18], CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 31 2021 *)
PROG
(PARI) lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 7), print1(n, ", ")); ); } \\ Michel Marcus, Sep 22 2014
Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0,7,...
+10
7
0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857, 4125, 4402, 4688, 4983, 5287, 5600, 5922, 6253, 6593, 6942, 7300, 7667, 8043, 8428, 8822, 9225
FORMULA
a(n) = n*(9*n+5)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(7+2*x)/(1-x)^3. (End)
EXAMPLE
The spiral begins:
.
15
/ \
16 14
/ \
17 3 13
/ / \ \
18 4 2 12
/ / \ \
19 5 0---1 11
/ / \
20 6---7---8---9--10
.
MATHEMATICA
CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 11 2020 *)
EXTENSIONS
Formula that confused indices corrected by R. J. Mathar, Jun 04 2010
0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510
COMMENTS
After 0, this sequence is the third column of the array in A185874. Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.
FORMULA
O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2. a(n+1) - 3*a(n) + 3*a(n-1) = 3* A105163(n) for n>0. Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)
EXAMPLE
The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
. 0;
. 1, 5;
. 4, 7, 10;
. 9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
Third column is included in A189834. Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.
MATHEMATICA
Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 21, 48}, 50] (* Harvey P. Dale, Jul 18 2019 *)
PROG
(PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
(Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
(Magma) [n*(n+1)*(n+5)/2: n in [0..50]];
CROSSREFS
Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).
a(n) = 81*n^2 - 44*n + 6.
+10
5
6, 43, 242, 603, 1126, 1811, 2658, 3667, 4838, 6171, 7666, 9323, 11142, 13123, 15266, 17571, 20038, 22667, 25458, 28411, 31526, 34803, 38242, 41843, 45606, 49531, 53618, 57867, 62278, 66851, 71586, 76483, 81542, 86763, 92146, 97691, 103398, 109267, 115298, 121491
COMMENTS
The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - a(n)* A156772(n)^2 = 1 for n > 0. For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-3; {1, 1, 3, 1, 9n-4, 1, 3, 1, 1, 18n-6}]. - Magus K. Chu, Sep 13 2022
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (6 + 25*x + 131*x^2)/(1-x)^3.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {6, 43, 242}, 40]
PROG
(Magma) [81*n^2 - 44*n + 6: n in [0..40] ];
Triangle read by rows where T(m,n)=2*m*n + m + n - 6.
+10
4
-2, 1, 6, 4, 11, 18, 7, 16, 25, 34, 10, 21, 32, 43, 54, 13, 26, 39, 52, 65, 78, 16, 31, 46, 61, 76, 91, 106, 19, 36, 53, 70, 87, 104, 121, 138, 22, 41, 60, 79, 98, 117, 136, 155, 174, 25, 46, 67, 88, 109, 130, 151, 172, 193, 214, 28, 51, 74, 97, 120, 143, 166, 189, 212
COMMENTS
Numbers n such that 2n+13 is not prime.
EXAMPLE
Triangle begins:
-2;
1, 6;
4, 11, 18;
7, 16, 25, 34;
10, 21, 32, 43, 54;
13, 26, 39, 52, 65, 78;
16, 31, 46, 61, 76, 91, 106;
19, 36, 53, 70, 87, 104, 121, 138;
22, 41, 60, 79, 98, 117, 136, 155, 174;
25, 46, 67, 88, 109, 130, 151, 172, 193, 214; etc.
MATHEMATICA
t[n_, k_]:= 2 n*k + n + k - 6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)
PROG
(Magma) [2*n*k + n + k - 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012
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