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a(n) = (a(n-1) XOR a(n-2)) + 1, a(0) = a(1) = 0.
+10
6
0, 0, 1, 2, 4, 7, 4, 4, 1, 6, 8, 15, 8, 8, 1, 10, 12, 7, 12, 12, 1, 14, 16, 31, 16, 16, 1, 18, 20, 7, 20, 20, 1, 22, 24, 15, 24, 24, 1, 26, 28, 7, 28, 28, 1, 30, 32, 63, 32, 32, 1, 34, 36, 7, 36, 36, 1, 38, 40, 15, 40, 40, 1, 42, 44, 7, 44, 44, 1, 46, 48, 31, 48, 48, 1, 50, 52, 7, 52, 52, 1
COMMENTS
The function moving to the next overlapping pair in the sequence is f:(i,j) = (j, (i XOR j) + 1) is one-to one. This means that the only possible trajectories for the sequence are loops, "lines", and "rays". The inverse is f^{-1}: (i,j) = (i XOR (j-1), i) is defined except when j = 0. Thus the only infinite non-repeating trajectories are those starting with (i,0) for some i. If we define the size of a pair (i,j) to be the largest power of two <= max(i,j). It is trivial to see that the size of f(i,j) is always >= the size of (i,j). Coupled with the fact there are only finitely many pairs with a given size, this means that "line" trajectories are not possible. Any trajectory that goes to a larger size must be part of a ray, so that tracing back will eventually reach zero. - Franklin T. Adams-Watters, Mar 03 2014
FORMULA
a(3n)=2n. a(3n+1)=4*floor((n+1)/2). a(6n+2)=1. a(6n+5)=2^( A001511(n+1)+2)-1. a(-n) = -a(n) if n == 0 (mod 3), a(-1-n) = -a(n) if n == 1 (mod 3), a(-2-n) = a(n) if n == 2 (mod 3). - Michael Somos, Mar 03 2014
EXAMPLE
G.f. = x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 4*x^6 + 4*x^7 + x^8 + 6*x^9 + 8*x^10 + ...
MATHEMATICA
a[ n_] := If[ n < 0, BitXor[ a[n + 1], a[n + 2] - 1], If[n < 2, 0, 1 + BitXor[ a[n - 1], a[n - 2]]]]; (* Michael Somos, Mar 03 2014 *) a[ n_] := If[ Mod[n, 3] == 0, 2 n/3, If[ Mod[n, 3] == 1, 4 Quotient[n + 3, 6], If[ n == -1, -1, 2^IntegerExponent[ Fibonacci[n + 1], 2] - 1]]]; (* Michael Somos, Mar 03 2014 *) nxt[{a_, b_}]:={b, BitXor[a, b]+1}; NestList[nxt, {0, 0}, 80][[All, 1]] (* Harvey P. Dale, Feb 26 2020 *)
Triangle T read by rows: T(n, k) = (n - k)*(1 - (-1)^k + 2*k)/4, with 0 <= k < n.
+10
5
0, 0, 1, 0, 2, 1, 0, 3, 2, 2, 0, 4, 3, 4, 2, 0, 5, 4, 6, 4, 3, 0, 6, 5, 8, 6, 6, 3, 0, 7, 6, 10, 8, 9, 6, 4, 0, 8, 7, 12, 10, 12, 9, 8, 4, 0, 9, 8, 14, 12, 15, 12, 12, 8, 5, 0, 10, 9, 16, 14, 18, 15, 16, 12, 10, 5, 0, 11, 10, 18, 16, 21, 18, 20, 16, 15, 10, 6
COMMENTS
T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A332566(n). The h-th subdiagonal of the triangle T gives 0 followed by the multiples of h+1 repeated.
For k > 0, the (2*k-1)-th and (2*k)-th columns of the triangle T give the multiples of k.
FORMULA
O.g.f.: y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2*(1 - y)^3*(1 + y)^2).
T(n, k) = k*(n - k)/2 for k even.
T(n, k) = (1 + k)*(n - k)/2 for k odd.
EXAMPLE
n\k| 0 1 2 3 4 5
---+------------
1 | 0
2 | 0 1
3 | 0 2 1
4 | 0 3 2 2
5 | 0 4 3 4 2
6 | 0 5 4 6 4 3
...
For n = 4 the matrix M(4) is
0 1 1 2
1 0 1 1
1 1 0 1
2 1 1 0
and therefore T(4, 0) = 0, T(4, 1) = 3, T(4, 2) = 2 and T(4, 3) = 2.
MATHEMATICA
T[n_, k_]:=(n-k)(1-(-1)^k+2k)/4; Flatten[Table[T[n, k], {n, 1, 12}, {k, 0, n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[y*(x*(2 + y + y^2) - (1 + y + 2*y^2))/((1 - x)^2 *(1 - y)^3 (1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 12]]
a(n) = (2*n^2 + 3 + (-1)^n)/2.
+10
4
2, 2, 6, 10, 18, 26, 38, 50, 66, 82, 102, 122, 146, 170, 198, 226, 258, 290, 326, 362, 402, 442, 486, 530, 578, 626, 678, 730, 786, 842, 902, 962, 1026, 1090, 1158, 1226, 1298, 1370, 1446, 1522, 1602, 1682, 1766, 1850, 1938, 2026, 2118
COMMENTS
For n>1, a(n) is the number of row vectors of length 2n with entries in [1,n], first entry 1, with maximum inner distance. That is, vectors where the modular distance between adjacent entries is at least (n-2)/2. Modular distance is the minimum of remainders of (x - y) and (y - x) when dividing by n. Geometrically, this metric counts how far the integers mod n are from each other if 1 and n are adjacent as on a circle. - Omar Aceval Garcia, Jan 30 2021
FORMULA
a(n+1) = 2* A080827(n+1) = (n+2)^2 - A042964(n+1) = a(n) + 2*n + 1 -(-1)^n. a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Colin Barker, Oct 15 2014 G.f.: 2*(1-x+x^2+x^3) / ((1-x)^3*(x+1)). - Colin Barker, Oct 15 2014 E.g.f.: cosh(x) + (1 + x + x^2)*exp(x). - G. C. Greubel, Dec 14 2021
MATHEMATICA
Table[n^2 + 3/2 + (-1)^n/2, {n, 0, 50}] (* Bruno Berselli, Oct 15 2014 *) CoefficientList[Series[2(x^3+x^2-x+1)/((1-x)^3 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2014 *) LinearRecurrence[{2, 0, -2, 1}, {2, 2, 6, 10}, 60] (* Harvey P. Dale, Apr 08 2019 *)
PROG
(PARI) Vec(-2*(x^3+x^2-x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014 (Sage) [(2*n^2 +3 +(-1)^n)/2 for n in (0..50)] # G. C. Greubel, Dec 14 2021
Maximum remainder when (k + 1)^n + (k - 1)^n is divided by k^2 for variable n and k > 2.
+10
2
6, 8, 20, 24, 42, 48, 72, 80, 110, 120, 156, 168, 210, 224, 272, 288, 342, 360, 420, 440, 506, 528, 600, 624, 702, 728, 812, 840, 930, 960, 1056, 1088, 1190, 1224, 1332, 1368, 1482, 1520, 1640, 1680, 1806, 1848, 1980, 2024, 2162, 2208, 2352, 2400, 2550, 2600
FORMULA
maxr(n) = n*n - 2*n if n is even, and n*n - n if n is odd.
G.f.: x^3*(-6-2*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 (proved by Iain Fox, Nov 26 2017) a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 7. - Colin Barker, Oct 29 2017 (proved by Iain Fox, Nov 26 2017) a(n) = n^2 - n*(3 + (-1)^n)/2. - Iain Fox, Nov 26 2017 E.g.f.: x*(exp(x)*x - sinh(x)).
(End)
EXAMPLE
For n = 3, maxr => 3*3 - 3 = 6 since 3 is odd.
For n = 4, maxr => 4*4 - 2*4 = 8 since 4 is even.
MATHEMATICA
LinearRecurrence[{1, 2, -2, -1, 1}, {6, 8, 20, 24, 42}, 50] (* Harvey P. Dale, Apr 18 2018 *)
PROG
(PARI) a(n) = if (n % 2, n^2 - n, n^2 - 2*n); \\ Michel Marcus, Aug 26 2013 (PARI) first(n) = Vec(x^3*(-6-2*x)/((x+1)^2*(x-1)^3) + O(x^(n+3))) \\ Iain Fox, Nov 26 2017
Modified eccentric connectivity index of the cycle graph with n vertices, C[n].
+10
1
12, 32, 40, 72, 84, 128, 144, 200, 220, 288, 312, 392, 420, 512, 544, 648, 684, 800, 840, 968, 1012, 1152, 1200, 1352, 1404, 1568, 1624, 1800, 1860, 2048, 2112, 2312, 2380, 2592, 2664, 2888, 2964, 3200, 3280, 3528, 3612, 3872, 3960, 4232, 4324, 4608, 4704
COMMENTS
The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity index.
FORMULA
a(n) = 2*n*(n-1) if n is odd; and a(n) = 2*n^2 if n is even (n>2).
G.f.: -4*x^3*(3+5*x-4*x^2-2*x^3+2*x^4)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 26 2014
EXAMPLE
a(3) = 3*4 = 12 because there are 3 vertices and each vertex has eccentricity 1 and the total degree of neighboring vertices is 4.
MAPLE
a:= n-> n*(2*n-1+(-1)^n):
MATHEMATICA
a[n_] := 2n(n-Boole[OddQ[n]]);
PROG
(PARI) a(n) = if (n % 2, 2*n*(n-1), 2*n^2); \\ Michel Marcus, Jun 20 2014
1, 1, 2, 2, 24, 24, 720, 720, 40320, 40320, 3628800, 3628800, 479001600, 479001600, 87178291200, 87178291200, 20922789888000, 20922789888000, 6402373705728000, 6402373705728000, 2432902008176640000, 2432902008176640000, 1124000727777607680000
COMMENTS
For n>1, a(n) is the product of the smallest parts in the partitions of 4*floor(n/2) = A168273(n) into two parts.
FORMULA
E.g.f.: log((1+x)/(1-x))/2+1/(1-x^2). - Robert Israel, Oct 19 2014
MATHEMATICA
Table[(2*Floor[n/2])!, {n, 0, 20}]
PROG
(Magma) [Factorial(2*Floor(n/2)) : n in [0..20]];
Irregular triangle of palindromic subsequences. Every row has 2*n+1 terms. From the second row, there are only two alternated numbers: 2*n+4 and 2*n+2.
+10
1
2, 4, 2, 4, 6, 4, 6, 4, 6, 8, 6, 8, 6, 8, 6, 8, 10, 8, 10, 8, 10, 8, 10, 8, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16, 14, 16
COMMENTS
Row sums = 2, 10, 26, 50, ... = A069894(n). 0, for b(n)
0, 1, 2, for c(n)
0, 1, 2, 3, 4, for d(n)
0, 1, 2, 3, 4, 5, 6,
etc,
a(n) is used for
1) b(n+1) = b(n) + (a(0)=2) i.e. 0, 2, 4, 6, ... = A005843(n). 2) c(n+3) = c(n) + (period 3:repeat 4, 2, 4) i.e. 0, 1, 2, 4, 3, 6, 8, ... = A265667(n). 3) d(n+5) = d(n) + (period 5:repeat 6, 4, 6, 4, 6) i.e. 0, 1, 2, 3, 4, 6, 5, 8, 7, 10, ... = A265734(n). Etc.
a(n) has a companion with the same terms,differently distributed,yielding permutations of the nonnegative numbers. See A265672. a(n) other writing (by pairs):
2, 4, 2, 4,
6, 4, 6, 4,
6, 8, 6, 8, 6, 8, 6, 8,
10 8, 10, 8, 10, 8, 10, 8,
10, 12, 10, 12, 10, 12, 10, 12, 10, 12, 10, 12,
14, 12, 14, 12, 14, 12, 14, 12, 14, 12, 14, 12,
etc.
Row sums: 12, 20, 56, 72, ... = 4* A074378(n+1). The last term of the successive rows is the number of their terms.
FORMULA
a(2n) = 4*n + 2 times 4*n + 2 = 2, 2, 6, 6, 6, 6, 6, 6, 10,....
a(2n+1) = 4*(n+1) times 4*(n+1) = 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 12, ....
EXAMPLE
The triangle is
2,
4, 2, 4,
6, 4, 6, 4, 6,
8, 6, 8, 6, 8, 6, 8,
etc.
MATHEMATICA
Table[2 (n - 1) + 2 (Boole@ OddQ@ k + 1), {n, 0, 7}, {k, 2 n + 1}] // Flatten (* Michael De Vlieger, Jan 19 2016 *)
CROSSREFS
Cf. A007395, A005843, A008586, A016825, A053186, A069894, A074378, A086520, A168273, A168276, A265667, A265672, A265734.
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