A105636 -id:A105636 - cp's OEIS frontend

A289705 Number of 4-cycles in the n-triangular honeycomb queen graph.

0, 0, 15, 96, 330, 855, 1866, 3624, 6468, 10818, 17193, 26208, 38598, 55209, 77028, 105168, 140904, 185652, 241011, 308736, 390786, 489291, 606606, 745272, 908076, 1098006, 1318317, 1572480, 1864254, 2197629, 2576904, 3006624, 3491664, 4037160, 4648599, 5331744, 6092730

Offset: 1

Eric W. Weisstein, Jul 14 2017

nonn

Eric Weisstein's World of Mathematics, Graph Cycle
Index entries for linear recurrences with constant coefficients, signature (4, -4, -4, 10, -4, -4, 4, -1).

Cf. A105636 (3-cycles), A289706 (5-cycles), A289707 (6-cycles).

Table[(24 n^5 + 170 n^4 - 660 n^3 + 160 n^2 + 606 n - 165 + (-1)^n (165 - 30 n))/320, {n, 20}]
LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {0, 0, 15, 96, 330, 855, 1866, 3624}, 20]
CoefficientList[Series[-((3 x^2 (-5 - 12 x - 2 x^2 + 7 x^3))/((-1 + x)^6 (1 + x)^2)), {x, 0, 20}], x]

a(n) = (24*n^5 + 170*n^4 - 660*n^3 + 160*n^2 + 606*n - 165 + (-1)^n*(165 - 30*n))/320.
a(n) = 4*a(n-1)-4*a(n-2)-4*a(n-3)+10*a(n-4)-4*a(n-5)-4*a(n-6)+4*a(n-7)-a(n-8).
G.f.: (-3*x^3*(-5 - 12*x - 2*x^2 + 7*x^3))/((-1 + x)^6*(1 + x)^2).

A303383 Total volume of all cubes with side length q such that n = p + q and p <= q.

Original entry on oeis.org

0, 1, 8, 35, 91, 216, 405, 748, 1196, 1925, 2800, 4131, 5643, 7840, 10241, 13616, 17200, 22113, 27216, 34075, 41075, 50336, 59653, 71820, 83916, 99541, 114920, 134603, 153811, 178200, 201825, 231616, 260288, 296225, 330616, 373491, 414315, 464968, 512981

Offset: 1

Wesley Ivan Hurt, Apr 22 2018

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A105636.
Subsequence of A217843.
After 8, all terms belong to A265377.

[0] cat [&+[(n-k)^3: k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Apr 23 2018

Table[Sum[(n - i)^3, {i, Floor[n/2]}], {n, 50}]

a(n) = Sum_{i=1..floor(n/2)} (n-i)^3.
From Bruno Berselli, Apr 23 2018: (Start)
G.f.: x*(1 + x + x^2)*(1 + 6*x + 16*x^2 + 6*x^3 + x^4)/((1 - x)^5*(1 + x)^4).
a(n) = (30*(n - 2)*(n + 1)*(n^2 - n + 2) + (2*n - 1)*(2*n^2 - 2*n - 1)*(-1)^n + 119)/128. Therefore:
a(n) = n^2*(3*n - 2)*(5*n - 6)/64 for n even;
a(n) = (n - 1)^2*(3*n - 1)*(5*n + 1)/64 for n odd. (End)
a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9). - Wesley Ivan Hurt, Apr 23 2021

cp's OEIS Frontend

A289705 Number of 4-cycles in the n-triangular honeycomb queen graph.

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