A309298 (1/6) times the sum of the elements of all subsets of [n] whose sum is divisible by six.

0, 0, 0, 1, 2, 7, 20, 49, 132, 330, 786, 1892, 4472, 10368, 23940, 54720, 123836, 278664, 622896, 1383672, 3058720, 6729184, 14738688, 32157312, 69907200, 151461952, 327158208, 704648448, 1513680192, 3243601280, 6934595840, 14793782400, 31496441856, 66929938944

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-12,14,-36,72,-60,72,-144,104,-48,96,-64).

Formula

G.f.: -x^3*(96*x^16-80*x^15+8*x^14-140*x^13+112*x^12+12*x^11 +40*x^10 -36*x^9 -32*x^8 +32*x^7 -14*x^6 +20*x^5 -21*x^4+12*x^3-7*x^2 +4*x-1) / ((2*x-1)^3 *(2*x^3-1)^3).

a(n) = 6*a(n-1) - 12*a(n-2) + 14*a(n-3) - 36*a(n-4) + 72*a(n-5) - 60*a(n-6) + 72*a(n-7) - 144*a(n-8) + 104*a(n-9) - 48*a(n-10) + 96*a(n-11) - 64*a(n-12). - Wesley Ivan Hurt, Jul 23 2025

Mathematica

CoefficientList[Series[-x^3*(96*x^16 - 80*x^15 + 8*x^14 - 140*x^13 + 112*x^12 + 12*x^11 + 40*x^10 - 36*x^9 - 32*x^8 + 32*x^7 - 14*x^6 + 20*x^5 - 21*x^4 + 12*x^3 - 7*x^2 + 4*x - 1)/((2*x - 1)^3*(2*x^3 - 1)^3), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jul 23 2025 *)
LinearRecurrence[{6, -12, 14, -36, 72, -60, 72, -144, 104, -48, 96, -64}, {0, 0, 0, 1, 2, 7, 20, 49, 132, 330, 786, 1892, 4472, 10368, 23940, 54720, 123836, 278664, 622896, 1383672}, 40] (* Harvey P. Dale, Aug 01 2025 *)

Crossrefs

Column k=6 of A309280.

Sequence in context: A270109 A360421 A123203 * A335927 A261054 A134311

Adjacent sequences: A309295 A309296 A309297 * A309299 A309300 A309301

Keyword

nonn,easy

Author

Alois P. Heinz, Jul 21 2019

Status

approved