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

Author

Alois P. Heinz, Jul 21 2019

Keywords

• nonn
• easy

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).

Crossrefs

Column k=6 of A309280.

Programs

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 *)

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