A152731 a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.

0, 0, 1, 63, 665, 3368, 11592, 31696, 74361, 156087, 300993, 542920, 927648, 1515416, 2383745, 3630375, 5376505, 7770336, 10990728, 15251160, 20803993, 27944847, 37017281, 48417776, 62600832, 80084368, 101455425, 127375983

Offset

1,4

Comments

0 + 0 + 1 = 1^6; 0 + 1 + 63 = 2^6; ...

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-21,21,-21,15,-6,1).

Formula

From R. J. Mathar, Dec 12 2008: (Start)

a(n) = -26*n/3 + 20*n^3/3 - 5*n^2 + 7/3 - 2*n^5 + n^6/3 + 5*n^4/3 - 7*A131713(n)/3.

G.f.: x^3*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)). (End)

Mathematica

k0=k1=0; lst={k0, k1}; Do[kt=k1; k1=n^6-k1-k0; k0=kt; AppendTo[lst, k1], {n, 1, 5!}]; lst
LinearRecurrence[{6, -15, 21, -21, 21, -21, 15, -6, 1}, {0, 0, 1, 63, 665, 3368, 11592, 31696, 74361}, 5000]
CoefficientList[Series[x^2*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)), {x, 0, 5000}], x] (* Stefano Spezia, Sep 02 2018 *)

Prog

(PARI) x='x+O('x^30); concat([0, 0], Vec(x^3*(1+x)*(x^4+56*x^3 +246*x^2 +56*x+1)/((1-x)^7*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0, 0] cat Coefficients(R!(x^3*(1+x)*(x^4+56*x^3+246*x^2+56*x+1)/((1-x)^7*(1 +x+ x^2)))); // G. C. Greubel, Sep 01 2018

Crossrefs

Cf. A152728, A152729, A152730, A152725, A152726, A000212.

Sequence in context: A181125 A228221 A022522 * A090028 A152725 A086578

Adjacent sequences: A152728 A152729 A152730 * A152732 A152733 A152734

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Status

approved