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A152729 a(n) + a(n+1) + a(n+2) = n^4, with a(0) = a(1) = a(2) = 0. 5
0, 0, 1, 15, 65, 176, 384, 736, 1281, 2079, 3201, 4720, 6720, 9296, 12545, 16575, 21505, 27456, 34560, 42960, 52801, 64239, 77441, 92576, 109824, 129376, 151425, 176175, 203841, 234640, 268800, 306560, 348161, 393855, 443905, 498576, 558144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n+2) - a(n-1) = n^4 - (n-1)^4 = A005917(n) for all n in Z. - Michael Somos, Sep 02 2018

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

FORMULA

a(n) = -1/3 + (4/3)*(n-1)^3 + (2/3)*(n-1)^2 - (4/3)*(n-1) - ((1/6)*i)*sqrt(3)*(-1/2 + (1/2)*i*sqrt(3))^(n-1) + (1/6)*(-1/2 - (1/2)*i*sqrt(3))^(n-1) + (1/3)*(n-1)^4 + (1/6)*(-1/2 + (1/2)*i*sqrt(3))^(n-1) + (1/6)*i*sqrt(3)*(-1/2 - (1/2)*i*sqrt(3))^(n-1) with n >= 0 and i=sqrt(-1). - Paolo P. Lava, Dec 19 2008

G.f.: -x^3*(x+1)*(x^2+10*x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Oct 28 2014

a(n) = a(2 - n) for all n in Z. - Michael Somos, Sep 02 2018

EXAMPLE

0 + 0 + 1 = 1^4; 0 + 1 + 15 = 2^4; 1 + 15 + 65 = 3^4; ...

G.f. = x^3 + 15*x^4 + 65*x^5 + 176*x^6 + 384*x^7 + 736*x^8 + 1281*x^9 + ... - Michael Somos, Sep 02 2018

MATHEMATICA

k0=k1=0; lst={k0, k1}; Do[kt=k1; k1=n^4-k1-k0; k0=kt; AppendTo[lst, k1], {n, 1, 4!}]; lst

LinearRecurrence[{4, -6, 5, -5, 6, -4, 1}, {0, 0, 1, 15, 65, 176, 384}, 50] (* G. C. Greubel, Sep 01 2018 *)

a[ n_] := With[ {m = Max[n, 2 - n]}, SeriesCoefficient[ x^3 (1 + x) (1 + 10 x + x^2) / ((1 - x)^5 (1 + x + x^2)), {x , 0, m}]]; (* Michael Somos, Sep 02 2018 *)

PROG

(PARI) concat([0, 0], Vec(-x^3*(x+1)*(x^2+10*x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014

(PARI) {a(n) = my(m = max(n, 2 - n)); polcoeff( x^3 * (1 + x) * (1 + 10*x + x^2) / ((1 - x)^5 * (1 + x + x^2)) + x * O(x^m), m)}; /* Michael Somos, Sep 02 2018 */

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0] cat Coefficients(R!(x^3*(x+1)*(x^2+10*x+1)/((1-x)^5*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018

CROSSREFS

Cf. A005917, A152728, A152725, A152726, A000212.

Sequence in context: A005917 A218216 A027455 * A055268 A090026 A027526

Adjacent sequences:  A152726 A152727 A152728 * A152730 A152731 A152732

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

STATUS

approved

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Last modified October 23 01:24 EDT 2018. Contains 316518 sequences. (Running on oeis4.)