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
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
KEYWORD
nonn,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, Dec 11 2008
EXTENSIONS
Definition adapted to offset by Georg Fischer, Jun 18 2021
STATUS
approved