A152728 a(n) + a(n+1) + a(n+2) = n^3.

0, 0, 0, 1, 7, 19, 38, 68, 110, 165, 237, 327, 436, 568, 724, 905, 1115, 1355, 1626, 1932, 2274, 2653, 3073, 3535, 4040, 4592, 5192, 5841, 6543, 7299, 8110, 8980, 9910, 10901, 11957, 13079, 14268, 15528, 16860, 18265, 19747, 21307, 22946, 24668, 26474

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

• nonn
• easy

Comments

The differences between the terms are (1) a(3*k) - a(3*k-1) = 9*k*(k-1)+1; (2) otherwise, a(n) - a(n-1) = (n-2)*(n-1). - J. M. Bergot, Jul 10 2013

Second differences give A047266. - J. M. Bergot, Dec 01 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A000578, A047266, A057078.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); [0,0,0] cat Coefficients(R!(x^3*(1+4*x+x^2)/((1+x+x^2)*(x-1)^4))); // G. C. Greubel, Sep 01 2018

Maple

seq(ceil((n^3 - 3*n^2 + n)/3), n=0..100); # Robert Israel, Dec 01 2014

Mathematica

k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^3-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
LinearRecurrence[{3,-3,2,-3,3,-1}, {0,0,0,1,7,19}, 50] (* G. C. Greubel, Sep 01 2018 *)

PARI

x='x+O('x^50); concat([0,0,0], Vec(x^3*(1+4*x+x^2)/((1+x+x^2)*(x -1)^4 ))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Aug 15 2010: (Start)

a(n) = ( (n-1)*(n^2-2*n-1) - A057078(n))/3.

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

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-5). - Charles R Greathouse IV, Jul 10 2013

a(3n) = n*(9n^2-9n+1), a(3n+1) = n*(9n^2-2), a(3n+2) = n*(9n^2+9n+1). - Ralf Stephan, Jul 12 2013

a(n) = ceiling((n^3 - 3*n^2 + n)/3). - Robert Israel, Dec 01 2014

E.g.f.: (3*exp(x)*(1 - x + x^3) - exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Mar 04 2023