A071408 a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)) with a(1)=0, a(2)=1, and where w is the imaginary cubic root of unity.

0, 1, 4, 7, 10, 15, 20, 25, 32, 39, 46, 55, 64, 73, 84, 95, 106, 119, 132, 145, 160, 175, 190, 207, 224, 241, 260, 279, 298, 319, 340, 361, 384, 407, 430, 455, 480, 505, 532, 559, 586, 615, 644, 673, 704, 735, 766, 799, 832, 865, 900, 935, 970, 1007, 1044

Offset: 1

Author

Robert G. Wilson v, Jun 24 2002

Keywords

• nonn
• easy

Comments

w = exp(2*Pi*i/3)= (-1 - sqrt(-3))/2. Beginning with a(2) the first differences are 3,3,3,5,5,5,7,7,7,9,9,9,11, etc.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A071618.

Programs

Mathematica

a[1] = 0; a[2] = 1; w = Exp[2Pi*I/3]; a[n_] := a[n] = Simplify[(2/3)(1 + w^n + w^(2n)) + 2a[n - 1] - a[n - 2]]; Table[ a[n], {n, 1, 60}]
Table[If[n<3,n-1,Floor[((n+1)^2-4)/3]],{n,1,100}] (*  Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,4,7,10},60] (* Harvey P. Dale, Jun 10 2016 *)

PARI

a(n)=n*(n+2)\3 - 1 \\ Charles R Greathouse IV, Mar 02 2017

Formula

a(n) = A032765(n)-1.

a(n) = floor((n-1)*(n+1)*(n+3)/(3*n+3)). - Gary Detlefs, Jul 13 2010

a(n) = (n-1)^2 - A030511(n-1). - Wesley Ivan Hurt, Jun 19 2013

G.f.: x^2*(1+x)*(x^2-x-1) / ( (1+x+x^2)*(x-1)^3 ). - R. J. Mathar, Jun 23 2013

a(n) = n + floor(n*(n-1)/3) - 1. - Bruno Berselli, Mar 02 2017