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

1,3

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

Table of n, a(n) for n=1..55.

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

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

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 *)

Prog

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

Crossrefs

Cf. A071618.

Sequence in context: A310705 A310706 A095875 * A137461 A137379 A342761

Adjacent sequences: A071405 A071406 A071407 * A071409 A071410 A071411

Keyword

nonn,easy

Author

Robert G. Wilson v, Jun 24 2002

Status

approved