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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 20 00:16 EDT 2021. Contains 345154 sequences. (Running on oeis4.)