A026039 a(n) = (d(n) - r(n))/5, where d = A026037 and r is the periodic sequence with fundamental period (1,2,0,2,0).

2, 4, 8, 13, 21, 31, 44, 61, 81, 106, 135, 169, 209, 254, 306, 364, 429, 502, 582, 671, 768, 874, 990, 1115, 1251, 1397, 1554, 1723, 1903, 2096, 2301, 2519, 2751, 2996, 3256, 3530, 3819, 4124, 4444, 4781, 5134, 5504, 5892, 6297, 6721, 7163, 7624, 8105, 8605, 9126, 9667, 10229, 10813, 11418, 12046, 12696, 13369

Offset: 3

Author

Clark Kimberling

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,1,-3,3,-1).

Crossrefs

Cf. A026037.

Programs

Magma

[Round((2*n-1)*(n^2-n+6)/30): n in [3..60]]; // Vincenzo Librandi, Jun 25 2011

Mathematica

f[n_] := Round[(2 n - 1)*(n^2 - n + 6)/30]; Array[f, 57, 3]
LinearRecurrence[{3,-3,1,0,1,-3,3,-1},{2,4,8,13,21,31,44,61},60] (* Harvey P. Dale, Sep 05 2023 *)

Formula

a(n) = (n + 3)*(2*n^2 + 9*n + 22)/30 - 1/5 - (-1/25*((5 - 5^(1/2))^(1/2) - (5 + 5^(1/2))^(1/2))*2^(1/2))*sin(2*n*Pi/5) - (1/25*((5 - 5^(1/2))^(1/2) + (5 + 5^(1/2))^(1/2))*2^(1/2))*sin(4*n*Pi/5). - Richard Choulet, Dec 14 2008

a(n) = round((2*n-1)*(n^2-n+6)/30) = floor((2*n^3-3*n^2+13*n)/30) = ceiling((n-1)*(2*n^2-n+12)/30) = round((n-1)*(2*n^2-n+12)/30). - Mircea Merca, Dec 03 2010

From R. J. Mathar, May 24 2010: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8).

G.f.: -x^3*(-2+2*x-2*x^2+x^3-2*x^4+3*x^5-3*x^6+x^7) / ( (x^4+x^3+x^2+x+1)*(x-1)^4 ). (End)

a(n) = a(n-5) + n^2 - 6*n + 13, n > 5, a(1)=0, a(2)=1. - Mircea Merca, Dec 03 2010