A168547 a(n) = 1 - 2*n^2 + 4*n*(1 + 2*n^2)/3.

1, 3, 17, 59, 145, 291, 513, 827, 1249, 1795, 2481, 3323, 4337, 5539, 6945, 8571, 10433, 12547, 14929, 17595, 20561, 23843, 27457, 31419, 35745, 40451, 45553, 51067, 57009, 63395, 70241, 77563, 85377, 93699, 102545, 111931, 121873, 132387, 143489, 155195

Offset: 0

Author

Paul Curtz, Nov 29 2009

Keywords

• nonn
• easy

Comments

Binomial transform of the quasi-finite sequence 1,2,12,16,0,... (0 continued).

A bisection of A168582.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Programs

Magma

[1-2*n^2+4*n*(1+2*n^2)/3: n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

Mathematica

Table[1-2*n^2+4*n*(1+2*n^2)/3, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,3,17,59},60] (* Harvey P. Dale, May 21 2023 *)

PARI

a(n)=1-2*n^2+4*n*(1+2*n^2)/3 \\ Charles R Greathouse IV, Jul 26 2016

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: (1 - x + 11*x^2 + 5*x^3)/(x-1)^4.

First differences: a(n+1) - a(n) = 2*A054569(n+1).

Second differences: a(n+2) - 2*a(n+1) + a(n) = 4*A004767(n).

Third differences: a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = 16.

a(n+1) = A166464(n) + A035597(n+1).

a(n) = 1 - 2*n^2 + 4*A005900(n). - R. J. Mathar, Dec 05 2009

E.g.f.: (1/3)*(3 + 6*x + 18*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Jul 26 2016

Extensions

Edited and extended by R. J. Mathar, Dec 05 2009