A168668 a(n) = n*(2 + 5*n).

0, 7, 24, 51, 88, 135, 192, 259, 336, 423, 520, 627, 744, 871, 1008, 1155, 1312, 1479, 1656, 1843, 2040, 2247, 2464, 2691, 2928, 3175, 3432, 3699, 3976, 4263, 4560, 4867, 5184, 5511, 5848, 6195, 6552, 6919, 7296, 7683, 8080, 8487, 8904, 9331, 9768, 10215, 10672

Offset: 0

Author

Paul Curtz, Dec 02 2009

Keywords

• nonn
• easy

Comments

Appears on the main diagonal of the following table of terms of the Hydrogen series, A169603:

0, 3, 8, 15, 24, ... A005563

0, 7, 16, 1, 40, 55, ... A061039

0, 11, 24, 39, 56, 3, 96, ... A061043

0, 15, 32, 51, 72, 95, 120, ... A061047

0, 19, 40, 63, 88, 115, 144, 175, 208, 1, ...

Links

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

Crossrefs

Cf. A000217, A000384, A001622, A010692, A131242, A254963 (comment).

Cf. A005563, A061039, A061043, A061047, A169603.

Programs

Magma

[n*(2+5*n): n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

Maple

A168668:=n->n*(2+5*n): seq(A168668(n), n=0..50); # Wesley Ivan Hurt, Mar 28 2015

Mathematica

f[n_]:=n*(2+5*n);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3,-3,1},{0,7,24},50] (* Harvey P. Dale, Sep 09 2021 *)

PARI

vector(50,n,n--;n*(2+5*n)) \\ Derek Orr, Jun 26 2015

Formula

G.f.: x*(7 + 3*x)/(1-x)^3.

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

First differences: a(n) - a(n-1) = 10*n-3.

Second differences: a(n) - 2*a(n-1) + a(n-2) = 10 = A010692(n).

a(n) = A131242(10n+6). - Philippe Deléham, Mar 27 2013

a(n) = A000384(n) + 6*A000217(n). - Luciano Ancora, Mar 28 2015

a(n) = A000217(n) + A000217(3*n). - Bruno Berselli, Jul 01 2016

E.g.f.: x*(7 + 5*x)*exp(x). - G. C. Greubel, Jul 29 2016

Sum_{n>=1} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/4 + sqrt(5)*log(phi)/4 - 5*log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

Extensions

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