A102083 a(n) = 8*n^2 + 4*n + 1.

1, 13, 41, 85, 145, 221, 313, 421, 545, 685, 841, 1013, 1201, 1405, 1625, 1861, 2113, 2381, 2665, 2965, 3281, 3613, 3961, 4325, 4705, 5101, 5513, 5941, 6385, 6845, 7321, 7813, 8321, 8845, 9385, 9941, 10513, 11101, 11705, 12325, 12961, 13613, 14281, 14965, 15665

Offset: 0

Author

N. J. A. Sloane, Feb 14 2005

Keywords

• nonn
• easy

Comments

If Y and Z are 2-blocks of a 2n-set X then, for n>=2, a(n-2) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Nov 18 2007

Equals binomial transform of [1, 12, 16, 0, 0, 0, ...]. - Gary W. Adamson, Jul 19 2008

Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 05 2011

First differences of A100157. - John Molokach, Jul 10 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Milan Janjic, Two Enumerative Functions
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A085601, A195241.

Programs

Mathematica

Table[8*n^2 + 4*n + 1, {n, 0, 300}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 13, 41}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

PARI

a(n)=8*n^2+4*n+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1+10*x+5*x^2)/(1-x)^3. - Paul Barry, Jun 04 2005

a(n) = 4*(4*n-1)+a(n-1) (with a(0)=1). - Vincenzo Librandi, Nov 16 2010

E.g.f.: (8*x^2 + 12*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017