A033585 a(n) = 2*n*(4*n + 1).

0, 10, 36, 78, 136, 210, 300, 406, 528, 666, 820, 990, 1176, 1378, 1596, 1830, 2080, 2346, 2628, 2926, 3240, 3570, 3916, 4278, 4656, 5050, 5460, 5886, 6328, 6786, 7260, 7750, 8256, 8778, 9316, 9870, 10440, 11026, 11628, 12246, 12880, 13530, 14196, 14878, 15576

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If Y is a fixed 3-subset of a (4n+1)-set X then a(n) is the number of (4n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Milan Janjic, Two Enumerative Functions.
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A081266, A144312, A144314. - Reinhard Zumkeller, Sep 17 2008

Cf. A000217.

Programs

Maple

seq(binomial(4*n+1,2), n=0..36); # Zerinvary Lajos, Jan 21 2007

Mathematica

f[n_]:=2*n*(4*n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)

PARI

a(n)=2*n*(4*n+1) \\ Charles R Greathouse IV, Jun 16 2017

Formula

a(n) = 2*A007742(n).

a(n) = A000217(4*n) = A014105(2*n). - Reinhard Zumkeller, Sep 17 2008

a(n) = 16*n + a(n-1) - 6 with a(0) = 0. - Vincenzo Librandi, Aug 05 2010

a(n) = A005843(n)*A016813(n). - Omar E. Pol, Oct 31 2013

G.f.: -2*x*(5+3*x)/(x-1)^3 . - R. J. Mathar, Feb 06 2017

E.g.f.: (8*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 18 2017

From Amiram Eldar, Jul 22 2020: (Start)

Sum_{n>=1} 1/a(n) = 2 - Pi/4 - 3*log(2)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/4 + sqrt(2)*arcsinh(1)/2 + log(2)/2 - 2. (End)