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

0,2

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).

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)

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 *)

Prog

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

Crossrefs

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

Cf. A000217.

Sequence in context: A271912 A288947 A328146 * A118629 A050509 A211057

Adjacent sequences: A033582 A033583 A033584 * A033586 A033587 A033588

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved