A005475 a(n) = n*(5*n+1)/2.

0, 3, 11, 24, 42, 65, 93, 126, 164, 207, 255, 308, 366, 429, 497, 570, 648, 731, 819, 912, 1010, 1113, 1221, 1334, 1452, 1575, 1703, 1836, 1974, 2117, 2265, 2418, 2576, 2739, 2907, 3080, 3258, 3441, 3629

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - Omar E. Pol, Sep 26 2011

For n >= 3, a(n) is the sum of the numbers appearing in the 3rd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
• Leo Tavares, Illustration: Hexagonal Trapeziums.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001622, A110449, A130520, A162147, A202803.

Cf. similar sequences listed in A022289.

Programs

Maple

seq(binomial(5*n+1,2)/5, n=0..34); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # Zerinvary Lajos, Apr 29 2007

Mathematica

Table[n (5 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)

PARI

a(n)=n*(5*n+1)/2; \\ Joerg Arndt, Mar 27 2013

Formula

a(n) = A110449(n, 2) for n>1.

a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010

a(n) = A130520(5*n+2). - Philippe Deléham, Mar 26 2013

a(n) = A202803(n)/2. - Philippe Deléham, Mar 27 2013

a(n) = A162147(n) - A162147(n-1). - J. M. Bergot, Jun 21 2013

a(n) = A000217(3*n) - A000217(2*n). - Bruno Berselli, Oct 13 2016

From G. C. Greubel, Aug 23 2017: (Start)

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

E.g.f.: (x/2)*(5*x+6)*exp(x). (End)

Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - R. J. Mathar, May 30 2022

Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 10 2022

Extensions

Incorrect comment deleted and minor errors corrected by Johannes W. Meijer, Feb 04 2010