A033571 a(n) = (2*n + 1)*(5*n + 1).

1, 18, 55, 112, 189, 286, 403, 540, 697, 874, 1071, 1288, 1525, 1782, 2059, 2356, 2673, 3010, 3367, 3744, 4141, 4558, 4995, 5452, 5929, 6426, 6943, 7480, 8037, 8614, 9211, 9828, 10465, 11122, 11799, 12496, 13213, 13950, 14707, 15484, 16281, 17098, 17935, 18792, 19669, 20566, 21483

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the diagonals in the spiral. - Omar E. Pol, Sep 10 2011

Also sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is a line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Leo Tavares, Illustration: Stellar Layers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A153127. - Reinhard Zumkeller, Dec 20 2008

Cf. A000566, A008596, A010010, A153126, A158186.

Cf. A001622, A003154, A007742, A019952.

Programs

GAP

List([0..50], n-> (2*n+1)*(5*n+1)); # G. C. Greubel, Oct 12 2019

Magma

[(2*n+1)*(5*n+1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

Maple

seq((2*n+1)*(5*n+1), n=0..50); # G. C. Greubel, Oct 12 2019

Mathematica

Table[(2*n+1)*(5*n+1), {n,0,50}] (* G. C. Greubel, Oct 12 2019 *)

PARI

a(n)=(2*n+1)*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[(2*n+1)*(5*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

Formula

a(n) = A153126(2*n) = A000566(2*n+1). - Reinhard Zumkeller, Dec 20 2008

From Reinhard Zumkeller, Mar 13 2009: (Start)

a(n) = A008596(n) + A158186(n), for n > 0.

a(n) = A010010(n) - A158186(n). (End)

a(n) = a(n-1) + 20*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010

From G. C. Greubel, Oct 12 2019: (Start)

G.f.: (1 + 15*x + 4*x^2)/(1-x)^3.

E.g.f.: (1 + 17*x + 10*x^2)*exp(x). (End)

a(n) = A003154(n+1) + A007742(n). - Leo Tavares, Mar 27 2022

Sum_{n>=0} 1/a(n) = sqrt(1+2/sqrt(5))*Pi/6 + sqrt(5)*log(phi)/6 + 5*log(5)/12 - 2*log(2)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

Extensions

Terms a(36) onward added by G. C. Greubel, Oct 12 2019