A014634 a(n) = (2*n+1)*(4*n+1).

1, 15, 45, 91, 153, 231, 325, 435, 561, 703, 861, 1035, 1225, 1431, 1653, 1891, 2145, 2415, 2701, 3003, 3321, 3655, 4005, 4371, 4753, 5151, 5565, 5995, 6441, 6903, 7381, 7875, 8385, 8911, 9453, 10011, 10585, 11175, 11781, 12403, 13041, 13695, 14365, 15051

Offset: 0

Author

Mohammad K. Azarian, N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Odd hexagonal numbers. Bisection of A000384. - Omar E. Pol, Apr 06 2008

Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

a(n) is also the sum of natural numbers which can be placed in a center box and expanded ones on 4 arms on N, S, E, W or NE, NW, SW, SE directions. See illustration in links. - Kival Ngaokrajang, Jul 08 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Kival Ngaokrajang, Illustration of initial terms.
• Leo Tavares, Illustration: Dual Square Stars
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000384, A005408, A016813, A033954, A157870, A194268.

Cf. A003154, A000290.

Programs

Magma

[(2*n+1)*(4*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Jul 09 2014

Maple

A014634:=n->(2*n+1)*(4*n+1); seq(A014634(k), k=0..100); # Wesley Ivan Hurt, Nov 04 2013

Mathematica

lst={};Do[a=(2*n+1)*(4*n+1);AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Table[(2 n + 1) (4 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jul 09 2014 *)
LinearRecurrence[{3,-3,1},{1,15,45},50] (* Harvey P. Dale, Aug 30 2021 *)

PARI

a(n)=(2*n+1)*(4*n+1) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = A157870(n)/2. - Vladimir Joseph Stephan Orlovsky, Mar 10 2009

a(n) = a(n-1) + 16*n-2 (with a(0)=1). - Vincenzo Librandi, Nov 20 2010

G.f.: (1+12*x+3*x^2)/(1-x)^3. - Colin Barker, Jan 08 2012

a(n) = A005408(n) * A016813(n). - Omar E. Pol, Nov 05 2013

a(n) = 2*A033954(n) + 1 = A194268(n) - n. - Wesley Ivan Hurt, Jul 14 2014

E.g.f.: (8*x^2 +14*x + 1)*exp(x). - G. C. Greubel, Jul 18 2017

From Amiram Eldar, Feb 28 2022: (Start)

Sum_{n>=0} 1/a(n) = Pi/4 + log(2)/2.

Sum_{n>=0} (-1)^n/a(n) = Pi*(sqrt(2)-1)/4 + log(sqrt(2)+1)/sqrt(2). (End)

a(n) = A003154(n+1) + 2*A000290(n). - Leo Tavares, Mar 26 2022

Extensions

More terms from Wesley Ivan Hurt, Jul 09 2014