A001539 a(n) = (4*n+1)*(4*n+3).

3, 35, 99, 195, 323, 483, 675, 899, 1155, 1443, 1763, 2115, 2499, 2915, 3363, 3843, 4355, 4899, 5475, 6083, 6723, 7395, 8099, 8835, 9603, 10403, 11235, 12099, 12995, 13923, 14883, 15875, 16899, 17955, 19043, 20163, 21315, 22499, 23715, 24963, 26243, 27555, 28899

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Sequence arises from reading the line from 3, in the direction 3, 35, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

log(sqrt(2)+1)/sqrt(2) = 0.62322524... = 2/3 - 2/35 + 2/99 - 2/195 + 2/323, ... = (1 - 1/3) + (1/7 - 1/5) + (1/9 - 1/11) + (1/15 - 1/13) + (1/17 - 1/19) + (1/23 - 1/21) + ... - Gary W. Adamson, Mar 01 2009

Numbers k such that k+1 is a square and k+5 is divisible by 8. - Bruno Berselli, Sep 27 2017

The concatenation of 8*A000217(n) and 99 is a term of the sequence. Example: for A000217(5) = 15, 8*15 = 120 and 12099 = a(27). In general, a(5*n+2) = 800*A000217(n) + 99. - Bruno Berselli, Sep 29 2017

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A000466.

Cf. A000217, A000290, A001533, A001538, A004767, A016286, A016813, A016826, A157142, A133766, A154633.

Programs

Mathematica

CoefficientList[Series[(3 + 26 x + 3 x^2)/(1 - x)^3, {x, 0, 41}], x] (* or *) Table[(4 n + 1) (4 n + 3), {n, 0, 41}] (* Michael De Vlieger, Sep 29 2017 *)

Maxima

makelist((4*n+1)*(4*n+3), n, 0, 30); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=(4*n+1)*(4*n+3) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = A016826(n) - 1 = (A001533(n)+5)/4 = (A001538(n)+16)/9.

Sum_{k>=0} 1/a(k) = Pi/8. - Benoit Cloitre, Aug 20 2002

G.f.: (3 + 26*x + 3*x^2)/(1 - x)^3. - Jaume Oliver Lafont, Mar 07 2009

a(n) = 32*n + a(n-1) for n > 0, a(0)=3. - Vincenzo Librandi, Nov 12 2010

a(n) = a(m) + 16*(n-m)*(n+m+1). The previous formula is obtained for m = n-1. - Bruno Berselli, Sep 29 2017

From Amiram Eldar, Feb 19 2023: (Start)

a(n) = A016813(n)*A004767(n).

Product_{n>=0} (1 - 1/a(n)) = sqrt(2)*cos(Pi/(2*sqrt(2))).

Product_{n>=0} (1 + 1/a(n)) = sqrt(2). (End)

From Elmo R. Oliveira, Oct 23 2024: (Start)

E.g.f.: exp(x)*(3 + 16*x*(2 + x)).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)