A051062 a(n) = 16*n + 8.

8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184, 200, 216, 232, 248, 264, 280, 296, 312, 328, 344, 360, 376, 392, 408, 424, 440, 456, 472, 488, 504, 520, 536, 552, 568, 584, 600, 616, 632, 648, 664, 680, 696, 712, 728, 744, 760, 776, 792, 808, 824, 840

Offset: 0

Author

N. J. A. Sloane, Gary W. Adamson, Dec 11 1999

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(97).

n such that 32 is the largest power of 2 dividing A003629(k)^n-1 for any k. - Benoit Cloitre, Mar 23 2002

Continued fraction expansion of tanh(1/8). - Benoit Cloitre, Dec 17 2002

If Y and Z are 2-blocks of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007

General form: (q*n+x)*q x=+1; q=2=A016825, q=3=A017197, q=4=A119413, ... x=-1; q=3=A017233, q=4=A098502, ... x=+2; q=4=A051062, ... - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

a(n)*n+1 = (4n+1)^2 and a(n)*(n+1)+1 = (4n+3)^2 are both perfect squares. - Carmine Suriano, Jun 01 2014

For all positive integers n, there are infinitely many positive integers k such that k*n + 1 and k*(n+1) + 1 are both perfect squares. Except for 8, all the numbers of this sequence are the smallest integers k which are solutions for getting two perfect squares. Example: a(1) = 24 and 24 * 1 + 1 = 25 = 5^2, then 24 * (1+1) + 1 = 49 = 7^2. [Reference AMM] - Bernard Schott, Sep 24 2017

Numbers k such that 3^k + 1 is divisible by 17*193. - Bruno Berselli, Aug 22 2018

References

• Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov 11 1999.

Links

• Mia Boudreau, Table of n, a(n) for n = 0..10000
• Mihaly Bencze, Problem 11508, The American Mathematical Monthly, Vol. 117, N° 5, May 2010, p. 459.
• Milan Janjić, Two Enumerative Functions. [Wayback Machine link]
• Tanya Khovanova, Recursive Sequences.
• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A005408, A008598, A017113, A106839, A139098, A244978.

Cf. A003484, A007814, A037227, A118413, A209675.

Cf. A003629, A016825, A017197, A017233, A098502, A119413.

Programs

Magma

[16*n+8: n in [0..50]]; // Wesley Ivan Hurt, Jun 01 2014

Maple

A051062:=n->16*n+8; seq(A051062(n), n=0..50); # Wesley Ivan Hurt, Jun 01 2014

Mathematica

Range[8, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
Table[16n+8, {n,0,50}] (* Wesley Ivan Hurt, Jun 01 2014 *)
LinearRecurrence[{2,-1},{8,24},60] (* or *) NestList[#+16&,8,60] (* Harvey P. Dale, Aug 18 2019 *)

PARI

a(n)=16*n+8 \\ Charles R Greathouse IV, May 09 2016

Formula

a(n) = A118413(n+1,4) for n>3. - Reinhard Zumkeller, Apr 27 2006

a(n) = 32*n - a(n-1) for n>0, a(0)=8. - Vincenzo Librandi, Aug 06 2010

A003484(a(n)) = 8; A209675(a(n)) = 9. - Reinhard Zumkeller, Mar 11 2012

A007814(a(n)) = 3; A037227(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012

a(-1 - n) = - a(n). - Michael Somos, Jun 02 2014

Sum_{n>=0} (-1)^n/a(n) = Pi/32 (A244978). - Amiram Eldar, Feb 28 2023

From Elmo R. Oliveira, Apr 16 2024: (Start)

G.f.: 8*(1+x)/(1-x)^2.

E.g.f.: 8*exp(x)*(1 + 2*x).

a(n) = 8*A005408(n) = A008598(n) + 8 = A139098(n+1) - A139098(n).

a(n) = 4*A016825(n) = 2*A017113(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

From Amiram Eldar, Nov 25 2024: (Start)

Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(7*Pi/32).

Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(7*Pi/32). (End)