A017113 a(n) = 8*n + 4.

4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(65).

n such that 16 is the largest power of 2 dividing A003629(k)^n - 1 for any k. - Benoit Cloitre, Mar 23 2002

Continued fraction expansion of tanh(1/4). - Benoit Cloitre, Dec 17 2002

Consider all primitive Pythagorean triples (a,b,c) with c - a = 8, sequence gives values for b. (Corresponding values for a are A078371(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004

Also numbers of the form a^2 + b^2 + c^2 + d^2, where a,b,c,d are odd integers. - Alexander Adamchuk, Dec 01 2006

If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 4-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007

A007814(a(n)) = 2; A037227(a(n)) = 5. - Reinhard Zumkeller, Jun 30 2012

Numbers k such that 3^k + 1 is divisible by 41. - Bruno Berselli, Aug 22 2018

Lexicographically smallest arithmetic progression of positive integers avoiding Fibonacci numbers. - Paolo Xausa, May 08 2023

From Martin Renner, May 24 2024: (Start)

Also number of points in a grid cross with equally long arms and a width of two points, e.g.:

* *

* * * *

* * * * * *

* * * * * * * * * * * * * * * * * * * *

* * * * * * * * * * * * * * * * * * * *

* * * * * *

* * * *

* *

etc. (End)

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1100 from Vincenzo Librandi)
• E. Catalan, Extrait d'une lettre, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206. [If N is a prime number of the form 4*m+1, then 8*N+4 is the sum of four odd squares.]
• Cody Clifton, Commutativity in non-Abelian Groups, May 06 2010.
• Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.2.
• Dr Barker, How to Avoid the Fibonacci Numbers, YouTube video, 2023.
• Meimei Gu and Rongxia Hao, 3-extra connectivity of 3-ary n-cube networks, arXiv:1309.5083 [cs.DM], Sep 19, 2013.
• Milan Janjic, Two Enumerative Functions.
• 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

First differences of A016742 (even squares).

Cf. A078370, A078371, A081770 (subsequence).

Cf. A003629, A005408, A007814, A019683, A037227, A051062, A118413.

Cf. A008586, A016825.

Programs

Haskell

a017113 = (+ 4) . (* 8)
a017113_list = [4, 12 ..]  -- Reinhard Zumkeller, Jul 13 2013

Magma

[8*n+4: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

Mathematica

LinearRecurrence[{2,-1}, {4,12}, 50] (* G. C. Greubel, Apr 26 2018 *)

PARI

a(n)=8*n+4 \\ Charles R Greathouse IV, Sep 23 2013

Formula

a(n) = A118413(n+1,3) for n > 2. - Reinhard Zumkeller, Apr 27 2006

a(n) = Sum_{k=0..4*n} (i^k+1)*(i^(4*n-k)+1), where i = sqrt(-1). - Bruno Berselli, Mar 19 2012

a(n) = 4*A005408(n). - Omar E. Pol, Apr 17 2016

E.g.f.: (8*x + 4)*exp(x). - G. C. Greubel, Apr 26 2018

G.f.: 4*(1+x)/(1-x)^2. - Wolfdieter Lang, Oct 27 2020

Sum_{n>=0} (-1)^n/a(n) = Pi/16 (A019683). - Amiram Eldar, Dec 11 2021

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(3*Pi/16).

Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(3*Pi/16). (End)

a(n) = 2*A016825(n) = A008586(2*n+1). - Elmo R. Oliveira, Apr 10 2025