OFFSET
0,1
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
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).
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)
MATHEMATICA
LinearRecurrence[{2, -1}, {4, 12}, 50] (* G. C. Greubel, Apr 26 2018 *)
PROG
(Magma) [8*n+4: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011
(Haskell)
a017113 = (+ 4) . (* 8)
a017113_list = [4, 12 ..] -- Reinhard Zumkeller, Jul 13 2013
(PARI) a(n)=8*n+4 \\ Charles R Greathouse IV, Sep 23 2013
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved