This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A017113 #177 Apr 10 2025 17:39:31
%S A017113 4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148,156,
%T A017113 164,172,180,188,196,204,212,220,228,236,244,252,260,268,276,284,292,
%U A017113 300,308,316,324,332,340,348,356,364,372,380,388,396,404,412,420,428,436,444,452,460,468
%N A017113 a(n) = 8*n + 4.
%C A017113 Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(65).
%C A017113 n such that 16 is the largest power of 2 dividing A003629(k)^n - 1 for any k. - _Benoit Cloitre_, Mar 23 2002
%C A017113 Continued fraction expansion of tanh(1/4). - _Benoit Cloitre_, Dec 17 2002
%C A017113 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
%C A017113 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
%C A017113 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
%C A017113 A007814(a(n)) = 2; A037227(a(n)) = 5. - _Reinhard Zumkeller_, Jun 30 2012
%C A017113 Numbers k such that 3^k + 1 is divisible by 41. - _Bruno Berselli_, Aug 22 2018
%C A017113 Lexicographically smallest arithmetic progression of positive integers avoiding Fibonacci numbers. - _Paolo Xausa_, May 08 2023
%C A017113 From _Martin Renner_, May 24 2024: (Start)
%C A017113 Also number of points in a grid cross with equally long arms and a width of two points, e.g.:
%C A017113 * *
%C A017113 * * * *
%C A017113 * * * * * *
%C A017113 * * * * * * * * * * * * * * * * * * * *
%C A017113 * * * * * * * * * * * * * * * * * * * *
%C A017113 * * * * * *
%C A017113 * * * *
%C A017113 * *
%C A017113 etc. (End)
%H A017113 Paolo Xausa, <a href="/A017113/b017113.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1100 from Vincenzo Librandi)
%H A017113 E. Catalan, <a href="https://doi.org/10.24033/bsmf.401">Extrait d'une lettre</a>, 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.]
%H A017113 Cody Clifton, <a href="http://www.whitman.edu/mathematics/SeniorProjectArchive/2010/SeniorProject_CodyClifton.pdf">Commutativity in non-Abelian Groups</a>, May 06 2010.
%H A017113 Colin Defant and Noah Kravitz, <a href="https://arxiv.org/abs/2201.03461">Loops and Regions in Hitomezashi Patterns</a>, arXiv:2201.03461 [math.CO], 2022. Theorem 1.2.
%H A017113 Dr Barker, <a href="https://www.youtube.com/watch?v=bdp7fqs6jTs">How to Avoid the Fibonacci Numbers</a>, YouTube video, 2023.
%H A017113 Meimei Gu and Rongxia Hao, <a href="http://arxiv.org/abs/1309.5083">3-extra connectivity of 3-ary n-cube networks</a>, arXiv:1309.5083 [cs.DM], Sep 19, 2013.
%H A017113 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A017113 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A017113 William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.
%H A017113 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.
%H A017113 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A017113 a(n) = A118413(n+1,3) for n > 2. - _Reinhard Zumkeller_, Apr 27 2006
%F A017113 a(n) = Sum_{k=0..4*n} (i^k+1)*(i^(4*n-k)+1), where i = sqrt(-1). - _Bruno Berselli_, Mar 19 2012
%F A017113 a(n) = 4*A005408(n). - _Omar E. Pol_, Apr 17 2016
%F A017113 E.g.f.: (8*x + 4)*exp(x). - _G. C. Greubel_, Apr 26 2018
%F A017113 G.f.: 4*(1+x)/(1-x)^2. - _Wolfdieter Lang_, Oct 27 2020
%F A017113 Sum_{n>=0} (-1)^n/a(n) = Pi/16 (A019683). - _Amiram Eldar_, Dec 11 2021
%F A017113 From _Amiram Eldar_, Nov 22 2024: (Start)
%F A017113 Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(3*Pi/16).
%F A017113 Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(3*Pi/16). (End)
%F A017113 a(n) = 2*A016825(n) = A008586(2*n+1). - _Elmo R. Oliveira_, Apr 10 2025
%t A017113 LinearRecurrence[{2,-1}, {4,12}, 50] (* _G. C. Greubel_, Apr 26 2018 *)
%o A017113 (Magma) [8*n+4: n in [0..50]]; // _Vincenzo Librandi_, Apr 26 2011
%o A017113 (Haskell)
%o A017113 a017113 = (+ 4) . (* 8)
%o A017113 a017113_list = [4, 12 ..] -- _Reinhard Zumkeller_, Jul 13 2013
%o A017113 (PARI) a(n)=8*n+4 \\ _Charles R Greathouse IV_, Sep 23 2013
%Y A017113 First differences of A016742 (even squares).
%Y A017113 Cf. A078370, A078371, A081770 (subsequence).
%Y A017113 Cf. A003629, A005408, A007814, A019683, A037227, A051062, A118413.
%Y A017113 Cf. A008586, A016825.
%K A017113 nonn,easy
%O A017113 0,1
%A A017113 _N. J. A. Sloane_