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A017113 a(n) = 8*n + 4. 40

%I

%S 4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148,156,

%T 164,172,180,188,196,204,212,220,228,236,244,252,260,268,276,284,292,

%U 300,308,316,324,332,340,348,356,364,372,380,388,396,404,412,420,428,436,444,452,460,468

%N a(n) = 8*n + 4.

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

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

%C Continued fraction expansion of tanh(1/4). - _Benoit Cloitre_, Dec 17 2002

%C 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 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 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 From the Cody Clifton's reference : "There are no 5/8 commutative groups of order 4 mod 8. The maximum commutativity of a non-Abelian group is 5/8, and this degree of commutativity only occurs when the order of the center of the group is equal to one fourth the order of the group [proof given]". - _Jonathan Vos Post_, May 23 2012

%C A007814(a(n)) = 2; A037227(a(n)) = 5. - _Reinhard Zumkeller_, Jun 30 2012

%C With the substitution n -> n-2 this is 8n-12, as in the paper by Gu and Hao. - _Jonathan Vos Post_, Sep 23 2013

%C Numbers k such that 3^k + 1 is divisible by 41. - _Bruno Berselli_, Aug 22 2018

%C Numbers whose even divisors are twice the odd ones. - _Paolo P. Lava_, Oct 17 2018

%H Vincenzo Librandi, <a href="/A017113/b017113.txt">Table of n, a(n) for n = 0..1100</a>

%H 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 Cody Clifton, <a href="http://www.whitman.edu/mathematics/SeniorProjectArchive/2010/SeniorProject_CodyClifton.pdf">Commutativity in non-Abelian Groups</a>, May 06 2010.

%H 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A118413(n+1,3) for n>2. - _Reinhard Zumkeller_, Apr 27 2006

%F a(n) = 16*n-a(n-1) with n>0, a(0)=4. - _Vincenzo Librandi_, Nov 19 2010

%F 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 G.f.: (1+x)/x^2*(1 - 1/(x*Q(0) + 1)) where Q(k)= 1 - 1/(9^k - 3*x*81^k/(3*x*9^k - 1/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k + 1/Q(k+1) ))))); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 12 2013

%F a(n) = 4*A005408(n). - _Omar E. Pol_, Apr 17 2016

%F E.g.f.: (8*x + 4)*exp(x). - _G. C. Greubel_, Apr 26 2018

%t lst={}; Do[AppendTo[lst,8*n+4], {n,0,6!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 26 2009 *)

%t LinearRecurrence[{2,-1}, {4,12}, 50] (* _G. C. Greubel_, Apr 26 2018 *)

%o (MAGMA) [8*n+4: n in [0..50]]; // _Vincenzo Librandi_, Apr 26 2011

%o (Haskell)

%o a017113 = (+ 4) . (* 8)

%o a017113_list = [4, 12 ..] -- _Reinhard Zumkeller_, Jul 13 2013

%o (PARI) a(n)=8*n+4 \\ _Charles R Greathouse IV_, Sep 23 2013

%Y First differences of A016742 (even squares). Cf. A078370, A078371.

%Y Cf. A081770 (subsequence).

%Y Cf. A051062.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

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Last modified November 18 17:33 EST 2019. Contains 329287 sequences. (Running on oeis4.)