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 A099587 #68 Jun 28 2025 12:39:28
%S A099587 0,1,2,3,4,4,0,-14,-48,-116,-232,-396,-560,-560,0,1912,6528,15760,
%T A099587 31520,53808,76096,76096,0,-259808,-887040,-2141504,-4283008,-7311552,
%U A099587 -10340096,-10340096,0,35303296,120532992,290992384,581984768
%N A099587 a(n) = coefficient of x in (1+x)^n mod (1+x^4).
%C A099587 {A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4.
%C A099587 For the definition, see [Erdelyi] and the Shevelev link. - _Vladimir Shevelev_, Jul 03 2017
%D A099587 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H A099587 G. C. Greubel, <a href="/A099587/b099587.txt">Table of n, a(n) for n = 0..1000</a>
%H A099587 Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%H A099587 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-July/017798.html">Coefficient of x^k in ((x+1)^n modulo x^N+1)</a>, seqfan, Thu Jul 20 2017.
%H A099587 G. Tollisen and T. Lengyel, <a href="http://www.emis.de/journals/INTEGERS/papers/e4/e4.Abstract.html">A Congruential Identity and the 2-adic Order of Lacunary Sums of Binomial Coefficients</a>, Integers 4 (2004), #A4.
%H A099587 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-2).
%F A099587 G.f.: x*(x-1)^2 / (2*x^4-4*x^3+6*x^2-4*x+1). - _Colin Barker_, Jul 15 2013
%F A099587 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4). - _G. C. Greubel_, Nov 09 2015
%F A099587 From _Vladimir Shevelev_, Jun 29 2017: (Start)
%F A099587 a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k+1).
%F A099587 a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-2)/8)/2), where round(x) is the integer nearest to x.
%F A099587 a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_4(n)*K_3(m) - K_3(n)*K_4(m), where K_1 is A099586, K_3=A099588, and K_4=A099589.
%F A099587 (End)
%F A099587 a(n) = A099589(n+2)-2*A099589(n+1)+A099589(n). - _R. J. Mathar_, Jun 28 2025
%t A099587 RecurrenceTable[{a[1]=1, a[2]=2, a[3]=3, a[4]=4, a[n] = 4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4]}, a, {n, 1, 100}] (* _G. C. Greubel_, Nov 09 2015 *)
%t A099587 a[n_] := n*HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, (4-n)/4}, {1/2, 3/4, 5/4}, -1]; Array[a, 40, 0] (* _Jean-François Alcover_, Jul 20 2017, from _Vladimir Shevelev_'s first formula *)
%t A099587 LinearRecurrence[{4,-6,4,-2},{0,1,2,3},50] (* _Harvey P. Dale_, Mar 27 2022 *)
%o A099587 (PARI) a(n) = polcoeff(((1+x)^n)%(x^4+1),1)
%Y A099587 Cf. A099586, A099588, A099589.
%Y A099587 Cf. A057681, A099586, A139398, A133476, A139714, A139748, A139761.
%K A099587 sign,easy
%O A099587 0,3
%A A099587 _Ralf Stephan_, Oct 24 2004
%E A099587 a(0)=0 added by _N. J. A. Sloane_, Jun 30 2017