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%I #54 May 10 2023 12:17:23
%S 1,1,1,0,-4,-14,-34,-68,-116,-164,-164,0,560,1912,4616,9232,15760,
%T 22288,22288,0,-76096,-259808,-627232,-1254464,-2141504,-3028544,
%U -3028544,0,10340096,35303296,85229696,170459392,290992384,411525376,411525376,0,-1405035520
%N Constant term in (1+x)^n mod (1+x^4).
%C Equals real part of term (1,1) in M^n, where M = a 2 X 2 matrix [1,1; i,1], where i = sqrt(-1). - _Gary W. Adamson_, Mar 25 2009
%C {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 For the definition, see [Erdelyi] and the Shevelev link. - _Vladimir Shevelev_, Jul 03 2017
%C a(n) = 0 if and only if n == 4 (mod 8). - _Robert Israel_, Jul 04 2017
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H Colin Barker, <a href="/A099586/b099586.txt">Table of n, a(n) for n = 1..1000</a>
%H John B. Dobson, <a href="http://arxiv.org/abs/1610.09361">A matrix variation on Ramus's identity for lacunary sums of binomial coefficients</a>, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
%H 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 Vladimir Shevelev, <a href="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 Gregory Tollisen and Tamás 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-2).
%F From _Colin Barker_, Nov 08 2015: (Start)
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-2*a(n-4) for n>4.
%F G.f.: -x*(2*x-1)*(x^2-x+1) / (2*x^4-4*x^3+6*x^2-4*x+1).
%F (End)
%F a(n) = (1/2)*((2+sqrt(2))^(n/2)*cos(n*Pi/8) + (2-sqrt(2))^(n/2)*cos(3*n*Pi/8)). - _G. C. Greubel_, Nov 10 2015
%F From _Vladimir Shevelev_, Jun 29 2017: (Start)
%F a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k).
%F a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n)/8)/2), where round(x) is the integer nearest to x.
%F a(n+m) = a(n)*a(m) - K_4(n)*K_2(m) - K_3(n)*K_3(m) - K_2(n)*K_4(m), where K_2 is A099587, K_3 is A099588 and K_4 is A099589.
%F (End)
%p seq(eval(rem((1+x)^n, 1+x^4,x),x=0),n=1..40); # _Robert Israel_, Jul 03 2017
%t RecurrenceTable[{a[n] == 4 * a[n - 1] - 6 * a[n - 2] + 4 * a[n - 3] - 2 * a[n - 4], a[1] = 1, a[2] = 1, a[3] = 1, a[4] = 0}, a, {n, 50}] (* _G. C. Greubel_, Nov 10 2015 *)
%t a[n_] := HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, -n/4}, {1/4, 1/2, 3/4}, -1]; Array[a, 40] (* _Jean-François Alcover_, Jul 20 2017, from _Vladimir Shevelev_'s first formula *)
%o (PARI) a(n) = polcoeff(((1+x)^n)%(x^4+1),0)
%o (PARI) Vec(-x*(2*x-1)*(x^2-x+1)/(2*x^4-4*x^3+6*x^2-4*x+1) + O(x^100)) \\ _Colin Barker_, Nov 08 2015
%o (PARI) a(n) = real(([1,1; I,1])^n)[1,1]; \\ _Michel Marcus_, Nov 08 2015
%Y Cf. A099587, A099588, A099589.
%K sign,easy
%O 1,5
%A _Ralf Stephan_, Oct 24 2004
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