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%I #34 May 24 2022 04:42:39
%S 1,2,4,15,35,144,342,1421,3381,14062,33464,139195,331255,1377884,
%T 3279082,13639641,32459561,135018522,321316524,1336545575,3180705675,
%U 13230437224,31485740222,130967826661,311676696541,1296447829382,3085281225184,12833510467155
%N Expansion of (1 + x - 8*x^2 + x^3 + x^4) / ( (1 - x)*(1 - 10*x^2 + x^4) ).
%C Previous name was: Solutions a(n) to (r(n)-2)*(r(n)-3) = 6*a(n)*(a(n)-1), where r(n) = A180483(n).
%C A combinatorial interpretation is provided in A180483.
%H Harvey P. Dale, <a href="/A181442/b181442.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-1,1).
%F G.f.: (1+x-8*x^2+x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ). - _R. J. Mathar_, Feb 05 2011
%F Explicit formulas: r=sqrt(6), s=5+2*r, t=5-2*r.
%F a(2*n) = (12 + (6 + r)*s^n + (6 - r)*t^n)/24.
%F a(2*n+1) = (12 + (18 + 7*r)*s^n + (18 - 7*r)*t^n)/24.
%F a(n) = 11*a(n-2) - 11*a(n-4) + a(n-6).
%F a(n) = +a(n-1) +10*a(n-2) -10*a(n-3) -a(n-4) +a(n-5).
%F a(n) = (b(n) + 3*b(n-1) - 3*b(n-2) - b(n-3) + 1)/2, where b(n) = ((1+(-1)^n)/2)* ChebyshevU(n/2, 5). - _G. C. Greubel_, Apr 26 2022
%e For n=3: a(3)=15; b(3)=38; binomial(38,4) = 73815 = binomial(38,2)*binomial(15,2).
%p n:=0: for s from 1 to 100 do r:=(sqrt(24*s^2-24*s+1)+5)/2: if (floor(r)=r) then a[n]:=s: b[n]:=r: n:=n+1: end if: end do:
%t LinearRecurrence[{1,10,-10,-1,1},{1,2,4,15,35},30] (* _Harvey P. Dale_, Dec 22 2012 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x-8*x^2+x^3+x^4)/((1-x)*(1-10*x^2+x^4)) )); // _G. C. Greubel_, Apr 26 2022
%o (SageMath)
%o def b(n): return ((1+(-1)^n)/2)*chebyshev_U(n//2, 5)
%o def A181442(n): return (b(n) + 3*b(n-1) - 3*b(n-2) - b(n-3) + 1)/2
%o [A181442(n) for n in (0..50)] # _G. C. Greubel_, Apr 26 2022
%Y Cf. A004189, A180483.
%K nonn
%O 0,2
%A _Paul Weisenhorn_, Jan 29 2011
%E New name using g.f. by _R. J. Mathar_ from _Joerg Arndt_, Apr 27 2022
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