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%I #24 Sep 08 2022 08:45:45
%S 1,0,-1,-2,-8,0,128,1024,16384,0,-4194304,-134217728,-8589934592,0,
%T 35184372088832,4503599627370496,1152921504606846976,0,
%U -75557863725914323419136,-38685626227668133590597632
%N Hankel transform of A114464.
%C Hankel transform of A114464(n+1) is A160637.
%H G. C. Greubel, <a href="/A160636/b160636.txt">Table of n, a(n) for n = 0..115</a>
%F a(n) = 2^floor(C(n,2)/2)*((sqrt(2)-1)*sin((3*n+1)*Pi/4)/2 +(sqrt(2)+1)*cos((n+1)*Pi/4)/2).
%F a(4k+1) = 0, a(n) = (-1)^floor((n+2)/4) * 2^A011848(n) if n !== 1 (mod 4), where A011848(n) = floor(C(n,2)/2). - _M. F. Hasler_, May 09 2018
%F a(n) = -a(2-n) * 2^A004524(n) for all n in Z. - _Michael Somos_, Mar 14 2020
%t Table[Round[2^Floor[Binomial[n, 2]/2]*((Sqrt[2]-1)*Sin[(3*n+1)*Pi/4]/2 + (Sqrt[2]+1)*Cos[(n+1)*Pi/4]/2)], {n, 0, 50}] (* _G. C. Greubel_, May 03 2018 *)
%t a[ n_] := -Sign[Mod[n - 1, 4]]*(-1)^Quotient[n - 1, 4]*2^Quotient[n (n - 1), 4]; (* _Michael Somos_, Mar 14 2020 *)
%o (Magma) R:= RealField(); [Round(2^Floor(Binomial(n,2)/2)*((Sqrt(2)/2 -1/2)*Sin(3*Pi(R)*n/4+Pi(R)/4)+(Sqrt(2)/2+1/2)*Cos(Pi(R)*n/4+Pi(R)/4))): n in [0..50]]; // _G. C. Greubel_, May 03 2018
%o (PARI) for(n=0,50, print1(round(2^floor(binomial(n,2)/2)*((sqrt(2)-1)*sin((3*n+1)*Pi/4)/2 +(sqrt(2)+1)*cos((n+1)*Pi/4)/2)), ", ")) \\ _G. C. Greubel_, May 03 2018
%o (PARI) A160636(n)=if(n%4!=1,(-1)^((n+2)\4)<<(binomial(n,2)\2),0) \\ _M. F. Hasler_, May 09 2018
%Y Cf. A004524, A160637.
%K easy,sign
%O 0,4
%A _Paul Barry_, May 21 2009
%E Comment with an incorrect formula deleted by _M. F. Hasler_, May 09 2018
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