A160636 Hankel transform of A114464.

1, 0, -1, -2, -8, 0, 128, 1024, 16384, 0, -4194304, -134217728, -8589934592, 0, 35184372088832, 4503599627370496, 1152921504606846976, 0, -75557863725914323419136, -38685626227668133590597632

Offset

0,4

Comments

Hankel transform of A114464(n+1) is A160637.

Links

G. C. Greubel, Table of n, a(n) for n = 0..115

Formula

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).

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

a(n) = -a(2-n) * 2^A004524(n) for all n in Z. - Michael Somos, Mar 14 2020

Mathematica

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 *)
a[ n_] := -Sign[Mod[n - 1, 4]]*(-1)^Quotient[n - 1, 4]*2^Quotient[n (n - 1), 4]; (* Michael Somos, Mar 14 2020 *)

Prog

(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
(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
(PARI) A160636(n)=if(n%4!=1, (-1)^((n+2)\4)<<(binomial(n, 2)\2), 0) \\ M. F. Hasler, May 09 2018

Crossrefs

Cf. A004524, A160637.

Sequence in context: A028256 A209455 A288873 * A282626 A206712 A293777

Adjacent sequences: A160633 A160634 A160635 * A160637 A160638 A160639

Keyword

easy,sign

Author

Paul Barry, May 21 2009

Extensions

Comment with an incorrect formula deleted by M. F. Hasler, May 09 2018

Status

approved