A339516 a(n+1) = (a(n) - 2*(n-1)) * (2*n-1), where a(1)=1.

1, 1, -3, -35, -287, -2655, -29315, -381251, -5718975, -97222847, -1847234435, -38791923555, -892214242271, -22305356057375, -602244613549827, -17465093792945795, -541417907581320575, -17866790950183580031, -625337683256425302275, -23137494280487736185507

Offset

1,3

Comments

The sequence appears when computing constants that encode all odd numbers starting from 3, then from 5, then from 7, etc. The general formula of the constant is a(n) + (2n-1)!!*sqrt(2*Pi*e)*erf(1/sqrt(2)), where n>0. For more information on how to generate the constant please watch the Grime-Haran Numberphile video.

Links

Table of n, a(n) for n=1..20.

James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.

Formula

Homogeneous recurrence: (-2*n+9)*a(n-4) + (6*n-20)*a(n-3) + (-6*n+12)*a(n-2) + 2*n*a(n-1) - a(n) = 0 with a(1)=a(2)=1, a(3)=-3, a(4)=-35. - Georg Fischer, Sep 01 2022

Maple

A339516 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        (2*n-3)*(procname(n-1)-2*(n-2)) ;
    end if;
end proc:
seq(A339516(n), n=1..30) ; # R. J. Mathar, Aug 24 2022

Mathematica

a[1]=1; a[n_]:=(a[n-1]-2(n-2))(2n-3); Array[a, 20] (* Stefano Spezia, Dec 08 2020 *)

Prog

(Python)
#generate first 50 numbers of the sequence
cnt = 50
i=0
seq = list()
seq.append(1)
i=1
while (i<cnt):
    seq.append((seq[i-1]-2*(i-1))*(2*i-1))
    i=i+1
print(seq)

Crossrefs

Cf. A047907, A093302, A001147.

Sequence in context: A271209 A202883 A069448 * A121078 A289950 A308033

Adjacent sequences: A339513 A339514 A339515 * A339517 A339518 A339519

Keyword

sign

Author

Kamil Zabkiewicz, Dec 07 2020

Status

approved