A329497 Numerator of the rational number alpha_n involved in the calculation of the second moment of the n-th term of Ulam's "history-dependent random sequence".

0, 2, 21, 223, 2603, 33623, 477977, 2474153, 125009419, 2268871883, 8825650861, 183051309251, 89588582779, 8538921797353, 51275872341193, 59590264767344263, 322723582109367139, 45705953681299945463, 79490015605061949133

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Mark Kac, A history-dependent random sequence defined by Ulam, Advances in Applied Mathematics 10.3 (1989): 270-277. See page 273, sigma_k and alpha_k.

Example

0, 2, 21/2, 223/6, 2603/24, 33623/120, 477977/720, 2474153/1680, ...

Maple

s:=[1]; a:=[0];
for N from 2 to 40 do # N = n+1
n:=N-1;
t1:=s[n]+(1/n)*add(s[k], k=1..n)+2*s[n]/n+(2/n)*a[n];
t2:=s[n]+(1/n)*add(s[k], k=1..n)+a[n]+(2/n)*add(a[k], k=1..n);
s:=[op(s), t1];
a:=[op(a), t2];
od:
s; # sigma_n
a; # alpha_n
sn:=map(numer, s); # A329495
sd:=map(denom, s); # A329496
an:=map(numer, a); # A329497
ad:=map(denom, a); # A329498

Crossrefs

Cf. A329495, A329496, A327498.

Sequence in context: A155872 A323478 A106412 * A058476 A099748 A023812

Adjacent sequences: A329494 A329495 A329496 * A329498 A329499 A329500

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 17 2019

Status

approved