A329495 Numerator of second moment of the n-th term of Ulam's "history-dependent random sequence".

1, 4, 25, 101, 655, 11077, 94199, 2005319, 62541113, 530615357, 3888615949, 2547768661, 1603125774433, 510235838011, 76538488318091, 10462945272539099, 182249306891497151, 7488308007918913129, 6810487755999748187

Offset: 1

Author

N. J. A. Sloane, Nov 17 2019

Keywords

• nonn
• frac

Examples

			1, 4, 25/2, 101/3, 655/8, 11077/60, 94199/240, 2005319/2520, ...
		

Links

• 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.

Crossrefs

Cf. A329496, A329497, A327498.

Programs

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