A036246 CONTINUANT transform of squares 1, 4, 9, ...

1, 5, 46, 741, 18571, 669297, 32814124, 2100773233, 170195445997, 17021645372933, 2059789285570890, 296626678767581093, 50131968501006775607, 9826162452876095600065, 2210936683865622516790232, 566009617232052240393899457, 163578990316746963096353733305

Offset: 1

Author

Jeff Burch

Keywords

• frac
• nonn

Comments

Denominator of fraction equal to the continued fraction [ 0, 1, 4, ...n^2 ].

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A036245, A073824.

Programs

Maple

a:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, n^2 *a(n-1) +a(n-2)))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 06 2013

Mathematica

Table[Denominator[FromContinuedFraction[Range[0,n]^2]],{n,20}] (* Harvey P. Dale, Jul 16 2017 *)

PARI

A036246(n) = my(v=vector(n+1)); for(i=1, n+1, if(i==1, v[i]=1, if(i==2, v[i]=1, v[i]=(i-1)^2*v[i-1]+v[i-2]))); v[n+1] \\ Jianing Song, Nov 30 2019

Formula

a(n) ~ c * n^(2*n + 1) / exp(2*n), where c = 8.1245591771139376779472290412302409841950717664641832772206241208918274428499... - Vaclav Kotesovec, Jun 05 2018

From Jianing Song, Nov 30 2019: (Start)

a(n) = n^2 * a(n-1) + a(n-2) for n > 2.

Lim_{n->oo} A036245(n)/a(n) = A073824. (End)