A121002 Numerators of partial sums of Catalan numbers scaled by powers of 1/5.

1, 6, 32, 33, 839, 4237, 21317, 107014, 4292, 2687362, 13453606, 67326816, 336842092, 336990672, 1685488248, 8429380209, 42153972579, 210795791853, 210814897401, 5270725887663, 26354942262399

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A121003.

This is the first member (p=0) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).

The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..n), n>=0, for p=0,1,...

For more details on this p-family and the other three ones see the W. Lang link under A120996.

The limit lim_{n->infinity} r(n) = (3 - phi) = (2*phi-1)/phi = 1.38196601125010 (maple10, 15 digits). This is the square of the dimensionless pentagon side length.

Examples

			Rationals r(n): [1, 6/5, 32/25, 33/25, 839/625, 4237/3125,
21317/15625, 107014/78125, 4292/3125, 2687362/1953125,...].
		

Links

• W. Lang: Rationals r(n), limit.

Crossrefs

Cf. A120786 (numerators, second member p=1).

Formula

a(n)=numerator(r(n)) with r(n) := rII(p=0,n) = sum(C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.