A120784 Numerators of partial sums of Catalan numbers scaled by powers of 1/16.

1, 17, 137, 4389, 35119, 561925, 4495433, 287708141, 2301665843, 36826655919, 294613251551, 9427624079025, 75420992684203, 1206735883132973, 9653887065398089, 1235697544380650237

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Denominators are given under A120785.

From the expansion of sqrt(3)/2 = sqrt(1-1/4) = 1-(1/8)*sum(C(k)/16^k,k=0..infinity) one has r:=limit(r(n),n to infinity)= 4*(2 - sqrt(3)) = 1.071796769..., with the partial sums r(n) defined below.

Examples

			Rationals r(n): [1, 17/16, 137/128, 4389/4096, 35119/32768, 561925/524288, 4495433/4194304, 287708141/268435456,...].
		

Links

• W. Lang: Rationals r(n) and limit.

Formula

a(n)=numerator(r(n)), with the rationals r(n):=sum(C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.