A120781 -id:A120781 - cp's OEIS frontend

A120780 Numerators of partial sums of Catalan numbers scaled by powers of 1/8.

1, 9, 37, 597, 2395, 19181, 76757, 2456653, 9827327, 78621047, 314488387, 5031843585, 20127426343, 161019596469, 644078720181, 41221047786429, 164884208824551, 1319073735418803, 5276295061084887, 84420721860989787

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are under A120781.

From the expansion of sqrt(2)/2 = sqrt(1-1/2) = 1-(1/4)*sum(C(k)/8^k,k=0..infinity) one has r:=limit(r(n),n to infinity)= 2*(2 - sqrt(2)) = 1.171572875..., with the partial sums r(n) defined below.

			Rationals r(n): [1, 9/8, 37/32, 597/512, 2395/2048, 19181/16384,
76757/65536, 2456653/2097152,...].

W. Lang: Rationals r(n) and limit.

a(n)=numerator(r(n)), with the rationals r(n):=sum(C(k)/8^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

A120789 Numerators of partial sums of Catalan numbers scaled by powers of -1/8.

Original entry on oeis.org

1, 7, 29, 459, 1843, 14723, 58925, 1885171, 7541399, 60328761, 241319243, 3861078495, 15444365983, 123554742139, 494219302861, 31630025688259, 126520120431871, 1012160898632573, 4048643713939967, 64778298539407877

Offset: 0

Wolfdieter Lang, Jul 20 2006

From the expansion of sqrt(3/2) = 1+(1/4)*Sum_{k=0..oo} C(k)/(-8)^k one has, with the partial sums r(n) are defined below, r := lim_{n->oo} r(n) = 2*(sqrt(6)-2) = 0.898979485...

Denominators are given under A120781 (but may differ for higher n values).

			Rationals r(n): [1, 7/8, 29/32, 459/512, 1843/2048, 14723/16384,
58925/65536, 1885171/2097152, 7541399/8388608,...].

W. Lang: Rationals r(n) and limit.

a(n)=numerator(r(n)), with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*C(k)/8^k with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

cp's OEIS Frontend

A120780 Numerators of partial sums of Catalan numbers scaled by powers of 1/8.

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