A123749 Numerators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).

1, 11, 35, 965, 8755, 8783, 237449, 2138185, 6415985, 519743405, 4677875401, 14033861347, 378916960525, 3410263045325, 3410267502725, 30692424759805, 276231889624955, 828695755304725, 67124359204727825, 604119244624305025

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given by A124396.

The sums over central binomial coefficients scaled by powers of 9, r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k have the limit s = lim_{n->infinity} r(n) = 3/sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/9.

Examples

			a(3) = 965 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = a(3)/A124396(3).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wolfdieter Lang, Rationals and more.

Crossrefs

Cf. A124396 (denominators).

Cf. A123747/A123748 partial sums for a series for sqrt(5).

Programs

GAP

List([0..20], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/9^k )) ); # G. C. Greubel, Aug 10 2019

Magma

[Numerator( (&+[Binomial(2*k,k)/9^k: k in [0..n]])): n in [0..20]]; // G. C. Greubel, Aug 10 2019

Maple

A123749:=n-> numer(sum(binomial(2*k,k)/9^k, k=0..n)); seq(A123749(n), n=0..20); # G. C. Greubel, Aug 10 2019

Mathematica

Table[Numerator[Sum[Binomial[2*k, k]/9^k, {k,0,n}]], {n, 0, 20}] (* G. C. Greubel, Aug 10 2019 *)

PARI

vector(20, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/9^k))) \\ G. C. Greubel, Aug 10 2019

Sage

[numerator( sum(binomial(2*k,k)/9^k for k in (0..n)) ) for n in (0..20)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.

r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.