A123747 Numerators of partial sums of a series for sqrt(5).

1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583, 1064706466190659, 1065035803419763, 5326468921246139

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given by A123748.

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

Examples

			a(3) = 9 because r(3) = 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).
		

Links

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

Crossrefs

Cf. A001077/A001076 continued fraction convergents for sqrt(5).

Cf. A123748, A123749.

Programs

GAP

List([0..25], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/5^k )) ); # G. C. Greubel, Aug 10 2019

Magma

[Numerator( (&+[Binomial(2*k,k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019

Maple

A123747:=n-> numer(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123747(n), n=0..25); # G. C. Greubel, Aug 10 2019

Mathematica

Table[Numerator[Sum[Binomial[2*k, k]/5^k, {k,0,n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)

PARI

vector(25, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/5^k))) \\ G. C. Greubel, Aug 10 2019

Sage

[numerator( sum(binomial(2*k,k)/5^k for k in (0..n)) ) for n in (0..25)] # 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)/5^k, in lowest terms.

r(n) = Sum_{k=0..n} ((2*k-1)!!/((2*k)!!))*(4/5)^k, n>=0, with the double factorials A001147 and A000165.