A123748 Denominators of partial sums of a series for sqrt(5).

1, 5, 25, 5, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 78125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125, 476837158203125, 2384185791015625

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators of sums over central binomial coefficients scaled by powers of 5.

Numerators are given by A123747.

For the rationals r(n) see the W. Lang link under A123747.

Examples

			a(3) = 5 because r(3) = 1+2/5+6/25+4/25 = 9/5 = A123747(3)/a(3).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A123747, A123749.

Programs

GAP

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

Magma

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

Maple

A123748:=n-> denom(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123748(n), n=0..25); # G. C. Greubel, Aug 10 2019

Mathematica

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

PARI

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

Sage

[denominator( 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) = denominator(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.