A124398 Denominators of partial sums of a series for sqrt(5)/3.

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

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

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

Numerators are given by A124397.

For the rationals r(n) see the W. Lang link under A124397.

r(n) is not 1/3 times the rational sequence A123747/A123748 which converges to sqrt(5).

Examples

			a(3) = 25 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = A124397(3)/a(3).
		

Links

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

Crossrefs

Cf. A124397 (numerators), A208899 (sqrt(5)/3).

Programs

GAP

List([0..20], n-> DenominatorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019

Magma

[Denominator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

Maple

seq(denom(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019

Mathematica

Table[Denominator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)

PARI

a(n) = denominator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ Michel Marcus, Aug 11 2019

Sage

[denominator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.

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