A124397 Numerators of partial sums of a series for sqrt(5)/3.

1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279, 14210752102407, 1777633916948199

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given by A124398.

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

Examples

			a(3) = 17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).
		

Links

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

Crossrefs

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

Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).

Cf. A124398 (denominators), A208899 (sqrt(5)/3).

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

[numerator(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) = numerator(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.