A120088 Numerators of partial sums of a series for sqrt(2).

3, 11, 23, 179, 365, 1439, 2911, 46147, 93009, 369605, 743409, 5917879, 11887761, 47365319, 95064943, 3032383331, 6082445497, 24264959593, 48649328861, 388310999293, 778263028691, 3106935548009, 6225306416473, 99433372856743, 199189221750317, 795541400400905

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k.

From the expansion of sqrt(1+x) = 1 + x*(Sum_{k>=0} C(k)*(-x/4)^k)/2, valid for |x|<=1, one finds for x=+1: sqrt(2) = 1 + (Sum_{k>=0} (-1)^k*C(k)/4^k)/2.

The denominators are given by 2*A120777(n).

The rationals r(n):=1 + (Sum_{k=0..n} (-1)^k*C(k)/4^k)/2, with the Catalan numbers C(n)=A000108(n), are A120088(n)/(2*A120777(n)), n>=0.

Examples

			Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768,...]
		

Links

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

Crossrefs

For similar partial sums with positive terms (not alternating) see rationals A119951/A120069.

For the partial sums (Sum_{k=0..n} (-1)^k*C(k)/4^k) see A120788(n)/A120777(n).

Programs

Magma

[Numerator(1 + (&+[(-1/4)^k*Binomial(2*k,k)/(2*(k+1)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, Mar 27 2018

Mathematica

r[n_]:= 1+Sum[(-1/4)^k*CatalanNumber[k]/2, {k, 0, n}]; Numerator[Table[ r[n], {n, 0, 50}]] (* G. C. Greubel, Mar 27 2018 *)

PARI

{r(n) = 1 + sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(2*(k+1)))};
for(n=0,30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 27 2018

Formula

a(n) = numerator(r(n)), with the rationals defined above.