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

0,1

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.

Links

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

W. Lang, Rationals r(n).

Formula

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

Example

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

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 *)

Prog

(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
(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

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).

Sequence in context: A096297 A081857 A168163 * A081737 A005475 A293404

Adjacent sequences: A120085 A120086 A120087 * A120089 A120090 A120091

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jul 20 2006

Status

approved