A121504 Numerators of partial sums of a series used for the series of sqrt(2) + sqrt(3) involving Catalan numbers.

3, 53, 433, 13941, 111759, 1789509, 14320329, 916611309, 7333257267, 117334608047, 938685468127, 30038055403185, 240304869286059, 3844880951681069, 30759058568289097, 3937160132112061181

Offset

0,1

Comments

The corresponding denominators are A120785(n).

The limit of this series is 4*(4-(sqrt(2)+sqrt(3))) = 3.414942518 (maple10, 10 digits).

See A121503 for the geometric interpretation of sqrt(2)+sqrt(3) and a Popper reference.

Links

G. C. Greubel, Table of n, a(n) for n = 0..830

W. Lang, Rationals r(n), limit.

Formula

a(n) = numerator(r(n)) with r(n):= sum(C(k)*(1+2^(k+1))/16^k,k=0..n), n>=0, with C(k)=A000108(k) (Catalan numbers).

Example

Rationals r(n): [3, 53/16, 433/128, 13941/4096, 111759/32768,
1789509/524288, 14320329/4194304, 916611309/268435456,...].

Mathematica

Table[Numerator[Sum[CatalanNumber[k]*(1 + 2^(k + 1))/16^k, {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Sep 27 2018 *)

Prog

(PARI) for(n=0, 30, print1(numerator(sum(k=0, n, binomial(2*k, k)*(1 + 2^(k+1))/(16^k*(k+1)))), ", ")) \\ G. C. Greubel, Sep 27 2018
(Magma) [Numerator( (&+[Binomial(2*k, k)*(1 + 2^(k+1))/(16^k*(k+1)): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Sep 27 2018

Crossrefs

A121503/(4*A120785) are the partial sums of a series for sqrt(2)+sqrt(3).

Sequence in context: A292530 A036941 A215435 * A099665 A173802 A001279

Adjacent sequences: A121501 A121502 A121503 * A121505 A121506 A121507

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 16 2006

Status

approved