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

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• easy

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.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..830
• W. Lang, Rationals r(n), limit.

Crossrefs

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

Programs

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

Mathematica

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

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

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).