A053988 Denominators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

2, 11, 108, 1501, 26910, 590519, 15326584, 459207001, 15597711450, 592253828099, 24859063068708, 1142924647332469, 57121373303554742, 3083411233744623599, 178780730183884614000, 11081321860167101444401, 731188462040844810716466, 51172111020998969648708219

Offset: 1

Author

Vladeta Jovovic, Apr 03 2000

Keywords

• easy
• frac
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..360

Crossrefs

Cf. A001497, A053987 (numerators), A161011 (tan(1/2)).

Programs

Magma

[(&+[ (-1)^k*Factorial(2*n-2*k)/(Factorial(n-2*k)*Factorial(2*k)): k in [0..Floor(n/2)]] ): n in [1..20]]; // G. C. Greubel, May 13 2020

Maple

a:= n -> add((-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!), k = 0..floor(n/2));
seq(a(n), n = 1..20); # G. C. Greubel, May 13 2020

Mathematica

Table[Sum[(-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!), {k,0,Floor[n/2]}], {n, 20}] (* G. C. Greubel, May 13 2020 *)

PARI

a(n)=sum(k=0,floor(n/2),(-1)^k*(2*n-2*k)!/(n-2*k)!/(2*k)!) \\ Benoit Cloitre, Jan 03 2006

Sage

[sum((-1)^k*factorial(2*n-2*k)/(factorial(n-2*k)*factorial(2*k)) for k in (0..floor(n/2))) for n in (1..20)] # G. C. Greubel, May 13 2020

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!) - Benoit Cloitre, Jan 03 2006

From G. C. Greubel, May 13 2020: (Start)

E.g.f.: cos((1 - sqrt(1-4*x))/2)/sqrt(1-4*x) - 1.

a(n) = 2*(2*n-1)*a(n-1) - a(n-2).

a(n) = ((-i)^n/2)*(y(n, 2*i) + (-1)^n*y(n, -2*i)), where y(n, x) are the Bessel Polynomials. (End)

a(n) ~ cos(1/2) * 2^(2*n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, May 14 2020

Extensions

More terms from G. C. Greubel, May 13 2020