A245930 G.f.: 1 / AGM((1 - 3*x)^2, (1 + x)^2).

1, 2, 3, 4, 9, 30, 91, 232, 549, 1378, 3839, 11100, 31301, 85694, 234207, 653328, 1856829, 5300010, 15062839, 42702596, 121448901, 347414166, 997886671, 2870139480, 8257776521, 23782773242, 68627659563, 198437633884, 574654851209, 1665825647430, 4833258038251, 14037680955552, 40816416373293

Offset

0,2

Comments

Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.

Self-convolution of A245931.

Limit a(n+1)/a(n) = 3.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 / AGM((1-x)^2 + 4*x^2, (1-x)^2 - 4*x^2).

G.f.: 1 / AGM((1-x)^2, sqrt((1-x)^4 - 16*x^4)).

Recurrence: n^2*a(n) = (5*n^2 - 5*n + 2)*a(n-1) - 2*(5*n^2 - 10*n + 6)*a(n-2) + 2*(5*n^2 - 15*n + 12)*a(n-3) + 11*(n-2)^2*a(n-4) - 15*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Aug 16 2014

a(n) ~ 3^(n+2) / (4*Pi*n). - Vaclav Kotesovec, Aug 16 2014

Example

G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 9*x^4 + 30*x^5 + 91*x^6 + 232*x^7 +...
where
1/A(x) = 1 - 2*x + x^2 - 4*x^4 - 8*x^5 - 12*x^6 - 16*x^7 - 40*x^8 - 144*x^9 - 448*x^10 - 1152*x^11 - 2732*x^12 - 6840*x^13 - 18964*x^14 +...
equals AGM((1 - 3*x)^2, (1 + x)^2).
SPECIFIC VALUES:
A(x) = 2 at x = 0.2650276124990406644...
A(x) = 3 at x = 0.31872724866867463...
A(x) = 4 at x = 0.32990867978741...

Prog

(PARI) {a(n)=local(A=1); A = 1 / agm((1-3*x)^2, (1+x)^2 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A245931, A245932, A227845.

Sequence in context: A007704 A328836 A059972 * A086432 A186928 A076018

Adjacent sequences: A245927 A245928 A245929 * A245931 A245932 A245933

Keyword

nonn

Author

Paul D. Hanna, Aug 14 2014

Status

approved