A233347 G.f. satisfies: 1-x = Sum_{n>=0} (-x)^n*A(x)^(n mod 3).

1, 1, 1, 4, 10, 29, 86, 266, 842, 2720, 8924, 29661, 99654, 337902, 1154814, 3973848, 13756956, 47878802, 167425436, 587954324, 2072659700, 7331934464, 26018377112, 92596633117, 330415480966, 1181911210006, 4237288522870, 15222929711704, 54796317032956, 197601339619494

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. satisfies: (1 + x^3*A(x)^3) / (1 + x*A(x)) = (1-x)*(1+x^3).

G.f.: (1 - sqrt(4*(1-x)*(1+x^3) - 3))/(2*x).

a(n) ~ sqrt(1-3*r+r^3) / (2*sqrt(Pi)*n^(3/2)*r^(n+1)), where r = (1-2*sqrt(1/4 + 3^(1/3)/2^(2/3)) + sqrt(2-2*6^(1/3) + 7/sqrt(1/4 + 3^(1/3)/2^(2/3))))/4 = 0.2634705866719... is the root of the equation r^4-r^3+r=1/4. - Vaclav Kotesovec, Dec 08 2013

Example

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 29*x^5 + 86*x^6 + 266*x^7 +...
such that, by definition,
1-x = 1 - x*A(x) + x^2*A(x)^2 - x^3 + x^4*A(x) - x^5*A(x)^2 + x^6 - x^7*A(x) + x^8*A(x)^2 - x^9 + x^10*A(x) - x^11*A(x)^2 + ....

Mathematica

CoefficientList[Series[(1 - Sqrt[4*(1-x)*(1+x^3) - 3])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 08 2013 *)

Prog

(PARI) /* By Definition: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(n=1, #A, (-x)^n*Ser(A)^(n%3)))[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Closed Form: */
{a(n)=local(A=1+x); polcoeff((1 - sqrt(4*(1-x)*(1+x^3) - 3 +x^2*O(x^n)))/(2*x), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A116388 A221420 A212262 * A264158 A152808 A243601

Adjacent sequences: A233344 A233345 A233346 * A233348 A233349 A233350

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2013

Status

approved