A268296 G.f.: 1/(1-x) = sqrt(1 + a(1)*x + x*sqrt(1 + a(2)*x + x*sqrt(1 + a(3)*x + x*sqrt(...) ) ) ).

1, 5, 33, 713, 259985, 33800019393, 571220656468785748897, 163146519188315271151004136501803467012673, 13308393361631661480558768899208178264126697372663086894845840270509196618153817473, 88556666933960837614967248701566245757216163828988828731024437766134512448872086575494410071109970379702252158611451372093516192879655317859248726893259500181148929

Offset

1,2

Comments

Compare g.f. to: 1+x = sqrt(1+x + x*sqrt(1+x + x*sqrt(1+x + x*sqrt(...)))).

Limit a(n+1)/a(n)^2 = 1/2.

Links

Table of n, a(n) for n=1..10.

Formula

a(n) ~ 2 * c^(2^n), where c = 1.4448099826529... . - Vaclav Kotesovec, Mar 26 2016

Example

G.f.: 1/(1-x) = sqrt(1 + x + x*sqrt(1 + 5*x + x*sqrt(1 + 33*x + x*sqrt(1 + 713*x + x*sqrt(1 + 259985*x + x*sqrt(1 + 33800019393*x + x*sqrt(1 + 571220656468785748897*x + x*sqrt(1 +...)))))))).
Illustrate the calculation of initial terms.
1: 1/(1-x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 +...
2: ((1/(1-x)^2 - 1 - x)/x)^2  =  1 + 6*x + 17*x^2 + 34*x^3 + 58*x^4 + 90*x^5 + 131*x^6 +...
3: (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2  =  1 + 34*x + 357*x^2 + 1272*x^3 + 3308*x^4 + 7266*x^5 + 14302*x^6 +...
4: (( (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2 - 1 - 33*x)/x)^2  =  1 + 714*x + 129993*x^2 + 914824*x^3 + 3994428*x^4 + 13632080*x^5 + 39691244*x^6 +...
5: (( (( (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2 - 1 - 33*x)/x)^2 - 1 - 713*x)/x)^2  =  1 + 259986*x + 16900009697*x^2 + 237849421320*x^3 + 1875425573144*x^4 + 10852626534712*x^5 +...

Mathematica

poly = 1/(1-x)^2; Table[coef = SeriesCoefficient[poly, {x, 0, 1}] - 1;  poly = ((poly - 1 - coef*x)/x)^2; coef, {n, 1, 10}] (* Vaclav Kotesovec, Mar 26 2016 *)

Prog

(PARI) {a(n) = my(F=1/(1-x +x^2*O(x^n))); for(i=1, n-1, F = ((F^2 - 1+x - polcoeff(F^2, 1)*x)/x)); polcoeff(F^2, 1)-1}
for(n=1, 12, print1(a(n), ", "))

Crossrefs

Sequence in context: A140424 A295090 A350876 * A212296 A276160 A145505

Adjacent sequences: A268293 A268294 A268295 * A268297 A268298 A268299

Keyword

nonn

Author

Paul D. Hanna, Mar 26 2016

Status

approved