A371711 Expansion of g.f. A(x) satisfying A(x) = 2*x + A(2*x^2)^(1/2) with A(0) = 1.

1, 2, 2, 0, 2, 0, -4, 0, 22, 0, -36, 0, -108, 0, 376, 0, 1894, 0, -4180, 0, -10628, 0, 25032, 0, -300676, 0, 709528, 0, 2579816, 0, -9878288, 0, 85643910, 0, -153029428, 0, -514327828, 0, 2085333416, 0, -10936571628, 0, 17723495752, 0, 60604096152, 0, -268963402096, 0, -1956018724132, 0, 5018953346552

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..518

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(2*x^2) = (A(x) - 2*x)^2.

(2) A(8*x^4) = (A(2*x^2) - 4*x^2)^2 = ((A(x) - 2*x)^2 - 4*x^2)^2.

(3) A(128*x^8) = (A(8*x^4) - 16*x^4)^2 = (((A(x) - 2*x)^2 - 4*x^2)^2 - 16*x^4)^2.

(4) 1 = (...(((((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 - (2*x)^16)^2 - (2*x)^32)^2 - (2*x)^64)^2 - ...), an infinite nested square.

(5) A(x) = lim_{n->oo} (2*x + sqrt((2*x)^2 + sqrt((2*x)^4 + sqrt((2*x)^8 + sqrt((2*x)^16 + sqrt((2*x)^32 + sqrt((2*x)^64 + ... + sqrt((2*x)^(2^n) + 1)...))))))), with 1 appearing in the final (n-th) nested radical only.

(6) A(x) = x/Series_Reversion(F(x)), where F(x) = x*A(F(x)) is the g.f. of A371712 defined by F( 2*x^2/(1-2*x)^2 ) = 2*F(x)^2.

(7) F( 2*x^2/(A(x) - 2*x)^2 ) = 2*x^2, where F(x/A(x)) = x and F(x) = x*A(F(x)) is the g.f. of A371712.

The radius of convergence is r = 1/2, with A(r) = (3 + sqrt(5))/2 = 2.6180339...

Example

G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^4 - 4*x^6 + 22*x^8 - 36*x^10 - 108*x^12 + 376*x^14 + 1894*x^16 - 4180*x^18 - 10628*x^20 + ...
where A(2*x^2) = (A(x) - 2*x)^2.
The g.f. A(x) satisfies the infinite nested square given by
1 = (...(((((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 - (2*x)^16)^2 - (2*x)^32)^2 - (2*x)^64)^2 - ...)
as illustrated in the following process.
(1) (A(x) - 2*x)^2 = 1 + 4*x^2 + 8*x^4 + 32*x^8 - 256*x^12 + 5632*x^16 - 36864*x^20 - 442368*x^24 + ...
(2) ((A(x) - 2*x)^2 - (2*x)^2)^2 = 1 + 16*x^4 + 128*x^8 + 8192*x^16 - 1048576*x^24 + 369098752*x^32 + ...
(3) (((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 = 1 + 256*x^8 + 32768*x^16 + 536870912*x^32 - 17592186044416*x^48 + ...
(4) ((((A(x) - 2*x)^2 - (2*x)^2)^2 - (2*x)^4)^2 - (2*x)^8)^2 = 1 + 65536*x^16 + 2147483648*x^32 + 2305843009213693952*x^64 + ...
One may continue indefinitely in this manner, the process tending to 1 as a limit for |x| <= 1/2.
RELATED SERIES.
Let F(x) be the g.f. of A371712, then F(x/A(x)) = x, where
F(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 74*x^5 + 292*x^6 + 1204*x^7 + 5112*x^8 + 22182*x^9 + 97964*x^10 + 439252*x^11 + 1995864*x^12 + ...
which satisfies F( 2*x^2/(1-2*x)^2 ) = 2*F(x)^2.

Prog

(PARI) /* By definition, A(x) = 2*x + A(2*x^2)^(1/2) with A(0) = 1 */
{a(n) = my(A=1); for(i=1, #binary(n), A = 2*x + sqrt(subst(A, x, 2*x^2) +O(x^(n+2))) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Cf. A371712.

Sequence in context: A353596 A182122 A104624 * A193863 A363566 A273496

Adjacent sequences: A371708 A371709 A371710 * A371712 A371713 A371714

Keyword

sign

Author

Paul D. Hanna, Apr 07 2024

Status

approved