A274717 G.f. satisfies: A(x) = A(x^2) + sqrt( A(x^2) ), where A(x) = Sum_{n>=1} a(n) * x^n / 2^A274716(n).

1, 1, 1, 1, 1, 1, 7, 1, -21, 1, 71, 1, 5, 7, 1095, 1, -15885, -21, 18443, 1, -55841, 71, 324945, 1, -2649857, 5, 6109987, 7, -18206579, 1095, 92290439, 1, -3700779069, -15885, 5957604211, -21, -29227819205, 18443, 88113645133, 1, -917331711003, -55841, 2134630503193, 71, -9943308217037, 324945, 29285764343377, 1, -616425445539209, -2649857, 1440419971225759, 5, -7198783835108021, 6109987, 19846350459729237

Offset

1,7

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1030

Formula

a(2*n) = a(n) for n>=1.

Example

G.f.: A(x) = x + x^2 + 1/2*x^3 + x^4 + 1/8*x^5 + 1/2*x^6 + 7/16*x^7 + x^8 - 21/128*x^9 + 1/8*x^10 + 71/256*x^11 + 1/2*x^12 + 5/1024*x^13 + 7/16*x^14 + 1095/2048*x^15 + x^16 - 15885/32768*x^17 - 21/128*x^18 + 18443/65536*x^19 + 1/8*x^20 - 55841/262144*x^21 + 71/256*x^22 + 324945/524288*x^23 + 1/2*x^24 - 2649857/4194304*x^25 + 5/1024*x^26 + 6109987/8388608*x^27 + 7/16*x^28 - 18206579/33554432*x^29 + 1095/2048*x^30 + 92290439/67108864*x^31 + x^32 +...+ a(n)*x^n/2^A274716(n) +...
such that ( A(x) - A(x^2) )^2 = A(x^2).
RELATED SERIES.
The following series forms an odd function:
A(x) - A(x^2) = x + 1/2*x^3 + 1/8*x^5 + 7/16*x^7 - 21/128*x^9 + 71/256*x^11 + 5/1024*x^13 + 1095/2048*x^15 - 15885/32768*x^17 + 18443/65536*x^19 - 55841/262144*x^21 + 324945/524288*x^23 - 2649857/4194304*x^25 + 6109987/8388608*x^27 - 18206579/33554432*x^29 +...
where ( A(x) - A(x^2) )^2 = A(x^2):
(A(x) - A(x^2))^2 = x^2 + x^4 + 1/2*x^6 + x^8 + 1/8*x^10 + 1/2*x^12 + 7/16*x^14 + x^16 - 21/128*x^18 + 1/8*x^20 + 71/256*x^22 + 1/2*x^24 + 5/1024*x^26 + 7/16*x^28 + 1095/2048*x^30 + x^32 +...

Prog

(PARI) {A274716(n) = if(n<3, 0, if(n%2==0, A274716(n/2), A274716(2*(n\4)+1) + n\2 ) )}
{a(n) = my(A=x+x^2); for(i=0, #binary(n),
A = subst(A, x, x^2) + sqrt( subst(A, x, x^2 +x^2*O(x^n)) ) );
2^A274716(n)*polcoeff(A, n)}
for(n=1, 65, print1(a(n), ", "))

Crossrefs

Cf. A274716.

Sequence in context: A229823 A229824 A229825 * A050310 A178445 A019431

Adjacent sequences: A274714 A274715 A274716 * A274718 A274719 A274720

Keyword

sign

Author

Paul D. Hanna, Jul 07 2016

Status

approved