A273926 Given G(x) such that G( G(x)^2 - G(x)^3 ) = x^2, then G(x) = Sum_{n>=1} A273925(n)*x^n / 2^a(n).

0, 1, 3, 2, 7, 4, 10, 5, 15, 8, 18, 9, 22, 11, 25, 12, 31, 16, 34, 17, 38, 19, 41, 20, 46, 23, 49, 24, 53, 26, 56, 27, 63, 32, 66, 33, 70, 35, 73, 36, 78, 39, 81, 40, 85, 42, 88, 43, 94, 47, 97, 48, 101, 50, 104, 51, 109, 54, 112, 55, 116, 57, 119, 58, 127, 64, 130, 65, 134, 67, 137, 68, 142, 71, 145, 72, 149, 74, 152, 75, 158, 79, 161, 80, 165, 82, 168, 83, 173, 86, 176, 87, 180, 89, 183, 90, 190, 95, 193, 96, 197, 98, 200, 99, 205, 102, 208, 103, 212, 105, 215, 106, 221, 110, 224, 111, 228, 113, 231, 114

Offset: 1

Author

Paul D. Hanna, Jun 04 2016

Keywords

• nonn

Comments

Terms appear to occur only once in the sequence.

Both bisections of this sequence appear to be monotonically increasing.

Examples

			G.f.: G(x) = x + 1/2*x^2 + 3/8*x^3 + 3/4*x^4 + 175/128*x^5 + 41/16*x^6 + 4947/1024*x^7 + 321/32*x^8 + 687611/32768*x^9 + 11403/256*x^10 + 25132181/262144*x^11 + 107305/512*x^12 + 1941554203/4194304*x^13 + 2111325/2048*x^14 + 77643067507/33554432*x^15 + 21427329/4096*x^16 + 25549683166419/2147483648*x^17 + 1782548851/65536*x^18 + 1073363084982753/17179869184*x^19 + 18891311061/131072*x^20 + 91744420207896017/274877906944*x^21 + 406630578535/524288*x^22 + 3975787925128277349/2199023255552*x^23 + 4432136534071/1048576*x^24 +...+ A273925(n)*x^n / 2^a(n) +...
such that G( G(x)^2 - G(x)^3 ) = x^2.
The bisections of this sequence begin:
odd bisection (cf. A120738): [0, 3, 7, 10, 15, 18, 22, 25, 31, 34, 38, 41, 46, 49, 53, 56, 63, 66, 70, 73, 78, 81, 85, 88, 94, 97, 101, 104, 109, 112, 116, 119, 127, 130, 134, 137, 142, 145, 149, 152, 158, 161, 165, 168, 173, 176, 180, 183, 190, 193, 197, 200, 205, 208, 212, 215, 221, 224, 228, 231, 236, 239, 243, 246, 255, ...].
even bisection (cf. A101925): [1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..261

Crossrefs

Cf. A273925, A120738, A101925, A000120, A005187.

Programs

PARI

{a(n) = my(A=x); for(i=0,n, A = serreverse( sqrt(subst(A,x,x^2 - x^3 +x^2*O(x^n) )) )); valuation(denominator(polcoeff(A,n)),2)}
for(n=1,60,print1(a(n),", "))

Formula

a(2*n-1) = A120738(n-1) = 4*(n-1) - A000120(n-1), for n>=0 (conjecture).

a(2*n) = A101925(n-1) = A005187(n-1) + 1, for n>=0 (conjecture).