A382316 G.f. satisfies A(x) = A(x^2) + A(x^2)^2*A(x^3)/A(x^6), with A(0) = 0 and A'(0) = 1.

1, 1, 2, 2, 5, 4, 9, 7, 20, 14, 31, 24, 60, 40, 90, 66, 167, 108, 241, 176, 414, 263, 586, 418, 987, 615, 1352, 966, 2185, 1350, 2965, 2079, 4710, 2886, 6262, 4392, 9681, 5906, 12780, 8843, 19492, 11826, 25325, 17512, 37876, 22920, 48956, 33474, 72418, 43680, 92482, 63163, 134737, 81120, 171400

Offset

1,3

Comments

Conjecture: there are no negative terms.

Links

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = A(x^2) + A(x^2)^2*A(x^3)/A(x^6).

(2) A(x) = Sum_{n>=1} A(x^(2^n))^2 * A(x^(3*2^(n-1))) / A(x^(3*2^n)).

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 2*x^4 + 5*x^5 + 4*x^6 + 9*x^7 + 7*x^8 + 20*x^9 + 14*x^10 + 31*x^11 + 24*x^12 + ...
where A(x) = A(x^2) + A(x^2)^2*A(x^3)/A(x^6).
Also,
A(x) = A(x^2)^2*A(x^3)/A(x^6) + A(x^4)^2*A(x^6)/A(x^12) + A(x^8)^2*A(x^12)/A(x^24) + A(x^16)^2*A(x^24)/A(x^48) + A(x^32)^2*A(x^48)/A(x^96) + A(x^64)^2*A(x^96)/A(x^192) + ...
SPECIFIC VALUES.
A(t) = 9 at t = 0.68847180027956205940267979038641905863045582714782...
A(t) = 8 at t = 0.68000436335045323090929950825639971249905277843093...
A(t) = 7 at t = 0.66985921620516731469547454989532308880630547524960...
A(t) = 6 at t = 0.65737286664573376894030760672820114313841705917430...
A(t) = 5 at t = 0.64144261483223982909521069091846348490606932294776...
A(t) = 4 at t = 0.62006687705991151801046877448774434542996717412639...
A(t) = 3 at t = 0.58912601200617113982185106132139917755192277860246...
A(t) = 2 at t = 0.53832059111937503579529574649620704260897492543302...
A(t) = 1 at t = 0.4312825248044065200519043095227488330708210749614664...
  where 1 = A(t^2) + A(t^2)^2*A(t^3)/A(t^6).
A(t) = -1 at t = -0.59571461954369933944437036806285058658364148215290...
A(t) = -2 at t = -0.66678513677193544830732191848117137374679072488651...
A(t) = -3 at t = -0.69705032454999534195395596764379153733430873493209...
A(3/4) = 26.8405229171592024154306450186566084468738368137910365...
A((3/4)^2) = 2.4046747047064122179702289067701576171879893457230...
A((3/4)^3) = 0.9466734991873289975804100911965455896223391986428...
A((3/4)^6) = 0.2240193181243728279435093925839312219634244267345...
  where A(3/4) = A((3/4)^2) + A((3/4)^2)^2 * A((3/4)^3)/A((3/4)^6).
A(2/3) = 6.72259693392064837541635094724488270079001880865422091...
A((2/3)^2) = 1.0809231663198130611363281762086318797796234978568...
A((2/3)^3) = 0.4683673295408771954688597756839493379598114762476...
A((2/3)^6) = 0.0969992271296495334648277546519292414772177869502...
  where A(2/3) = A((2/3)^2) + A((2/3)^2)^2 * A((2/3)^3)/A((2/3)^6).
A(1/2) = 1.53138505185900906832355981120204708472786982201094570...
A(1/2^2) = 0.358177624519110738761904166640236187849903606250688...
A(1/2^3) = 0.145192252825982457842220275286639707061805131839559...
A(1/2^6) = 0.015876893945713689638350838775507416799939590279746...
  where A(1/2) = A(1/2^2) + A(1/2^2)^2 * A(1/2^3) / A(1/2^6).
A(1/3) = 0.57598415529791326182863188692753472229624603083869819...
A(1/3^2) = 0.126599409101471530593231501159700920592291661250562...
A(1/3^3) = 0.038514512700385872313441260828942975227010687913862...
A(1/3^6) = 0.001373628958361319304620348266623005975061138211202...
  where A(1/3) = A(1/3^2) + A(1/3^2)^2 * A(1/3^3) / A(1/3^6).
A(1/4) = 0.35817762451911073876190416664023618784990360625068836...
A(1/4^2) = 0.066930091081345926072225127505578452254382097136108...
A(1/4^3) = 0.015876893945713689638350838775507416799939590279746...
A(1/4^6) = 0.000244200258755715612372492881920620370720397134924...
  where A(1/4) = A(1/4^2) + A(1/4^2)^2 * A(1/4^3) / A(1/4^6).

Prog

(PARI) {a(n) = my(A=[0, 1]); for(m=1, n, A=concat(A, 0);
A[#A] = polcoef( subst(Ser(A), x, x^2)^2*subst(Ser(A), x, x^3)/subst(Ser(A), x, x^6) + subst(Ser(A), x, x^2) - Ser(A), #A-1) ); H=A; A[n+1]}
for(n=1, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A222882 A236335 A016724 * A058657 A321285 A091434

Adjacent sequences: A382313 A382314 A382315 * A382317 A382318 A382319

Keyword

nonn

Author

Paul D. Hanna, May 08 2025

Status

approved