A377255 G.f. A(x) satisfies A(x)^2 = A(x^2) + 2*A(x^3).

1, 1, 0, 0, 1, -1, 1, -1, 1, 0, -2, 4, -6, 9, -9, 4, 7, -28, 58, -89, 107, -86, -13, 233, -597, 1064, -1483, 1540, -701, -1725, 6342, -13197, 20956, -25817, 20555, 5544, -64631, 164340, -296076, 417478, -432575, 178788, 560242, -1985957, 4126485, -6565542, 8076278, -6271393, -2464675, 22301182

Offset

1,11

Links

Paul D. Hanna, Table of n, a(n) for n = 1..4100

Formula

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

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

(2) A(x)^4 = A(x^4) + 4*A(x^2)*A(x^3) + 6*A(x^6) + 8*A(x^9).

(3) A(x)^6 = A(x^2)*A(x^4) + 6*A(x^3)*A(x^4) + 14*A(x^2)*A(x^6) + 20*A(x^3)*A(x^6) + 24*A(x^2)*A(x^9) + 16*A(x^3)*A(x^9).

Example

G.f.: A(x) = x + x^2 + x^5 - x^6 + x^7 - x^8 + x^9 - 2*x^11 + 4*x^12 - 6*x^13 + 9*x^14 - 9*x^15 + 4*x^16 + 7*x^17 - 28*x^18 + 58*x^19 - 89*x^20 + 107*x^21 - 86*x^22 - 13*x^23 + 233*x^24 + ...
where A(x)^2 = A(x^2) + 2*A(x^3), with A(x)^2 starting as
A(x)^2 = x^2 + 2*x^3 + x^4 + 2*x^6 + x^10 - x^12 + x^14 + 2*x^15 - x^16 - x^18 + 2*x^21 - 2*x^22 + 2*x^24 - 6*x^26 + 2*x^27 + 9*x^28 - 9*x^30 + 4*x^32 - 4*x^33 + 7*x^34 - 20*x^36 + ...
SPECIFIC VALUES.
A(t) = 1 at t = 0.5961279711567386344661607121872765340939629312983557348...
  where 1 = A(t^2) + 2*A(t^3)
  with A(t^2) = 0.48584863781523105586522683132298453370018867420365...
A(t) = 3/4 at t = 0.49033493066608907250741527002971672524477120833459...
A(t) = 2/3 at t = 0.45065864825914969582287502929418473633530906069240...
A(t) = 1/2 at t = 0.36333158898920999946871502379628293458672414274111...
A(t) = 1/3 at t = 0.26310835228397618711419831037322632867088816952916...
A(Phi) = 1.05761472477916031953536531352124685545841595352791...
  where Phi = (sqrt(5) - 1)/2.
A(3/5) = 1.01002217201836051776551980621440910635351807228749...
A(1/2) = 0.77109400600780738196546666957984770864912034983630...
  where A(1/2) = sqrt( A(1/4) + 2*A(1/8) ).
A(2/5) = 0.56734858743763718207207305803814791968268623416723...
A(1/3) = 0.44753740023994743528805975081933178883640829094372...
  where A(1/3) = sqrt( A(1/9) + 2*A(1/27) ).
A(1/4) = 0.31328171134072218873594020054568132198802257842731...
  where A(1/4) = sqrt( A(1/16) + 2*A(1/64) ).
A(1/5) = 0.24026672371879473703677685329112971095482956823328...
A(1/8) = 0.14065212738022314906818422523953979263643369573594...
A(1/9) = 0.12347203190902362722290808353234748838680201328604...
A(1/16) = 0.0664071475765825099743123343265025607965378788165...
A(1/25) = 0.0416000984615477684536424445378822652386391825446...
A(1/27) = 0.0384088463522536377428198560108166851971212063906...
A(1/64) = 0.0158691415419945358337499110058877254527680188361...

Prog

(PARI) {a(n) = my(A, V=[0, 1]); for(i=1, n+1, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( (subst(A, x, x^2) + 2*subst(A, x, x^3) - A^2 )/2, #V) ); polcoef(A, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Cf. A377256, A377257, A377258.

Sequence in context: A176461 A255249 A330394 * A084407 A114526 A333387

Adjacent sequences: A377252 A377253 A377254 * A377256 A377257 A377258

Keyword

sign

Author

Paul D. Hanna, Nov 27 2024

Status

approved