A112104 Unique sequence of 1's and 2's where g.f. A(x) satisfies A(x) = B(B(x)) such that B(x) is an integer series, with A(0) = 0.
1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 +... then A(x) = B(B(x)) where B(x) = x + x^2 + x^5 - 3*x^6 + 7*x^7 - 10*x^8 - 5*x^9 +... is the g.f. of A112105.
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..512
Programs
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Mathematica
kmax = 105; A[x_] := Sum[a[k] x^k, {k, kmax}]; B[x_] := Sum[b[k] x^k, {k, kmax}]; sol = {a[1] -> 1, b[1] -> 1}; Do[Print["k = ", k]; sc = SeriesCoefficient[B[(B[x] /. sol) + O[x]^(k+1)] - A[x], {x, 0, k}] /. sol; r = Reduce[(a[k] == 1 || a[k] == 2) && sc == 0, {a[k], b[k]}, Integers]; sol = Join[r // ToRules, sol], {k, 2, kmax}]; a /@ Range[kmax] /. sol (* Jean-François Alcover, Nov 05 2019 *) -
PARI
{a(n,m=2)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}