A206743 -id:A206743 - cp's OEIS frontend

A274142 Number of integers in n-th generation of tree T(1/2) defined in Comments.

1, 1, 1, 2, 2, 4, 5, 8, 11, 17, 25, 37, 54, 81, 119, 177, 261, 388, 574, 851, 1260, 1868, 2767, 4101, 6077, 9006, 13347, 19781, 29315, 43448, 64392, 95436, 141444, 209636, 310705, 460501, 682519, 1011581, 1499295, 2222155, 3293534, 4881472, 7235018, 10723311, 15893460, 23556367, 34913897, 51747400

Offset: 0

Clark Kimberling, Jun 11 2016

nonn

Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
Guide to related sequences:
r           sequence
1/2         A274142
1/3         A274143
1/4         A274144
2/3         A274145
3/4         A274146
-1/2        A274147
-1/3        A274148
-1/4        A274149
-2/3        A274150
-3/4        A274151
3/2         A274152
5/2         A274153
-3/2        A274154
-5/2        A274155
2^(1/2)     A000045 (Fibonacci numbers)
2^(1/3)     A000930
2^(1/4)     A003269
2^(-1/2)    A274156
3^(-1/2)    A274157
2^(-1/3)    A274158
3^(-1/3)    A274159
i           A274160
2i          A206743
3i          A274162
4i          A274163
i/2         A274149
i/3         A274165
i+1         A274166
i-1         A274167
(-1+3i)/2   A274168

			If r = 1/2, then g(3) = {3,2r,r+1, r^2}, in which the integers are 3 and 1, so that a(3) = 2.

Kenny Lau, Table of n, a(n) for n = 0..5847

Cf. A274143-A274160, A274162, A274163, A274165-A274168.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 1/2, {k, 1, z}];
Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
(* second program: *)
T[0] = {0}; T[n_] := T[n] = Complement[Join[T[n-1]+1, x*T[n-1]], T[n-1]]; Reap[For[n = 0, n <= 25, n++, cnt = Count[T[n] /. x -> 1/2, Integer]; Print[n, " ", cnt]; Sow[cnt]]][[2, 1]] (* _Jean-François Alcover, Jun 14 2016 *)

More terms from Jean-François Alcover, Jun 14 2016

More terms from Kenny Lau, Jul 04 2016

A206741 G.f.: 1/(1 - x/(1 - x/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^8/(1 -...- x^Fibonacci(n)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 102, 231, 524, 1189, 2698, 6124, 13900, 31551, 71618, 162566, 369013, 837633, 1901368, 4315978, 9796979, 22238489, 50479892, 114585999, 260102617, 590415686, 1340204451, 3042175244, 6905536091, 15675109089, 35581458383, 80767551510

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 102*x^7 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A206742, A206743.

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Fibonacci[Range[nmax + 1]])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

{a(n)=local(CF=1+x*O(x^n),M=ceil(log(n+1)/log(1.6))); for(k=0, M, CF=1/(1-x^fibonacci(M-k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,50,print1(a(n),", "))

a(n) ~ c * d^n, where d = 2.2699337019511296354569330617166782764872939098477919669570757033487700138... and c = 0.3272015736512679060779796519077970622372291004190408455581585307453... - Vaclav Kotesovec, Aug 25 2017

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 9, 11, 15, 23, 28, 43, 57, 78, 113, 149, 214, 293, 403, 569, 774, 1086, 1502, 2072, 2896, 3986, 5548, 7691, 10636, 14797, 20459, 28400, 39386, 54542, 75724, 104886, 145468, 201733, 279545, 387786, 537472, 745233, 1033383, 1432415, 1986394

Offset: 0

Ilya Gutkovskiy, Apr 18 2017

nonn

			G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 5*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..5000

Cf. A000040, A206739, A206741, A206742, A206743, A227310, A269353.

R:= 1:
for i from numtheory:-pi(50) to 1 by -1 do
  R:= series(1-x^ithprime(i)/R, x, 51);
od:
R:= series(1/R, x, 51):
seq(coeff(R,x,j),j=0..50); # Robert Israel, Apr 20 2017

nmax = 50; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.3864622092472465020397266918102624708859968795203700659786636158522760956... and c = 0.15945087310540003725148530084775272562567007586487061850065597143186... - Vaclav Kotesovec, Aug 25 2017

A294922 Expansion of 1/(1 + x/(1 + x^3/(1 + x^4/(1 + x^7/(1 + x^11/(1 + ... + x^Lucas(k)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -6, 8, -11, 16, -22, 31, -44, 61, -85, 119, -166, 232, -325, 454, -634, 886, -1237, 1728, -2415, 3373, -4712, 6583, -9194, 12843, -17941, 25060, -35006, 48899, -68303, 95409, -133272, 186159, -260036, 363230, -507373, 708720, -989969, 1382827, -1931590

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

sign

Cf. A000204, A007325, A206741, A206742, A206743, A279586.

nmax = 45; CoefficientList[Series[1/(1 + ContinuedFractionK[x^LucasL[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 + x/(1 + x^3/(1 + x^4/(1 + x^7/(1 + x^11/(1 + ... + x^A000204(k)/(1 + ...))))))), a continued fraction.

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A274142 Number of integers in n-th generation of tree T(1/2) defined in Comments.

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A206741 G.f.: 1/(1 - x/(1 - x/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^8/(1 -...- x^Fibonacci(n)/(1 -...)))))))), a continued fraction.

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A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

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A294922 Expansion of 1/(1 + x/(1 + x^3/(1 + x^4/(1 + x^7/(1 + x^11/(1 + ... + x^Lucas(k)/(1 + ...))))))), a continued fraction.

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