cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A206742 G.f.: 1/(1 - x/(1 - x^3/(1 - x^4/(1 - x^7/(1 - x^11/(1 - x^18/(1 -...- x^Lucas(n)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 34, 53, 80, 121, 187, 287, 436, 666, 1023, 1564, 2386, 3652, 5593, 8548, 13065, 19995, 30590, 46767, 71524, 109425, 167361, 255934, 391466, 598795, 915805, 1400649, 2142358, 3276767, 5011632, 7665186, 11724011, 17931702, 27426003
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Examples

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

Links

Crossrefs

Cf. A206741.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(LucasL[Range[nmax + 1]])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
  • PARI
    {Lucas(n)=polcoeff(x*(1+2*x)/(1-x-x^2+x*O(x^n)),n)}
{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^Lucas(M-k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

Formula

a(n) ~ c * d^n, where d = 1.52948673740109160123259225872298170871226757805081837... and c = 0.3181991399535991335364627230448471420031275308618... - Vaclav Kotesovec, Aug 25 2017

A206743 G.f.: 1/(1 - x/(1 - x^2/(1 - x^5/(1 - x^12/(1 - x^29/(1 - x^70/(1 -...- x^Pell(n)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 99, 164, 272, 450, 746, 1235, 2046, 3389, 5613, 9299, 15402, 25514, 42262, 70005, 115962, 192084, 318182, 527053, 873043, 1446161, 2395504, 3968060, 6572925, 10887788, 18035177, 29874537, 49485965, 81971484, 135782448
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

From Clark Kimberling, Jun 12 2016: (Start)
Number of real integers in n-th generation of tree T(2i) defined as follows.
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.
For r = 2i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
See A274142 for a guide to related sequences. (End)

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +...

Links

Crossrefs

Cf. A000621, A206741, A274142.

Programs

  • Maple
    A206743 := proc(r)
local gs,n,gs2,el,a ;
gs := [2,r] ;
for n from 3 do
gs2 := [] ;
for el in gs do
gs2 := [op(gs2),el+1,r*el] ;
end do:
gs := gs2 ;
a := 0 ;
for el in gs do
if type(el,'realcons') then
a := a+1 :
end if;
end do:
print(n,a) ;
end do:
end proc: # R. J. Mathar, Jun 16 2016
  • Mathematica
    z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]]; u = Table[t[[k]] /. x -> 2 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}] (* Clark Kimberling, Jun 12 2016 *)
  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=local(CF=1+x*O(x^n),M=ceil(log(2*n+1)/log(2.4))); for(k=0, M, CF=1/(1-x^Pell(M-k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))
  • Python
    N = 1000
pell = [0,1]
c = 2
while c < N:
    pell.append(c)
    c = pell[-1]*2 + pell[-2]
pell.reverse()
gf = [0]*(N+1)
for p in pell:
    gf = [-x for x in gf]
    gf[0] += 1
    quotient = [0]*(N+1)
    remainder = [0]*(N+1)
    remainder[p] = 1
    for n in range(N+1):
        q = remainder[n]//gf[0]
        for i in range(n,N+1):
            remainder[i] -= q*gf[i-n]
        quotient[n] = q
    gf = quotient
for i in range(N+1):
    print(i,gf[i])
# Kenny Lau, Aug 01 2017

Formula

a(n) ~ c * d^n, where d = 1.6564594309887754808836889708489581749625897572527517021957723319642053908... and c = 0.3844078703275069072126260832303344589497793302955451672191630264983... - 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

Views

Author

Ilya Gutkovskiy, Apr 18 2017

Keywords

Examples

			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 + ...

Links

Crossrefs

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

Programs

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

Formula

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

A307082 Expansion of 1/(1 - x/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 5*x/(1 - 8*x/(1 - ... - Fibonacci(k)*x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 6, 26, 164, 1540, 22068, 492616, 17378968, 977896328, 88256247312, 12819022165520, 3002745820555664, 1135759674922075168, 694219521332053782624, 686053892556368634929824, 1096476587053610841771551296, 2834651494015025836540377942080
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

Links

Crossrefs

Cf. A000045, A001622, A206741, A307083.

Programs

  • Maple
    b:= proc(x, y) option remember; (F->
     `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-x+1), 0)+
     `if`(y>x, b(x, y-1), 0)))(combinat[fibonacci])
    end:
a:= n-> b(n$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 12 2023
  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-Fibonacci[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * phi^(n*(n+1)/2) / 5^(n/2), where phi = A001622 is the golden ratio and c = 10.15498753508843821457456033641336796744756370048241257586748102558791... - Vaclav Kotesovec, Sep 18 2021

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

Views

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

Crossrefs

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

Programs

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

Formula

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.
Showing 1-5 of 5 results.

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