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.

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A111330 Let qf(a,q) = Product_{j >= 0} (1-a*q^j); g.f. is qf(q,q^4)/qf(q^3,q^4).

Original entry on oeis.org

1, -1, 0, 1, -1, -1, 2, 0, -2, 1, 1, -1, -1, 1, 2, -2, -2, 3, 1, -4, 0, 5, -1, -5, 2, 5, -4, -5, 6, 4, -6, -4, 7, 4, -10, -2, 12, 0, -13, 2, 13, -4, -14, 6, 17, -10, -17, 14, 15, -17, -15, 21, 15, -26, -13, 31, 9, -35, -5, 39, 2, -44, 3, 49, -12, -52, 21, 53, -27, -55, 35, 57, -47, -57, 59, 55, -69, -52, 80, 49, -95, -43, 110, 34, -122
Offset: 0

Views

Author

N. J. A. Sloane, Nov 09 2005

Keywords

Crossrefs

Cf. A111317, A111335, A111374, A224704, A284313, A284316.

Formula

From Peter Bala, Nov 28 2020: (Start)
O.g.f.: A(x) = F(x)/G(x) where F(x) = Product_{k >= 0} 1 - x^(4*k+1) (see A284313) and G(x) = Product_{k >= 0} 1 - x^(4*k+3) (see A284316).
Continued fraction representations: A(x) = 1 - x/(1 + x^2 - x^3/(1 + x^4 - x^5/(1 + x^6 - ... ))).
A(x) = 1 - x/(1 - x^2*(x - 1)/(1 - x^5/(1 - x^4*(x^3 - 1)/(1 - x^9/(1 - x^6*(x^5 - 1)/(1 - ... )))))). Cf. A224704. (End)

A206737 G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 28, 39, 54, 76, 107, 150, 210, 294, 412, 578, 811, 1137, 1593, 2233, 3131, 4390, 6155, 8629, 12097, 16959, 23777, 33336, 46737, 65524, 91863, 128790, 180563, 253149, 354912, 497581, 697602, 978031, 1371190, 1922395
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n - 2, 1, n^3 - 2, 1, n^4 - 2, 1, n^6 - 2, 1, n^7 - 2, 1, n^9 - 2, 1, n^10 - 2, 1, ...] for n >= 3 and A(-1/n) = [0; 1, n - 1, 1, n^3 - 1, n^4 - 1, 1, n^6 - 1, n^7 - 1, 1, n^9 - 1, n^10 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - Peter Bala, Dec 15 2015

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...
Simple continued fraction expansions: A(1/10) = 1.11112345816325284441923227158 ... = [1, 8, 1, 998, 1, 9998, 1, 999998, 1, 9999998, 1, 999999998, 1, 9999999998, 1, ...]; A(-1/10) = 0.909082643877542661578687284018 ... = [0, 1, 9, 1, 999, 9999, 1, 999999, 9999999, 1, 999999999, 9999999999, 1, ...]. - _Peter Bala_, Dec 15 2015

Links

Crossrefs

Cf. A206738, A143951, A005169, A111317, A143951, A224704.

Programs

  • Maple
    N:= 100: # to get a(0) .. a(N)
C:= [0,[1,1],seq([-x^i,1],i=1..N,3)]:
S:= series(numtheory:-cfrac(C),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Dec 28 2015
  • Mathematica
    max = 15; CF = 1+x*O[x]^max; M = Sqrt[max+1]//Floor; For[k=0, k <= M, k++, CF = 1/(1-x^(3M-3k+1)*CF)]; CoefficientList[CF, x] (* Jean-François Alcover, Dec 29 2015, adapted from PARI *)
nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
  • PARI
    {a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^(3*M-3*k+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

Formula

a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - Vaclav Kotesovec, Aug 25 2017
From Peter Bala, Jul 03 2019: (Start)
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2+n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206738.
D(q) has a simple real zero at x = 0.7132721628.... The constants c and d quoted in the above asymptotic approximation for a(n) are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)

A206738 G.f.: 1/(1 - x^2/(1 - x^5/(1 - x^8/(1 - x^11/(1 - x^14/(1 - x^17/(1 -...- x^(3*n-1)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 9, 11, 14, 18, 22, 29, 35, 46, 56, 73, 90, 116, 144, 184, 231, 292, 370, 465, 591, 742, 942, 1185, 1502, 1893, 2395, 3023, 3819, 4826, 6093, 7702, 9724, 12290, 15519, 19611, 24767, 31294, 39527, 49937, 63082
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n^2 - 2, 1, n^3 - 2, 1, n^5 - 2, 1, n^6 - 2, 1, n^8 - 2, 1, n^9 - 2, 1, n^11 - 2, 1, n^12 - 2, 1, ...] for n >= 2 and A(-1/n) = [ 1, n^2 - 1, n^3 - 1, 1, n^5 - 1, n^6 - 1, 1, n^8 - 1, n^9 - 1, 1, n^11 - 1, n^12 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - Peter Bala, Dec 15 2015

Examples

			G.f.: A(x) = 1 + x^2 + x^4 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 3*x^11 + ...
Simple continued fraction expansions: A(1/2) = 1.34788543155288690684 ... = [1; 2, 1, 6, 1, 30, 1, 62, 1, 254, 1, 510, 1, 2046, 1, 4094, 1, ...] and A(-1/2) = 1.3199498363818812865 ... = [1; 3, 7, 1, 31, 63, 1, 255, 511, 1, 2047, 4095, 1, ...]. - _Peter Bala_, Dec 15 2015

Links

Crossrefs

Cf. A206737, A143951, A005169, A111317, A143951, A224704.

Programs

  • Maple
    N:= 100:
C:= [0,[1,1],seq([-x^i,1],i=2..N,3)]:
S:= series(numtheory:-cfrac(C),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 18 2024
  • Mathematica
    nmax = 60; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
  • PARI
    {a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^(3*M-3*k+2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

Formula

a(n) ~ c * d^n, where d = 1.26326802855134275222... and c = 0.16506173508242936... - Vaclav Kotesovec, Aug 25 2017
From Peter Bala, Jul 03 2019: (Start)
O.g.f. as a ratio of q series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2+2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206737.
D(q) has a simple real zero at x = 0.79159764784576529644 .... The constants c and d quoted in the above asymptotic approximation are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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