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 1-6 of 6 results.

A101912 G.f. satisfies: A(x) = 1/(1 + x*A(x^2)) and also the continued fraction: 1 + x*A(x^3) = [1; 1/x, 1/x^2, 1/x^4, 1/x^8, ..., 1/x^(2^(n-1)), ...].

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, -2, 3, -1, -3, 6, -4, -4, 12, -10, -5, 23, -25, -2, 43, -57, 12, 74, -124, 56, 120, -258, 172, 170, -516, 454, 187, -989, 1095, 40, -1811, 2487, -604, -3128, 5375, -2567, -4991, 11140, -7704, -6976, 22164, -20062, -7220, 42288, -48020, -36, 76928, -108334, 29476, 131898, -233020, 117166
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2004

Keywords

Comments

Sequence appears to have a rational g.f. - Ralf Stephan, May 17 2007
The conjecture is wrong. The g.f. is dependent on the number of terms. - Johannes W. Meijer, Aug 08 2011

Crossrefs

Cf. A101913, A101914, A101915, A101916, A101917, A101918.

Programs

  • Maple
    nmax:=57: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101912(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^2)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101912(k):=x(k): od: seq(A101912(n), n=0..nmax); # Johannes W. Meijer, Aug 08 2011
  • Mathematica
    m = 58; A[] = 0; Do[A[x] = 1/(1 + x A[x^2]) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)
  • PARI
    {a(n)=local(A);A=1-x;for(i=1,n\2+1, A=1/(1+x*subst(A,x,x^2)+x*O(x^n)));polcoeff(A,n,x)}
  • PARI
    {a(n)=local(M=contfracpnqn(concat(1, vector(#binary(n)+1,n,1/x^(2^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(3*n+1)),3*n+1)}

Formula

G.f.: 1/(1 + x/(1 + x^2/(1 + x^4/(1 + x^8/(1 + ...))))) (continued fraction). - Joerg Arndt, Oct 19 2012
G.f. A(x) = 1/B(x) where B(x) is the g.f. of A218031. - Joerg Arndt, Oct 19 2012
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/2)} a(k) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 01 2022

A101917 G.f. satisfies: A(x) = 1/(1 + x*A(x^7)) and also the continued fraction: 1 + x*A(x^8) = [1; 1/x, 1/x^7, 1/x^49, 1/x^343, ..., 1/x^(7^(n-1)), ...].

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 2, -3, 4, -5, 6, -7, 8, -10, 13, -17, 22, -28, 35, -43, 53, -66, 83, -105, 133, -168, 211, -264, 330, -413, 518, -651, 819, -1030, 1294, -1624, 2037, -2555, 3206, -4025, 5055, -6349, 7973, -10010, 12565, -15771, 19796, -24851, 31200, -39173, 49183, -61748, 77519, -97315, 122166
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2004

Keywords

Crossrefs

Cf. A101912, A101913, A101914, A101915, A101916, A101918.

Programs

  • Maple
    nmax:=57: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101917(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^7)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101917(k):=x(k): od: seq(A101917(n), n=0..nmax); # Johannes W. Meijer, Aug 08 2011
  • Mathematica
    m = 57; A[] = 0; Do[A[x] = 1/(1 + x A[x^7]) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)
  • PARI
    a(n)=local(A);A=1-x;for(i=1,n\7+1, A=1/(1+x*subst(A,x,x^7)+x*O(x^n)));polcoeff(A,n,x)
  • PARI
    a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(7))+1,n,1/x^(7^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(8*n+1)),8*n+1)

Formula

This was conjectured to have g.f. (1+x^7) / (1+x+x^7) by Ralf Stephan, May 17 2007, but this is wrong. This g.f. produces a sequence which differs at a(57) = -153367. The g.f. gives a(57) = -153366. - Johannes W. Meijer, Aug 08 2011
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/7)} a(k) * a(n-7*k-1). - Ilya Gutkovskiy, Mar 01 2022

A101914 G.f. satisfies: A(x) = 1/(1 + x*A(x^4)) and also the continued fraction: 1 + x*A(x^5) = [1; 1/x, 1/x^4, 1/x^16, 1/x^64, ..., 1/x^(4^(n-1)), ...].

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -1, 2, -3, 3, -2, 0, 3, -6, 8, -8, 5, 1, -9, 17, -22, 20, -10, -8, 31, -51, 60, -50, 16, 38, -100, 150, -163, 119, -11, -147, 315, -432, 433, -268, -70, 522, -964, 1222, -1118, 542, 484, -1756, 2887, -3385, 2793, -879, -2176, 5678, -8472, 9186, -6672, 542, 8372, -17816, 24384, -24350, 14952
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2004

Keywords

Comments

Sequence appears to have a rational g.f. - Ralf Stephan, May 17 2007
The conjecture is wrong. The g.f. is dependent on the number of terms. With twenty terms the g.f. is (1 + x^4)/(1 + x + x^4). - Johannes W. Meijer, Aug 08 2011

Crossrefs

Cf. A101912, A101913, A101915, A101916, A101917, A101918.

Programs

  • Maple
    nmax:=62: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101914(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^4)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101914(k):=x(k): od: seq(A101914(n), n=0..nmax); # Johannes W. Meijer, Aug 08 2011
  • Mathematica
    m = 63; A[] = 0; Do[A[x] = 1/(1 + x A[x^4]) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)
  • PARI
    a(n)=local(A);A=1-x;for(i=1,n\4+1, A=1/(1+x*subst(A,x,x^4)+x*O(x^n)));polcoeff(A,n,x)
  • PARI
    a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(4))+1,n,1/x^(4^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(5*n+1)),5*n+1)

Formula

a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/4)} a(k) * a(n-4*k-1). - Ilya Gutkovskiy, Mar 01 2022

A101915 G.f. satisfies: A(x) = 1/(1 + x*A(x^5)) and also the continued fraction: 1+x*A(x^6) = [1;1/x,1/x^5,1/x^25,1/x^125,...,1/x^(5^(n-1)),...].

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 2, -3, 4, -5, 6, -8, 11, -15, 20, -26, 34, -45, 60, -80, 106, -140, 185, -245, 325, -431, 571, -756, 1001, -1326, 1757, -2329, 3086, -4088, 5415, -7173, 9504, -12593, 16685, -22105, 29284, -38796, 51400, -68100, 90225, -119535, 158365, -209810, 277970, -368275, 487916, -646421, 856416
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2004

Keywords

Crossrefs

Cf. A101912, A101913, A101914, A101916, A101917, A101918.

Programs

  • PARI
    a(n)=local(A);A=1-x;for(i=1,n\5+1, A=1/(1+x*subst(A,x,x^5)+x*O(x^n)));polcoeff(A,n,x)
  • PARI
    a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(5))+1,n,1/x^(5^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(6*n+1)),6*n+1)

Formula

a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/5)} a(k) * a(n-5*k-1). - Ilya Gutkovskiy, Mar 01 2022

A036675 G.f. satisfies A(x) = 1 + x*A(x)^2*A(x^2).

Original entry on oeis.org

1, 1, 2, 6, 18, 59, 198, 690, 2450, 8878, 32632, 121518, 457262, 1736526, 6646340, 25613086, 99298674, 387021728, 1515594560, 5960406102, 23530528512, 93216984177, 370450977206, 1476458287082, 5900150928510, 23635544130948
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000621, A101913

Programs

  • Maple
    A := 1; f := proc(n) global A; coeff(series( 1+x*(A*subs(x=x^2,A)), x, n+1), x,n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;
  • Mathematica
    terms = 26; A[] = 0; Do[A[x] = 1 + x*A[x]^2*A[x^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
  • Maxima
    T(n,m):=if m=n then 1 else sum((m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
makelist(T(2*n+1,1),n,0,30); /* Vladimir Kruchinin, Mar 18 2015 */
  • PARI
    a(n)=local(A,m); if(n<0,0,m=2; A=1+O(x); while(m<=n+1,m*=2; A=2/(1+sqrt(1-4*x*subst(A,x,x^2)))); polcoeff(A,n))

Formula

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2/(1-x^2)) (continued fraction); more generally g.f. C(x/(1-x^2/(1-x^2))) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011]
a(n) ~ c * d^n / n^(3/2), where d = 4.250770453055989899189676464071962617426..., c = 0.600960911911396921862654605015399962... . - Vaclav Kotesovec, Aug 10 2014
a(n) = T(2*n+1,1), where T(n,m) = sum(i=1..n-m, (m*binomial(m+2*i-1,i))/(m+i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i)), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015

A217925 G.f. A(x) satisfies A(x) = 1 + x*A(x)*A(x^2)^2.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 19, 40, 77, 155, 306, 610, 1207, 2400, 4760, 9456, 18765, 37257, 73955, 146813, 291434, 578524, 1148434, 2279720, 4525487, 8983421, 17832976, 35399824, 70271944, 139495472, 276910976, 549691232, 1091185133, 2166094309, 4299884233, 8535634803, 16943967775
Offset: 0

Views

Author

Joerg Arndt, Oct 15 2012

Keywords

Comments

What does this sequence count?

Links

Crossrefs

Cf. A000108 (A(x) = 1 + x*A(x)^2), A000621 (A(x) = 1 + x*A(x)*A(x^2)).
Cf. A036675 (A(x) = 1 + x*A(x)^2*A(x^2)), A101913 (A(x) = 1 + x*A(x)*A(x^3); for abs. values).

Programs

  • Maxima
    T(n,m):=if n=m then 1 else sum(binomial(m+k-1,k)*T((n-m)/2,2*k),k,1,(n-m)/4);
makelist(T(4*n+1,1),n,0,25); /* Vladimir Kruchinin, Mar 25 2015 */
  • PARI
    N=66;  R=O('x^N);  x='x+R;
F = 1 + x;
{ for (k=1,N+1, F = 1 + x * F * subst(F,'x,'x^2)^2 + R; ); }
Vec(F+O('x^N))

Formula

a(n) ~ c * d^n, where d = 1.985085392419660786124534041173530134614822710253953085885966352..., c = 0.322822740100478716884116064042886830242825005622702339543369128... . - Vaclav Kotesovec, Aug 10 2014
a(n) = T(4*n+1,1), where T(n,m) = Sum_{k=1..(n-m)/4} C(m+k-1,k)*T((n-m)/2,2*k). - Vladimir Kruchinin, Mar 25 2015
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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