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 1-2 of 2 results.

A208887 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*(A(x) + A(-x)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 16, 22, 32, 46, 64, 92, 128, 179, 256, 358, 512, 730, 1024, 1460, 2048, 2878, 4096, 5756, 8192, 11644, 16384, 23288, 32768, 46147, 65536, 92294, 131072, 186018, 262144, 372036, 524288, 739210, 1048576, 1478420, 2097152, 2973636
Offset: 0

Views

Author

Paul D. Hanna, Mar 07 2012

Keywords

Comments

Limit a(n)^(1/n) = sqrt(2).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 +...
Related series:
A(x)*A(-x) = 1 + 3*x^2 + 6*x^4 + 11*x^6 + 22*x^8 + 46*x^10 + 92*x^12 +...
A(x)+A(-x) = 2 + 4*x^2 + 8*x^4 + 16*x^6 + 32*x^8 + 64*x^10 + 128*x^12 +...

Links

Crossrefs

Cf. A208888, A209199, A233895, A233896.

Programs

  • Mathematica
    CoefficientList[Series[(Sqrt[1+4*x^4] - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 19 2013 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,-x)+x^2*(A+subst(A,x,-x+x*O(x^n))));polcoeff(A,n)}
for(n=0,60,print1(a(n),", "))
  • PARI
    Vec((sqrt(1+4*x^4) - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)) + O(x^50)) \\ G. C. Greubel, Feb 01 2017

Formula

G.f.: A(x) = (sqrt(1+4*x^4) - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)).
Recurrence: (n+1)*a(n) = 2*(n+1)*a(n-2) - 4*(n-5)*a(n-4) + 8*(n-5)*a(n-6). - Vaclav Kotesovec, Aug 19 2013
a(n) ~ 2^(n/2) * (1 - 2*sin(Pi*n/4)*sin(Pi*n/2)/(sqrt(Pi)*n^(3/2))). - Vaclav Kotesovec, Aug 19 2013

A208889 G.f. satisfies: A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)*A(-x).

Original entry on oeis.org

1, 1, 3, 2, 5, -2, 3, -20, -2, -38, 75, 84, 612, 596, 2293, 534, 4722, -6354, 6716, -27696, 44962, -8232, 391279, 310072, 1975046, 906028, 6363470, -2135522, 15639296, -21729494, 57269753, -46268934, 370270484, 130699742, 2081100420, 915348168, 8523787094
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 2*x^3 + 5*x^4 - 2*x^5 + 3*x^6 - 20*x^7 +...
Related series:
A(x)/A(-x) = 1 + 2*x + 2*x^2 - 2*x^4 - 12*x^5 - 20*x^6 - 38*x^7 - 38*x^8 +...
A(x)*A(-x) = 1 + 5*x^2 + 15*x^4 + 36*x^6 + 87*x^8 + 320*x^10 + 1567*x^12 +...

Links

Crossrefs

Cf. A209199.

Programs

  • Mathematica
    terms = 37; A[] = 1; Do[A[x] = 1 + x*A[x]/A[-x] + x^2*A[x]*A[-x] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 09 2018 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A/subst(A,x,-x+x*O(x^n))+x^2*A*subst(A,x,-x+x*O(x^n)));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))
Showing 1-2 of 2 results.

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

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