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

A233895 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*(A(x)^2 + A(-x)^2).

Original entry on oeis.org

1, 1, 2, 3, 10, 18, 60, 115, 410, 822, 2996, 6174, 22980, 48324, 182328, 389187, 1484410, 3205710, 12329988, 26876586, 104080812, 228606012, 890262984, 1967830254, 7699472676, 17110322908, 67215426440, 150058534620, 591517612616, 1325828841480, 5241992235888
Offset: 0

Views

Author

Paul D. Hanna, Dec 17 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 18*x^5 + 60*x^6 + 115*x^7 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 30*x^4 + 68*x^5 + 205*x^6 + 482*x^7 +...
A(x)*A(-x) = 1 + 3*x^2 + 18*x^4 + 115*x^6 + 822*x^8 + 6174*x^10 +...
A(x)^2+A(-x)^2 = 2 + 10*x^2 + 60*x^4 + 410*x^6 + 2996*x^8 + 22980*x^10 +...

Links

Crossrefs

Cf. A208887, A143926, A233896.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*subst(A, x, -x)+x^2*(A^2+subst(A^2, x, -x+x*O(x^n))) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

Recurrence: n*(n+1)*(n+2)*(81*n^6 - 756*n^5 + 2565*n^4 - 3630*n^3 + 1150*n^2 + 1998*n - 1648)*a(n) = - 6*n*(n+1)*(81*n^5 - 1026*n^4 + 4851*n^3 - 10458*n^2 + 10280*n - 3872)*a(n-1) + 12*n*(486*n^8 - 4536*n^7 + 14634*n^6 - 14085*n^5 - 23214*n^4 + 69355*n^3 - 66518*n^2 + 29510*n - 6544)*a(n-2) - 72*(81*n^7 - 297*n^6 - 2124*n^5 + 13899*n^4 - 28141*n^3 + 20794*n^2 + 540*n - 4800)*a(n-3) - 144*(n-3)*(243*n^8 - 2268*n^7 + 7182*n^6 - 5976*n^5 - 13488*n^4 + 33301*n^3 - 30472*n^2 + 20528*n - 10400)*a(n-4) - 864*(n-4)*(n-3)*(n-2)*(108*n^3 - 441*n^2 + 75*n + 200)*a(n-5) - 1728*(n-5)*(n-4)*(n-3)*(81*n^6 - 270*n^5 + 690*n^3 - 695*n^2 + 374*n - 240)*a(n-6). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/n^(3/2), where d = sqrt(24 - 3*I*2^(2/3)*3^(5/6)*(3 + I*sqrt(3))^(1/3) + 6*I*2^(1/3)*3^(1/6)*(3 + I*sqrt(3))^(2/3) - 3*2^(2/3)*(9 + 3*I*sqrt(3))^(1/3)) = 3.12769717670219... is the root of the equation 1728 + 432*d^2 - 72*d^4 + d^6 = 0 and c = sqrt((34 - 4*sqrt(247) * sin(arccsc(494 * sqrt(247)/7687)/3)) / Pi) = 1.281119572461999772722... if n is even, and c = 2*sqrt(6 - sqrt(129) * sin(arcsin(323*sqrt(3/43)/86)/3)) / sqrt(Pi) = 0.970593260725094233562... if n is odd. - Vaclav Kotesovec, Dec 21 2013, updated Mar 18 2024
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