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 1-3 of 3 results.

A087221 Number of compositions (ordered partitions) of n into powers of 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 96, 133, 184, 254, 352, 488, 676, 935, 1294, 1792, 2482, 3436, 4756, 6584, 9116, 12621, 17473, 24190, 33490, 46365, 64190, 88868, 123034, 170334, 235818, 326478, 451994, 625764, 866338, 1199400, 1660510
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2003

Keywords

Comments

Series trisections have a common ratio:
sum(k>=0, a(3k+1)*x^k) / sum(k>=0, a(3k)*x^k)
= sum(k>=0, a(3k+2)*x^k) / sum(k>=0, a(3k+1)*x^k)
= sum(k>=0, a(3k+3)*x^k) / sum(k>=0, a(3k+2)*x^k)
= sum(k>=0, x^((4^n-1)/3) ) = (1 + x + x^5 + x^21 + x^85 + x^341 +...).

Examples

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

Links

Crossrefs

Cf. A078932, A087222, A087232, A087224. Different from A003269.

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n=0, 1, add(a(n-4^i), i=0..ilog[4](n)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 11 2014
  • Mathematica
    a[n_] := a[n] = If[n==0, 1, Sum[a[n-4^i], {i, 0, Log[4, n]}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)
  • PARI
    a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=n,m*=4; A=1/(1/subst(A,x,x^4)-x)); polcoeff(A,n))
  • PARI
    N=66; x='x+O('x^N);
Vec( 1/( 1 - sum(k=0, ceil(log(N)/log(4)), x^(4^k)) ) )
/* Joerg Arndt, Oct 21 2012 */

Formula

G.f.: 1/( 1 - sum(k>=0, x^(4^k) ) ). [Joerg Arndt, Oct 21 2012]
G.f. satisfies A(x) = A(x^4)/(1 - x*A(x^4)), A(0) = 1.
a(n) ~ c * d^n, where d=1.384450093664460722709070772652942206959424183007359023442195..., c=0.526605891697738213614083414993893445498621299371909641096106... - Vaclav Kotesovec, May 01 2014

A087224 G.f. satisfies A(x) = f(x)^2 + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).

Original entry on oeis.org

1, 3, 7, 19, 50, 133, 352, 935, 2482, 6584, 17473, 46365, 123034, 326478, 866338, 2298895, 6100296, 16187616, 42955106, 113984740, 302467434, 802621041, 2129817812, 5651638433, 14997065388, 39795888008, 105601506802
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2003

Keywords

Examples

			Given f(x) = 1 + x + x^5 + x^21 + x^85 + x^341 + ...
so that f(x)^2 = 1 + 2x + x^2 + 2x^5 + 2x^6 + x^10 + 2x^21 + ...
and f(x)^3 = 1 + 3x + 3x^2 + x^3 + 3x^5 + 6x^6 + 3x^7 + 3x^10 + ...
then A(x) = (1 + 2x + x^2 + 2x^5 + ...) + x*A(x)*(1 + 3x + 3x^2 + x^3 + 3x^5 + ...)
= 1 + 3x + 7x^2 + 19x^3 + 50x^4 + 133x^5 + 352x^6 + ...

Crossrefs

Cf. A087221, A087222, A087223.

Programs

  • PARI
    a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=3*n+3,m*=4; A=1/(1/subst(A,x,x^4)-x)); polcoeff(A,3*n+2))

Formula

a(n) = A087221(3n+2).

A087223 G.f. satisfies A(x) = f(x) + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).

Original entry on oeis.org

1, 2, 5, 14, 36, 96, 254, 676, 1792, 4756, 12621, 33490, 88868, 235818, 625764, 1660510, 4406296, 11692452, 31026836, 82332140, 218474784, 579739960, 1538385398, 4082226194, 10832507040, 28744906148, 76276860598, 202406625820
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2003

Keywords

Examples

			Given f(x) = 1 + x + x^5 + x^21 + x^85 + x^341 + ...
so that f(x)^3 = 1 + 3x + 3x^2 + x^3 + 3x^5 + 6x^6 + 3x^7 + 3x^10 + ...
then A(x) = (1 + x + x^5 + ...) + x*A(x)*(1 + 3x + 3x^2 + x^3 + 3x^5 + 6x^6 + ...)
= 1 + 2x + 5x^2 + 14x^3 + 36x^4 + 96x^5 + 254x^6 + ...

Crossrefs

Cf. A087221, A087222, A087224.

Programs

  • PARI
    a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=3*n+3,m*=4; A=1/(1/subst(A,x,x^4)-x)); polcoeff(A,3*n+1))

Formula

a(n) = A087221(3n+1).
Showing 1-3 of 3 results.

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