A089405 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A073288/A073289.
1, 1, 2, 4, 9, 20, 47, 112, 279, 712, 1868, 5020, 13792, 38630, 110105, 318756, 935817, 2782424, 8368484, 25434314, 78047606
Offset: 0
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.
A(x) = A(x^2) + x*A(x^2)^2 + x^2*A(x^2)^3 + x^3*A(x^2)^4 + ... = 1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + 18x^6 + 31x^7 + .... From _Joerg Arndt_, Dec 28 2012: (Start) There are a(6)=18 compositions of 6 into powers of 2: [ 1] [ 1 1 1 1 1 1 ] [ 2] [ 1 1 1 1 2 ] [ 3] [ 1 1 1 2 1 ] [ 4] [ 1 1 2 1 1 ] [ 5] [ 1 1 2 2 ] [ 6] [ 1 1 4 ] [ 7] [ 1 2 1 1 1 ] [ 8] [ 1 2 1 2 ] [ 9] [ 1 2 2 1 ] [10] [ 1 4 1 ] [11] [ 2 1 1 1 1 ] [12] [ 2 1 1 2 ] [13] [ 2 1 2 1 ] [14] [ 2 2 1 1 ] [15] [ 2 2 2 ] [16] [ 2 4 ] [17] [ 4 1 1 ] [18] [ 4 2 ] (End)
a:= proc(n) option remember;
`if`(n=0, 1, add(a(n-2^i), i=0..ilog2(n)))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Jan 11 2014
CoefficientList[Series[1/(1 - Sum[x^(2^i), {i, 0, 20}]), {x, 0, 20}], x]
a[0] = 1; a[n_] := a[n] = Sum[a[n-2^k], {k, 0, Log[2, n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
{a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = 1 /(1 / subst(A, x, x^2) - x)); polcoeff(A, n))}; /* Michael Somos, Dec 20 2002 */
N=66; x='x+O('x^N);
Vec( 1/(1-sum(k=0,ceil(log(N)/log(2)), x^(2^k) ) ) )
/* Joerg Arndt, Oct 21 2012 */
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