A235669 -id:A235669 - cp's OEIS frontend

A235684 Number of compositions of n into powers of 3 and doubled powers of 3.

1, 1, 2, 4, 7, 13, 25, 46, 86, 162, 302, 565, 1058, 1978, 3700, 6923, 12949, 24223, 45316, 84769, 158575, 296645, 554923, 1038079, 1941911, 3632677, 6795551, 12712263, 23780486, 44485521, 83217888, 155673480, 291214232, 544766722, 1019080592, 1906366927

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 13 2014

nonn

a(n+1)/a(n) tends to 1.87067337504749000600807516613083316430149226... (used Richardson's extrapolation) - Vaclav Kotesovec, Jan 14 2014

			a(3) = 4: 1+1+1, 2+1, 1+2, 3, thus we have 4 compositions with the allowed parts.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A078932, A235669, A235773.

a:= proc(n) option remember; `if`(n=0, 1, `if`(n<0, 0,
      add(a(n-3^i)+a(n-2*3^i), i=0..ilog[3](n))))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 13 2014

a[n_] := a[n] = If[n == 0, 1, If[n < 0, 0, Sum[a[n - 3^i] + a[n - 2*3^i], {i, 0, Log[3, n]}]]];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

A235773 Number of compositions of n into distinct powers of 3 and doubled powers of 3.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 7, 2, 2, 9, 8, 8, 32, 6, 6, 26, 6, 6, 31, 26, 26, 128, 6, 6, 26, 6, 6, 33, 32, 32, 158, 30, 30, 152, 30, 30, 176, 150, 150, 870, 24, 24, 126, 24, 24, 146, 126, 126, 750, 24, 24, 126, 24, 24, 151, 146, 146, 872, 126, 126, 770, 126, 126, 872

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 15 2014

			Let n=5. We have only two allowed compositions 2+3 = 3+2. So a(5) = 2.
For n=6, we have compositions 6 = 1+2+3 = 1+3+2 = 2+3+1 = 2+1+3 = 3+2+1 = 3+1+2. Thus a(6) = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A235684, A078932, A235669.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      expand(b(n, i-1)+`if`(3*3^i>n, 0, b(n-3*3^i, i-1)*x^2)
      +add(`if`(j*3^i>n, 0, b(n-j*3^i, i-1))*x, j=1..2))))
    end:
a:= n->(p->add(coeff(p, x, j)*j!, j=0..degree(p)))(b(n, ilog[3](n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 15 2014

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<0, 0, Expand[b[n, i-1] + If[3^(i+1) > n, 0, b[n-3^(i+1), i-1]x^2] + Sum[If[3^i j > n, 0, b[n-3^i j, i-1]]x, {j, 1, 2}]]]];
a[n_] := With[{p = b[n, Log[3, n] // Floor]}, Sum[Coefficient[p, x, j] j!, {j, 0, Exponent[p, x]}]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A235684 Number of compositions of n into powers of 3 and doubled powers of 3.

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