|
|
A117535
|
|
Number of ways of writing n as a sum of powers of 3, each power being used at most 4 times.
|
|
7
|
|
|
1, 1, 1, 2, 2, 1, 2, 2, 1, 3, 3, 2, 4, 4, 2, 3, 3, 1, 3, 3, 2, 4, 4, 2, 3, 3, 1, 4, 4, 3, 6, 6, 3, 5, 5, 2, 6, 6, 4, 8, 8, 4, 6, 6, 2, 5, 5, 3, 6, 6, 3, 4, 4, 1, 4, 4, 3, 6, 6, 3, 5, 5, 2, 6, 6, 4, 8, 8, 4, 6, 6, 2, 5, 5, 3, 6, 6, 3, 4, 4, 1, 5, 5, 4, 8, 8, 4, 7, 7, 3, 9, 9, 6, 12, 12, 6, 9, 9, 3, 8, 8, 5, 10, 10
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
It seems that this sequence can be calculated by constructing an insertion tree in which the insertion rules depend on the "age" of a term at a particular stage of the calculation. See the link for a discussion of this concept.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: product((1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))+x^(4*(3^j))), j=0..infinity). - Emeric Deutsch, Apr 02 2006
For n>=1, a(3*n+2) = a(n); a(3*n+1) = a(n) + a(n-1); a(3*n) = a(n) + a(n-1). - Tom Edgar, Jun 21 2017
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4) * A(x^3). - Ilya Gutkovskiy, Jul 09 2019
|
|
EXAMPLE
|
a(12) = 4 because 12=9+3=9+1+1+1=3+3+3+3=3+3+3+1+1+1.
|
|
MAPLE
|
g:= product((1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))+x^(4*(3^j))), j=0..10): gser:= series(g, x=0, 106): seq(coeff(gser, x, n), n=0..103); # Emeric Deutsch, Apr 02 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
add(`if`(n-j*3^i<0, 0, b(n-j*3^i, i-1)), j=0..4)))
end:
a:= n-> b(n, ilog[3](n)):
|
|
MATHEMATICA
|
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 0, 0, Sum[If[n - j*3^i < 0, 0, b[n - j*3^i, i - 1]], {j, 0, 4}]]]; a[n_] := b[n, Floor[Log[3, n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|