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A054390
Number of ways of writing n as a sum of powers of 3, each power being used at most three times.
11
1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 5, 2, 2, 4, 2, 2, 5, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 4, 3, 3, 5, 2, 2, 4, 2, 2, 5, 3, 3, 4, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 1, 5, 4, 4, 7, 3, 3, 6, 3, 3, 8, 5, 5, 7, 2, 2, 4, 2, 2, 6, 4, 4, 6, 2, 2
OFFSET
0,4
COMMENTS
Let M be an infinite matrix with (1, 1, 1, 1, 0, 0, 0, ...) in each column shifted down thrice from the previous column (for k>0). Then A054390 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Apr 14 2010
Conjecture: Number of ways of partitioning n into distinct parts of A038754. - R. J. Mathar, Mar 01 2023
LINKS
Karl Dilcher, Larry Ericksen, Polynomials Characterizing Hyper b-ary Representations, J. Int. Seq., Vol. 21 (2018), Article 18.4.3.
Timothy B. Flowers, Extending a Recent Result on Hyper m-ary Partition Sequences, Journal of Integer Sequences, Vol. 20 (2017), #17.6.7.
FORMULA
a(0)=1, a(1)=1, a(2)=1 and, for n>0, a(3n)=a(n)+a(n-1), a(3n+1)=a(n), a(3n+2)=a(n).
G.f.: Product_{j >= 0} (1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))). - Emeric Deutsch, Apr 02 2006
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3) * A(x^3). - Ilya Gutkovskiy, Jul 09 2019
EXAMPLE
a(33) = 4 because we have 33 = 27+3+3 = 27+3+1+1+1 = 9+9+9+3+3 = 9+9+9+3+1+1+1.
MAPLE
a[0]:=1: a[1]:=1: a[2]:=1: for n from 1 to 35 do a[3*n]:=a[n]+a[n-1]: a[3*n+1]:=a[n]: a[3*n+2]:=a[n] od: A:=[seq(a[n], n=0..104)]; # Emeric Deutsch, Apr 02 2006
g:=product((1+x^(3^j)+x^(2*(3^j))+x^(3*(3^j))), j=0..10): gser:=series(g, x=0, 125): seq(coeff(gser, x, n), n=0..104); # Emeric Deutsch, Apr 02 2006
# third 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..3)))
end:
a:= n-> b(n, ilog[3](n)):
seq(a(n), n=0..100); # Alois P. Heinz, Jun 21 2012
MATHEMATICA
a[0]=1; a[1]=1; a[2]=1; For[n=1, n <= 35, n++, a[3*n] = a[n] + a[n-1]; a[3*n+1] = a[n]; a[3*n+2] = a[n]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 20 2016, after Emeric Deutsch *)
CROSSREFS
Cf. A002487.
Sequence in context: A284312 A261612 A184241 * A161068 A161107 A161042
KEYWORD
nonn,look
AUTHOR
John W. Layman, May 09 2000
STATUS
approved