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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. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Let M = 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->inf} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Apr 14 2010

LINKS

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

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-Fran├žois Alcover, Dec 20 2016, after Emeric Deutsch *)

CROSSREFS

Cf. A002487.

Sequence in context: A284312 A261612 A184241 * A161068 A161107 A161042

Adjacent sequences:  A054387 A054388 A054389 * A054391 A054392 A054393

KEYWORD

nonn,look

AUTHOR

John W. Layman, May 09 2000

STATUS

approved

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Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)