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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 (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 October 16 20:36 EDT 2021. Contains 348047 sequences. (Running on oeis4.)