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A078932 Number of compositions (ordered partitions) of n into powers of 3. 7
1, 1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 44, 66, 99, 147, 219, 327, 487, 726, 1083, 1614, 2406, 3588, 5349, 7974, 11889, 17725, 26426, 39399, 58739, 87573, 130563, 194655, 290208, 432669, 645062, 961716, 1433814, 2137659, 3187014, 4751490, 7083951 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from T. D. Noe)

FORMULA

G.f.: 1/( 1 - sum(k>=0, x^(3^k) ) ). [Joerg Arndt, Oct 21 2012]

G.f. satisfies A(x) = A(x^3)/(1 - x*A(x^3)), A(0) = 1.

Sum(k>=0, a(2k+1)*x^k) / sum(k>=0, a(2k)*x^k) = sum(k>=0, x^((3^n-1)/2)) = (1 +2x +4x^2 +9x^3 +20x^4 +...)/(1 +x +3x^2 +6x^3 +13x^4 +...) = (1 +x +x^4 +x^13 +x^40 +x^121 +...).

a(n) ~ c * d^n, where d=1.4908903146089481048158292585129929112464706408636716058683929302099..., c=0.5482795768884593030933437319550701222657139895191578491936872735719... - Vaclav Kotesovec, May 01 2014

EXAMPLE

A(x) = A(x^3) + x*A(x^3)^2 + x^2*A(x^3)^3 + x^3*A(x^3)^4 + ... = 1 +x + x^2 +2x^3 +3x^4 +4x^5 +6x^6 +9x^7 + 13x^8 +...

MAPLE

a:= proc(n) option remember;

      `if`(n=0, 1, add(a(n-3^i), i=0..ilog[3](n)))

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Jan 11 2014

MATHEMATICA

a[n_] := a[n] = If[n == 0, 1, Sum[a[n-3^i], {i, 0, Log[3, n]}]]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Mar 23 2015, after Alois P. Heinz *)

PROG

(PARI) a(n)=local(A, m); if(n<1, n==0, m=1; A=1+O(x); while(m<=n, m*=3; A=1/(1/subst(A, x, x^3)-x)); polcoeff(A, n))

(PARI)

N=66; x='x+O('x^N);

Vec( 1/( 1 - sum(k=0, ceil(log(N)/log(3)), x^(3^k)) ) )

/* Joerg Arndt, Oct 21 2012 */

CROSSREFS

Cf. A023359.

Cf. A087218, A087219.

Sequence in context: A061418 A136423 A215245 * A206740 A172161 A117791

Adjacent sequences:  A078929 A078930 A078931 * A078933 A078934 A078935

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 16 2002

EXTENSIONS

New description from T. D. Noe, Jan 29 2007

STATUS

approved

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Last modified September 17 06:41 EDT 2019. Contains 327119 sequences. (Running on oeis4.)