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A078932 Number of compositions (ordered partitions) of n into powers of 3.

%I A078932 #23 Oct 27 2023 19:29:51
%S A078932 1,1,1,2,3,4,6,9,13,20,30,44,66,99,147,219,327,487,726,1083,1614,2406,
%T A078932 3588,5349,7974,11889,17725,26426,39399,58739,87573,130563,194655,
%U A078932 290208,432669,645062,961716,1433814,2137659,3187014,4751490,7083951
%N A078932 Number of compositions (ordered partitions) of n into powers of 3.
%H A078932 Alois P. Heinz, <a href="/A078932/b078932.txt">Table of n, a(n) for n = 0..5000</a> (first 501 terms from T. D. Noe)
%F A078932 G.f.: 1/( 1 - sum(k>=0, x^(3^k) ) ). [_Joerg Arndt_, Oct 21 2012]
%F A078932 G.f. satisfies A(x) = A(x^3)/(1 - x*A(x^3)), A(0) = 1.
%F A078932 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 +...).
%F A078932 a(n) ~ c * d^n, where d=1.4908903146089481048158292585129929112464706408636716058683929302099..., c=0.5482795768884593030933437319550701222657139895191578491936872735719... - _Vaclav Kotesovec_, May 01 2014
%e A078932 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 +...
%p A078932 a:= proc(n) option remember;
%p A078932       `if`(n=0, 1, add(a(n-3^i), i=0..ilog[3](n)))
%p A078932     end:
%p A078932 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 11 2014
%t A078932 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-François Alcover_, Mar 23 2015, after _Alois P. Heinz_ *)
%o A078932 (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))
%o A078932 (PARI)
%o A078932 N=66; x='x+O('x^N);
%o A078932 Vec( 1/( 1 - sum(k=0, ceil(log(N)/log(3)), x^(3^k)) ) )
%o A078932 /* _Joerg Arndt_, Oct 21 2012 */
%Y A078932 Cf. A023359.
%Y A078932 Cf. A087218, A087219.
%K A078932 nonn
%O A078932 0,4
%A A078932 _Paul D. Hanna_, Dec 16 2002
%E A078932 New description from _T. D. Noe_, Jan 29 2007