A161809 - OEIS

%I #22 Nov 01 2024 04:40:15

%S 1,3,6,12,21,33,51,75,105,147,201,267,354,462,591,753,948,1176,1455,

%T 1785,2166,2622,3153,3759,4470,5286,6207,7275,8490,9852,11415,13179,

%U 15144,17376,19875,22641,25761,29235,33063,37353,42105,47319,53124

%N G.f.: A(x) = exp( Sum_{n>=1} 3*A038500(n) * x^n/n ), where A038500 is the highest power of 3 dividing n.

%H Vaclav Kotesovec, <a href="/A161809/b161809.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Paul D. Hanna_, Jul 27 2009: (Start)

%F G.f. satisfies: A(x) = A(x^3)*(1+x+x^2)/(1-x)^2.

%F Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:

%F T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and

%F T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).

%F (End)

%e G.f.: A(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 + ...

%e log(A(x)) = 3*x + 3*x^2/2 + 9*x^3/3 + 3*x^4/4 + 3*x^5/5 + 9*x^6/6 + ...

%e From _Paul D. Hanna_, Jul 27 2009: (Start)

%e TRISECTIONS begin:

%e T_0(x) = 1 + 12*x + 51*x^2 + 147*x^3 + 354*x^4 + 753*x^5 + ...

%e T_1(x) = 3 + 21*x + 75*x^2 + 201*x^3 + 462*x^4 + 948*x^5 + ...

%e T_2(x) = 6 + 33*x + 105*x^2 + 267*x^3 + 591*x^4 + 1176*x^5 + ...

%e (End)

%t nmax = 50; CoefficientList[Series[Exp[Sum[3^(IntegerExponent[k, 3] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 01 2024 *)

%o (PARI) {a(n)=local(L=sum(m=1, n,3*3^valuation(m,3)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

%o (PARI) {a(n)=local(A=1+x);for(i=0,n\3,A=subst(A,x,x^3+x*O(x^n))*(1+x+x^2)/(1-x+x*O(x^n))^2);polcoeff(A,n)} \\ _Paul D. Hanna_, Jul 27 2009

%Y Cf. A038500, A182000, A182185, A000123.

%Y Partial sums of A309677.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 20 2009