A161809 G.f.: A(x) = exp( Sum_{n>=1} 3*A038500(n) * x^n/n ), where A038500 is the highest power of 3 dividing n.

1, 3, 6, 12, 21, 33, 51, 75, 105, 147, 201, 267, 354, 462, 591, 753, 948, 1176, 1455, 1785, 2166, 2622, 3153, 3759, 4470, 5286, 6207, 7275, 8490, 9852, 11415, 13179, 15144, 17376, 19875, 22641, 25761, 29235, 33063, 37353, 42105, 47319, 53124

Offset: 0

Author

Paul D. Hanna, Jul 20 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 + ...
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 + ...
From _Paul D. Hanna_, Jul 27 2009: (Start)
TRISECTIONS begin:
T_0(x) = 1 + 12*x + 51*x^2 + 147*x^3 + 354*x^4 + 753*x^5 + ...
T_1(x) = 3 + 21*x + 75*x^2 + 201*x^3 + 462*x^4 + 948*x^5 + ...
T_2(x) = 6 + 33*x + 105*x^2 + 267*x^3 + 591*x^4 + 1176*x^5 + ...
(End)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A038500, A182000, A182185, A000123.

Partial sums of A309677.

Programs

Mathematica

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 *)

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)}

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

Formula

From Paul D. Hanna, Jul 27 2009: (Start)

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

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

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

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

(End)