A182185 G.f.: exp( Sum_{n>=1} 3^b(n) * x^n/n ) where b(n) = highest exponent of 3 in 2^n+1.
1, 3, 5, 9, 15, 21, 29, 39, 49, 63, 81, 99, 123, 153, 183, 219, 261, 303, 353, 411, 469, 537, 615, 693, 781, 879, 977, 1089, 1215, 1341, 1485, 1647, 1809, 1989, 2187, 2385, 2607, 2853, 3099, 3375, 3681, 3987, 4323, 4689, 5055, 5457, 5895, 6333, 6813, 7335, 7857, 8421
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 3*x + 5*x^2 + 9*x^3 + 15*x^4 + 21*x^5 + 29*x^6 + 39*x^7 +... The g.f. satisfies: A(x)/A(x^3) = 1 + 3*x + 5*x^2 + 6*x^3 + 6*x^4 + 6*x^5 +...+ 6*x^n +... The logarithm of the g.f. begins: log(A(x)) = 3*x + x^2/2 + 9*x^3/3 + x^4/4 + 3*x^5/5 + x^6/6 + 3*x^7/7 + x^8/8 + 27*x^9/9 + x^10/10 + 3*x^11/11 + x^12/12 +...+ 3^b(n)*x^n/n +... where b(n) = highest exponent of 3 in 2^n+1, for n>=1, and begins: b = [1,0,2,0,1,0,1,0,3,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,4,...]. The g.f.s of the TRISECTIONS begin: T_0(x) = 1 + 9*x + 29*x^2 + 63*x^3 + 123*x^4 + 219*x^5 + 353*x^6 +... T_1(x) = 3 + 15*x + 39*x^2 + 81*x^3 + 153*x^4 + 261*x^5 + 411*x^6 +... T_2(x) = 5 + 21*x + 49*x^2 + 99*x^3 + 183*x^4 + 303*x^5 + 469*x^6 +... where T_1(x)/T_0(x) = 3*(1+x)/(1+5*x), T_2(x)/T_0(x) = (5+x)/(1+5*x).
Programs
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PARI
{a(n)=polcoeff(exp(sum(m=1, n+1, 3^valuation(2^m+1, 3)*x^m/m)+x*O(x^n)), n)} for(n=0, 65, print1(a(n), ", ")) -
PARI
{a(n)=local(A=1+x);for(i=1,ceil(log(n+1)/log(3)),A=(1-x^2)*(1-x^3)/(1-x)^3*subst(A,x,x^3+x*O(x^n)));polcoeff(A,n)}
Formula
G.f. satisfies: A(x) = (1-x^2)*(1-x^3)/(1-x)^3 * A(x^3).
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
(1) T_1(x)/T_0(x) = 3*(1+x)/(1+5*x),
(2) T_2(x)/T_0(x) = (5+x)/(1+5*x),
(3) T_0(x)/T_0(x^3) = (1+x)*(1+5*x)*(1-x^3)^2 / ((1-x)^3*(1+5*x^3)),
(4) T_1(x)/T_1(x^3) = (1+x)^2*(1-x^3)^2 / ((1-x)^3*(1+x^3)),
(5) T_2(x)/T_2(x^3) = (1+x)*(5+x)*(1-x^3)^2 / ((1-x)^3*(5+x^3)),
(6) A(x) = (1-x)/(1+5*x)*T_0(x) = (1-x)/(1+x)*T_1(x)/3 = (1-x)/(5+x)*T_2(x).