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

0,2

Links

Table of n, a(n) for n=0..51.

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).

Example

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).

Prog

(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)}

Crossrefs

Cf. A182000, A161809.

Sequence in context: A045602 A029470 A323646 * A161388 A229552 A029518

Adjacent sequences: A182182 A182183 A182184 * A182186 A182187 A182188

Keyword

nonn

Author

Paul D. Hanna, Apr 18 2012

Status

approved