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A182000
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G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.
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3
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1, 2, 6, 10, 22, 34, 62, 90, 150, 210, 326, 442, 654, 866, 1230, 1594, 2198, 2802, 3766, 4730, 6230, 7730, 9998, 12266, 15630, 18994, 23878, 28762, 35742, 42722, 52526, 62330, 75926, 89522, 108118, 126714, 151878, 177042, 210702, 244362, 288982, 333602, 392182
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. satisfies: A(x) = (1+x^2)/(1-x)^2 * A(x^2).
Define BISECTIONS: A(x) = B_0(x^2) + x*B_1(x^2), then: B_1(x)/B_0(x) = 2/(1+x).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 62*x^6 +...
The g.f. satisfies:
A(x)/A(x^2) = 1 + 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 +...+ 2*n*x^n +...
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 16*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 32*x^8/8 + 2*x^9/9 + 8*x^10/10 + 2*x^11/11 + 16*x^12/12 +...+ 2^A090740(n)*x^n/n +...
where the highest exponents of 2 in 3^n-1, for n>=1, begins:
A090740 = [1,3,1,4,1,3,1,5,1,3,1,4,1,3,1,6,1,3,1,4,1,3,1,5,1,3,1,4,1,...].
The g.f.s of the BISECTIONS begin:
B_0(x) = 1 + 6*x + 22*x^2 + 62*x^3 + 150*x^4 + 326*x^5 + 654*x^6 +...
B_1(x) = 2 + 10*x + 34*x^2 + 90*x^3 + 210*x^4 + 442*x^5 + 866*x^6 +...
where 2 * B_0(x) / B_1(x) = 1+x.
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 2^valuation(3^m-1, 2)*x^m/m)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, #binary(n)+1, A=(1+x^2)/(1-x)^2*subst(A, x, x^2+x*O(x^n))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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