OFFSET
1,4
COMMENTS
From Petros Hadjicostas, Jan 14 2018: (Start)
For this sequence, if (b(n): n>=1) = BIK((a(n): n>=1)), then b(n) = a(n+2) for n>=1.
Let A(x) = Sum_{n>=1} a(n)*x^n be the g.f. for this sequence. For an explanation on how to derive the formula BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1 - A(x^2))) from Bower's formulae in the link below about transforms, see the comments for sequence A001224. (For that sequence, the roles of sequences (a(n): n>=1) and (b(n): n>=1) are reversed.)
(End)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
FORMULA
From Petros Hadjicostas, Jan 14 2018: (Start)
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then (A(x) - a(1)*x - a(2)*x^2)/x^2 = BIK(A(x)) = (1/2)*(A(x)/(1-A(x)) + (A(x) + A(x^2))/(1-A(x^2))). Here, a(1) = a(2) = 1.
In general, we have:
a(3) = a(1),
a(4) = (1/2)*(a(1)^2 + a(1) + 2*a(2)),
a(5) = (1/2)*(a(1)^2 + a(1) + 2*a(2) + 2)*a(1),
a(6) = (1/2)*(a(1)^4 + 4*a(1)^2 + (3*a(1)^2 + a(1) + 3)*a(2) + a(2)^2 + a(1)),
a(7) = (1/2)*(a(1)^4 + 4*a(1)^2*a(2) + 6*a(1)^2 + 3*a(2)^2 + 3*a(1) + 7*a(2) + 2)*a(1),
and so on. No pattern is apparent here.
(End)
MATHEMATICA
m = 36; a[1] = a[2] = 1; A[_] = 0;
Do[A[x_] = x^2 (a[1]/x + a[2] + (1/2)(A[x]/(1 - A[x]) + (A[x] + A[x^2])/(1 - A[x^2]))) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Sep 17 2019 *)
PROG
(PARI)
BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
seq(n)={my(p=1+O(x^(n%2))); for(i=1, n\2, p=1+x*BIK(x*p)); Vec(p)} \\ Andrew Howroyd, Aug 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Name edited by Petros Hadjicostas, Jan 14 2018
a(31)-a(35) from Petros Hadjicostas, Jan 14 2018
STATUS
approved