A032131 Shifts left 2 places under the "BIK" (reversible, indistinct, unlabeled) transform with a(1) = a(2) = 1.

1, 1, 1, 2, 3, 7, 13, 31, 66, 160, 369, 907, 2191, 5461, 13558, 34209, 86426, 220359, 563475, 1449282, 3739365, 9688104, 25173917, 65621067, 171498288, 449361649, 1180078602, 3105740797, 8189749105, 21636207962, 57257857968, 151771200002, 402899862910, 1071076810324, 2851165864937

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

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)

Crossrefs

Cf. A001224, A032128, A032130.

Programs

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 *)

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

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)

Extensions

Name edited by Petros Hadjicostas, Jan 14 2018

a(31)-a(35) from Petros Hadjicostas, Jan 14 2018