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

%I A032131 #34 Sep 17 2019 03:48:57
%S A032131 1,1,1,2,3,7,13,31,66,160,369,907,2191,5461,13558,34209,86426,220359,
%T A032131 563475,1449282,3739365,9688104,25173917,65621067,171498288,449361649,
%U A032131 1180078602,3105740797,8189749105,21636207962,57257857968,151771200002,402899862910,1071076810324,2851165864937
%N A032131 Shifts left 2 places under the "BIK" (reversible, indistinct, unlabeled) transform with a(1) = a(2) = 1.
%C A032131 From _Petros Hadjicostas_, Jan 14 2018: (Start)
%C A032131 For this sequence, if (b(n): n>=1) = BIK((a(n): n>=1)), then b(n) = a(n+2) for n>=1.
%C A032131 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.)
%C A032131 (End)
%H A032131 Andrew Howroyd, <a href="/A032131/b032131.txt">Table of n, a(n) for n = 1..200</a>
%H A032131 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032131 From _Petros Hadjicostas_, Jan 14 2018: (Start)
%F A032131 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.
%F A032131 In general, we have:
%F A032131 a(3) = a(1),
%F A032131 a(4) = (1/2)*(a(1)^2 + a(1) + 2*a(2)),
%F A032131 a(5) = (1/2)*(a(1)^2 + a(1) + 2*a(2) + 2)*a(1),
%F A032131 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)),
%F A032131 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),
%F A032131 and so on. No pattern is apparent here.
%F A032131 (End)
%t A032131 m = 36; a[1] = a[2] = 1; A[_] = 0;
%t A032131 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}];
%t A032131 CoefficientList[A[x], x] // Rest (* _Jean-François Alcover_, Sep 17 2019 *)
%o A032131 (PARI)
%o A032131 BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
%o A032131 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
%Y A032131 Cf. A001224, A032128, A032130.
%K A032131 nonn
%O A032131 1,4
%A A032131 _Christian G. Bower_
%E A032131 Name edited by _Petros Hadjicostas_, Jan 14 2018
%E A032131 a(31)-a(35) from _Petros Hadjicostas_, Jan 14 2018