cp's OEIS Frontend

A293350 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293350 #7 Nov 02 2017 09:19:44
%S A293350 1,3,10,23,46,85,150,257,432,718,1182,1935,3155,5131,8330,13508,21888,
%T A293350 35449,57393,92901,150356,243323,393748,637143,1030966,1668187,
%U A293350 2699234,4367505,7066826,11434421,18501340,29935857,48437296,78373255,126810656,205184019
%N A293350 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293350 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A293076 for a guide to related sequences.
%C A293350 Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%e A293350 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293350 a(2)  = a(1) + a(0) + b(0) + 4 = 10;
%e A293350 a(3) = a(2) + a(1) + b(1) + 6 = 23.
%e A293350 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14,...)
%t A293350 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293350 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293350 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 2n;
%t A293350 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293350 Table[a[n], {n, 0, 40}]  (* A293350 *)
%t A293350 Table[b[n], {n, 0, 10}]
%Y A293350 Cf. A001622 (golden ratio), A293076.
%K A293350 nonn,easy
%O A293350 0,2
%A A293350 _Clark Kimberling_, Oct 28 2017