A294416 - OEIS

%I #9 Nov 01 2017 20:54:06

%S 1,3,12,27,54,99,174,297,498,825,1357,2220,3618,5882,9547,15479,25079,

%T 40614,65752,106428,172245,278741,451057,729872,1181007,1910961,

%U 3092053,5003102,8095246,13098442,21193785,34292327,55486215,89778648,145264972,235043732

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

%C The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A294414 for a guide to related sequences.

%C Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%e a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

%e a(2)  = a(1) + a(0) + b(1) + b(0) + 2 = 12

%e Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14,...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n;

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 40}]  (* A294416 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A293076, A293765, A294414.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 31 2017