%I #9 Feb 22 2018 22:41:29
%S 1,2,9,18,35,62,108,182,303,499,817,1332,2166,3516,5702,9239,14963,
%T 24225,39212,63462,102700,166189,268917,435135,704082,1139248,1843362,
%U 2982643,4826039,7808717,12634793,20443548,33078380,53521968,86600389,140122399
%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%C See A295862 for a guide to related sequences.
%H Clark Kimberling, <a href="/A295956/b295956.txt">Table of n, a(n) for n = 0..2000</a>
%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) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e b(3) = 6 (least "new number")
%e a(2) = a(1) + a(0) + b(2) + 1 = 9
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
%t j = 1; While[j < 6, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t Table[a[n], {n, 0, k}]; (* A295956 *)
%Y Cf. A001622, A000045, A295862.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Dec 08 2017