%I #8 Feb 22 2018 22:41:46

%S 1,4,11,22,41,72,123,206,342,562,919,1497,2433,3948,6400,10368,16789,

%T 27179,43992,71196,115214,186437,301679,488145,789854,1278030,2067916,

%U 3345979,5413929,8759943,14173908,22933888,37107834,60041761,97149635,157191437

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, 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="/A295957/b295957.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) = 4, b(0) = 2, b(1) = 3, b(2) = 5

%e b(3) = 6 (least "new number")

%e a(2) = a(1) + a(0) + b(2) + 1 = 11

%e Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ...)

%t a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; 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}]; (* A295957 *)

%Y Cf. A001622, A000045, A295862.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 08 2017