A295950 - OEIS

%I #12 Feb 22 2018 22:34:37

%S 1,4,10,20,37,65,111,187,310,510,834,1359,2209,3585,5812,9416,15249,

%T 24687,39959,64670,104654,169350,274031,443409,717469,1160908,1878408,

%U 3039348,4917789,7957171,12874995,20832202,33707235,54539476,88246751,142786268

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), 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="/A295950/b295950.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.

%F a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2) + f(n-2)*b(3) + ... + f(2)*b(n-1) + f(1)*b(n), where f(n) = A000045(n), the n-th Fibonacci number.

%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) = 10;

%e Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15,  ...)

%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];

%t j = 1; While[j < 5, 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}]   (* A295950 *)

%t Table[b[n], {n, 0, 20}]  (* complement *)

%Y Cf. A001622, A000045, A295862.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 08 2017