A295952 - OEIS

%I #9 Feb 22 2018 22:40:19

%S 1,5,10,21,38,67,114,192,318,523,855,1393,2264,3674,5956,9649,15625,

%T 25296,40944,66264,107233,173523,280783,454334,735146,1189510,1924687,

%U 3114229,5038949,8153212,13192196,21345444,34537677,55883160,90420877,146304078

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, 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="/A295952/b295952.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) = 5, b(0) = 2, b(1) = 3, b(2) = 4;

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

%e a(2) = a(1) + a(0) + b(2) = 10;

%e Complement: (b(n)) = (2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, ...)

%t a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;

%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}]   (* A295952 *)

%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