A295949 - OEIS

%I #10 Feb 22 2018 22:33:59

%S 1,2,8,16,31,56,97,164,273,450,737,1202,1956,3176,5151,8347,13519,

%T 21888,35430,57342,92797,150165,242989,393182,636200,1029412,1665644,

%U 2695089,4360767,7055891,11416694,18472622,29889354,48362015,78251409,126613465,204864916

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), 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="/A295949/b295949.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) = 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) = 8;

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17,  ...)

%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];

%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}]  (* A295949 *)

%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