login
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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.
2

%I #4 Dec 14 2017 14:23:17

%S 1,2,13,33,74,147,275,492,855,1455,2450,4070,6712,11003,17967,29255,

%T 47542,77154,125092,202683,328255,531463,860290,1392374,2253336,

%U 3646435,5900551,9547823,15449270,24998079,40448399,65447594,105897177,171346025,277244528

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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). See A296245 for a guide to related sequences.

%H Clark Kimberling, <a href="/A296293/b296293.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5

%e a(2) = a(0) + a(1) + 2*b(2) = 13

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, ...)

%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] + n*b[n];

%t j = 1; While[j < 10, 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}]; (* A296293 *)

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

%Y Cf. A001622, A296245.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 14 2017