A296556 - OEIS

%I #4 Dec 20 2017 14:41:15

%S 1,3,11,23,45,81,141,239,400,661,1085,1772,2885,4687,7604,12325,19965,

%T 32328,52333,84704,137082,221833,358964,580848,939865,1520768,2460690,

%U 3981517,6442268,10423848,16866181,27290096,44156346,71446513,115602932,187049520

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + n, where a(0) = 1, a(1) = 3, b(0) = 2, 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="/A296556/b296556.txt">Table of n, a(n) for n = 0..1000</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) = 3, b(0) = 2, b(1) = 4, b(2) = 5

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

%e Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, ...)

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

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + n;

%t j = 1; While[j < 16, k = a[j] - j - 1;

%t  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t u = Table[a[n], {n, 0, k}];  (* A296556 *)

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

%Y Cf. A001622, A296245, A296493, A296565, A296566.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 20 2017