A296558 - OEIS

%I #4 Dec 20 2017 23:33:45

%S 1,3,7,13,24,41,69,114,187,306,498,809,1312,2126,3443,5574,9022,14601,

%T 23628,38235,61869,100110,161985,262101,424092,686199,1110297,1796502,

%U 2906805,4703313,7610124,12313443,19923573,32237022,52160601,84397630,136558238

%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="/A296558/b296558.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 = 7

%e Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, ...)

%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}];  (* A296558 *)

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

%Y Cf. A001622, A296245, A296493, A296569, A296570.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 20 2017