# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a296271 Showing 1-1 of 1 %I A296271 #6 Dec 12 2017 16:38:03 %S A296271 3,4,12,28,75,151,289,520,908,1558,2620,4373,7217,11845,19350,31518, %T A296271 51228,83145,134813,218441,353782,572798,927204,1500677,2428635, %U A296271 3930122,6359656,10290738,16651417,26943243,43595815,70540282,114137392,184679042,298817877 %N A296271 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. %C A296271 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 A296271 Clark Kimberling, Table of n, a(n) for n = 0..1000 %H A296271 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. %e A296271 a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5; %e A296271 a(2) = a(0) + a(1) + b(0)*b(2) = 11; %e A296271 Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...) %t A296271 a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5; %t A296271 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n]; %t A296271 j = 1; While[j < 10, k = a[j] - j - 1; %t A296271 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; %t A296271 Table[a[n], {n, 0, k}]; (* A296271 *) %t A296271 Table[b[n], {n, 0, 20}] (* complement *) %Y A296271 Cf. A001622, A296245. %K A296271 nonn,easy %O A296271 0,1 %A A296271 _Clark Kimberling_, Dec 12 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE