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%I #10 Dec 12 2017 08:13:40
%S 2,3,21,49,106,204,374,659,1133,1913,3190,5272,8658,14155,23069,37513,
%T 60906,98780,160086,259350,419965,679891,1100481,1781048,2882258,
%U 4664090,7547189,12212179,19760329,31973532,51734950,83709638,135445813,219156747,354603929
%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, 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="/A296254/b296254.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.
%F a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.
%e a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4;
%e a(2) = a(0) + a(1) + b(1)^2 = 21;
%e Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)
%t a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^2;
%t j = 1; While[j < 6 , 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}] (* A296254 *)
%t Table[b[n], {n, 0, 20}] (* complement *)
%Y Cf. A001622, A296245.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Dec 11 2017
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