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 A296249 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 2, 4, 31, 71, 151, 286, 518, 904, 1543, 2591, 4303, 7090, 11618, 18964, 30871, 50159, 81391, 131950, 213782, 346216, 560527, 907319, 1468471, 2376466, 3845666, 6222916, 10069423, 16293239, 26363686, 42658014, 69022856, 111682095, 180706247, 292389711 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number. EXAMPLE a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5; a(2) = a(0) + a(1) + b(2)^2 = 31; Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...) MATHEMATICA a = 2; a = 4; b = 1; b = 3; b = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2; j = 1; While[j < 6 , k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; Table[a[n], {n, 0, k}]     (* A296249 *) Table[b[n], {n, 0, 20}]  (* complement *) CROSSREFS Cf. A001622, A296245. Sequence in context: A132473 A053037 A018294 * A181620 A220283 A188113 Adjacent sequences:  A296246 A296247 A296248 * A296250 A296251 A296252 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 10 2017 STATUS approved

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Last modified October 20 16:08 EDT 2019. Contains 328268 sequences. (Running on oeis4.)