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 A296269 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 2, 3, 10, 37, 82, 167, 312, 567, 987, 1697, 2852, 4744, 7820, 12819, 20927, 34069, 55356, 89824, 145620, 235927, 382075, 618577, 1001276, 1620528, 2622532, 4243843, 6867215, 11111957, 17980132, 29093112, 47074332, 76168599, 123244155, 199414084, 322659643 (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. EXAMPLE a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5; a(2) = a(0) + a(1) + b(0)*b(2) = 10; Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, ...) MATHEMATICA a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n]; j = 1; While[j < 10, 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}]; (* A296269 *) Table[b[n], {n, 0, 20}] (* complement *) CROSSREFS Cf. A001622, A296245. Sequence in context: A278051 A060604 A075890 * A141102 A144720 A164933 Adjacent sequences: A296266 A296267 A296268 * A296270 A296271 A296272 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 12 2017 STATUS approved

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