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 A295358 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 3, 5, 16, 30, 55, 95, 161, 268, 442, 724, 1181, 1921, 3120, 5061, 8201, 13283, 21506, 34812, 56342, 91179, 147547, 238753, 386328, 625110, 1011468, 1636610, 2648112, 4284756, 6932903, 11217695, 18150635, 29368368, 47519042, 76887450 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). EXAMPLE a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that b(3) = 7 (least "new number") a(3) = a(1) + a(0) + b(2) + b(1) - b(0) = 16 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; a = 5; b = 2; b = 4; b = 6; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - b[n - 3]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 32; u = Table[a[n], {n, 0, z}]   (* A295358 *) v = Table[b[n], {n, 0, 10}]  (* complement *) CROSSREFS Cf. A001622, A295357. Sequence in context: A331638 A080056 A019096 * A077551 A106588 A123785 Adjacent sequences:  A295355 A295356 A295357 * A295359 A295360 A295361 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 21 2017 STATUS approved

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)