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 A295757 Solution of the complementary equation a(n) = a(n-1) + a(n-3) + a(n-4) + b(n-1), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences. 1
 1, 2, 3, 4, 15, 29, 46, 76, 132, 220, 356, 580, 949, 1543, 2498, 4047, 6560, 10623, 17191, 27822, 45030, 72870, 117910, 190790, 308720, 499531, 808263, 1307806, 2116091, 3423920, 5540025, 8963959, 14504008, 23467992, 37972016, 61440024, 99412066, 160852117 (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. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045). LINKS Table of n, a(n) for n=0..37. Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that b(4) = 9 (least "new number") a(4) = a(3) + a(1) + a(0) + b(3) = 15 Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 2; a = 3; a = 4; b = 5; b = 6; b = 7; b = 8; a[n_] := a[n] = a[n - 1] + a[n - 3] + a[n - 4] + b[n - 1]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 36; Table[a[n], {n, 0, z}] (* A295757 *) Table[b[n], {n, 0, 20}] (*complement *) CROSSREFS Cf. A001622, A000045, A293411, A295754. Sequence in context: A269725 A270929 A103096 * A005645 A362639 A348628 Adjacent sequences: A295754 A295755 A295756 * A295758 A295759 A295760 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 01 2017 STATUS approved

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Last modified September 29 19:36 EDT 2023. Contains 365776 sequences. (Running on oeis4.)