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 A295754 Solution of the complementary equation a(n) = a(n-1) + a(n-3) + a(n-4) + b(n-4), 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. 4
 1, 2, 3, 4, 12, 23, 37, 61, 105, 175, 284, 463, 757, 1231, 1994, 3231, 5237, 8481, 13726, 22215, 35955, 58186, 94152, 152348, 246516, 398882, 645411, 1044305, 1689734, 2734059, 4423808, 7157881, 11581709, 18739612, 30321339, 49060968, 79382329, 128443321 (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 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(0) = 12 Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...) 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 - 4]; 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}]   (* A295754 *) Table[b[n], {n, 0, 20}]  (*complement *) CROSSREFS Cf. A001622, A000045, A293411, A295755. Sequence in context: A117555 A243392 A239943 * A059810 A037397 A168047 Adjacent sequences:  A295751 A295752 A295753 * A295755 A295756 A295757 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 30 2017 STATUS approved

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Last modified September 22 03:54 EDT 2020. Contains 337289 sequences. (Running on oeis4.)