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 A295621 Solution of the complementary equation a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*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. 3
 1, 2, 3, 4, 13, 22, 55, 96, 201, 346, 659, 1117, 2015, 3372, 5882, 9752, 16643, 27411, 46093, 75559, 125754, 205448, 339432, 553177, 909097, 1478897, 2421000, 3933174, 6420218, 10419979, 16972319, 27525507, 44762106, 72554068, 117844772, 190931789, 309833797 (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. 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) + 3*a(2) -2*a(1) - 2*a(0) + b(1) = 13 Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 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] + 3*a[n - 2] - 2*a[n - 3] - 2 a[n - 4] + b[n - 3]; 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}]   (* A295621 *) Table[b[n], {n, 0, 20}]  (*complement *) CROSSREFS Cf. A001622, A000045, A295619, A295620. Sequence in context: A236440 A162222 A010346 * A295755 A089142 A123215 Adjacent sequences:  A295618 A295619 A295620 * A295622 A295623 A295624 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 25 2017 STATUS approved

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Last modified May 31 15:27 EDT 2020. Contains 334748 sequences. (Running on oeis4.)