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A295620
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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.
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4
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1, 2, 3, 4, 12, 20, 49, 85, 177, 304, 578, 979, 1765, 2953, 5150, 8538, 14570, 23997, 40352, 66149, 110094, 179867, 297172, 484313, 795934, 1294823, 2119684, 3443689, 5621258, 9123343, 14860404, 24100573, 39192618, 63526879, 103182816, 167177109, 271286602
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OFFSET
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0,2
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COMMENTS
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The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.
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LINKS
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EXAMPLE
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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(0) = 12
Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)
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MATHEMATICA
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mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; a[3] = 4;
b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8;
a[n_] := a[n] = a[n - 1] + 3*a[n - 2] - 2*a[n - 3] - 2 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}] (* A295620 *)
Table[b[n], {n, 0, 20}] (*complement *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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