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A294532
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Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3.
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35
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1, 2, 6, 12, 23, 42, 73, 124, 207, 342, 562, 918, 1495, 2429, 3941, 6388, 10348, 16756, 27125, 43903, 71052, 114980, 186058, 301065, 487151, 788245, 1275426, 2063702, 3339160, 5402895, 8742089, 14145019, 22887144, 37032200, 59919382, 96951621, 156871043
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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, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3:
a(n) = a(n-1) + a(n-2) + b(n-2) A294532
a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533
a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534
a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535
a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536
a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538
a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540
a(n) = a(n-1) + a(n-2) + b(n-1) A294541
a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542
a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543
a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544
a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545
a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546
a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547
a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548
a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558
a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294559
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294560
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294561
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1 A294564
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 3 A294565
Conjecture: for every sequence listed here, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
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LINKS
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EXAMPLE
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a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(0) + a(1) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 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] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294532 *)
Table[b[n], {n, 0, 10}]
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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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