A294551 -id:A294551 - cp's OEIS frontend

A294532 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.

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

Offset: 0

Clark Kimberling, Nov 03 2017

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).

			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, ...)

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A293076, A294413.

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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A294532 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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