A293406 -id:A293406 - cp's OEIS frontend

A293358 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 16, 30, 53, 92, 155, 258, 425, 696, 1135, 1846, 2998, 4862, 7879, 12761, 20661, 33444, 54128, 87596, 141749, 229371, 371147, 600546, 971722, 1572299, 2544053, 4116385, 6660472, 10776892, 17437400, 28214329, 45651767, 73866135, 119517942

Offset: 0

Clark Kimberling, Oct 29 2017

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
A293358:  a(n) = a(n-1) + a(n-2) + b(n-1)
A293406:  a(n) = a(n-1) + a(n-2) + b(n-1) + 1
A293765:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2
A293766:  a(n) = a(n-1) + a(n-2) + b(n-1) + 3
A293767:  a(n) = a(n-1) + a(n-2) + b(n-1) - 1
A294365:  a(n) = a(n-1) + a(n-2) + b(n-1) + n
A294366:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2n
A294367:  a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1
A294368:  a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) = 8;
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...)

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

Cf. A001622 (golden ratio), A293076.

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293358 *)
Table[b[n], {n, 0, 10}]

cp's OEIS Frontend

A293358 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

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