A294422 -id:A294422 - cp's OEIS frontend

A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 10, 22, 43, 78, 136, 231, 387, 641, 1053, 1721, 2803, 4555, 7391, 11981, 19409, 31429, 50879, 82352, 133278, 215679, 349008, 564740, 913803, 1478600, 2392462, 3871123, 6263648, 10134836, 16398551, 26533456, 42932078, 69465607, 112397760, 181863444

Offset: 0

Clark Kimberling, Oct 31 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:
A294413:  a(n) = a(n-1) + a(n-2) - b(n-1) + 6
A294414:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2)
A294415:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1
A294416:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n
A294417:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n
A294418:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2)
A294419:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2)
A294420:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2)
A294421:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2)
A294422:  a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1
A294423:  a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n
A294424:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 1
A294425:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2
A294426:  a(n) = a(n-1) + 2*a(n-2) + b(n-1) + b(n-2) - 3
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) + b(0) = 6
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, ...)

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

Cf. A293076, A293765, A022940.

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 - 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}]  (* A294414 *)
Table[b[n], {n, 0, 10}]

Definition corrected by Georg Fischer, Sep 27 2020

cp's OEIS Frontend

A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

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