A294413 -id:A294413 - 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}]

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.

Original entry on oeis.org

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

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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