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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. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
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, ...)
MATHEMATICA
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}]
CROSSREFS
Sequence in context: A006503 A248851 A023554 * A299336 A222629 A070880
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 31 2017
EXTENSIONS
Definition corrected by Georg Fischer, Sep 27 2020
STATUS
approved

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Last modified June 22 15:32 EDT 2024. Contains 373587 sequences. (Running on oeis4.)