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A294557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
1, 2, 11, 24, 49, 90, 159, 272, 457, 759, 1250, 2046, 3336, 5425, 8807, 14281, 23140, 37476, 60674, 98211, 158949, 257228, 416249, 673552, 1089879, 1763512, 2853475, 4617074, 7470639, 12087806, 19558541, 31646446, 51205089, 82851640, 134056837, 216908588 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + b(1) + b(0) + 1 = 11
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, ...)
MATHEMATICA
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 - 1] + b[n - 2] + 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}] (* A294557 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A297545 A256905 A294547 * A009189 A012213 A012251
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 15 2017
EXTENSIONS
Definition corrected by Georg Fischer, Sep 27 2020
STATUS
approved

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Last modified February 23 04:31 EST 2024. Contains 370267 sequences. (Running on oeis4.)