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A294866 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 7, 17, 33, 57, 90, 133, 187, 253, 332, 425, 533, 657, 799, 960, 1141, 1343, 1567, 1814, 2085, 2381, 2703, 3052, 3429, 3835, 4271, 4738, 5237, 5770, 6338, 6942, 7583, 8262, 8980, 9738, 10537, 11378, 12262, 13190, 14163, 15182, 16248, 17362, 18525, 19738 (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 A294860 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2*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, 18}] (* A294866 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A294860.
Sequence in context: A086513 A166381 A083723 * A045947 A321123 A145066
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 16 2017
EXTENSIONS
Definition corrected by Georg Fischer, Sep 27 2020
STATUS
approved

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Last modified June 14 13:49 EDT 2024. Contains 373400 sequences. (Running on oeis4.)