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A294479
Solution of the complementary equation a(n) = a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 3, b(0) = 2.
2
1, 3, 7, 11, 17, 24, 32, 41, 52, 63, 76, 89, 104, 120, 137, 155, 174, 194, 215, 238, 261, 286, 311, 338, 365, 394, 424, 455, 487, 520, 554, 589, 625, 662, 701, 740, 781, 822, 865, 908, 953, 998, 1045, 1092, 1142, 1191, 1243, 1294, 1348, 1401, 1457, 1512
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294476 for a guide to related sequences.
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(0) + b(1) + 2 = 7
Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 12, 13, 14, ...)
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 - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294479 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 01 2017
STATUS
approved