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A294864 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 3
1, 2, 6, 9, 15, 21, 29, 38, 48, 59, 71, 84, 99, 114, 131, 148, 167, 187, 208, 230, 253, 277, 302, 328, 356, 384, 414, 444, 476, 508, 542, 576, 613, 649, 688, 726, 767, 807, 850, 892, 937, 982, 1029, 1076, 1125, 1174, 1225, 1276, 1329, 1382, 1437, 1493, 1550 (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) = a(0) + b(0) + 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + b[n - 2] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A294864 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A294860.
Sequence in context: A260699 A084265 A084140 * A103139 A181025 A345051
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 16 2017
STATUS
approved

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Last modified May 20 11:05 EDT 2024. Contains 372712 sequences. (Running on oeis4.)