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A294548
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 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, 8, 17, 34, 62, 110, 188, 316, 524, 862, 1410, 2298, 3736, 6065, 9834, 15934, 25805, 41778, 67624, 109445, 177114, 286606, 463769, 750426, 1214248, 1964729, 3179034, 5143822, 8322917, 13466803, 21789786, 35256657, 57046513, 92303242, 149349829
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) + 1 = 8.
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 18, ...).
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] + 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}] (* A294548 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A357576 A034972 A294537 * A061150 A160189 A281470
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 04 2017
STATUS
approved