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A305329 Solution (a(n)) of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-2); see Comments. 3
1, 2, 6, 14, 27, 47, 75, 112, 159, 217, 287, 370, 468, 582, 713, 862, 1030, 1218, 1427, 1658, 1912, 2190, 2493, 2822, 3179, 3565, 3981, 4428, 4907, 5419, 5965, 6546, 7163, 7817, 8509, 9240, 10011, 10823, 11677, 12574, 13515, 14501, 15533, 16613, 17742, 18921 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Define sequences a(n) and b(n) recursively, starting with a(0) = 1, a(1) = 2:
b(n) = least new;
a(n) = 2*a(n-1) - a(n-2) + b(n-2),
where "least new" means the least positive integer not yet placed. It appears that a(n)/a(n-1) -> 1, that {a(n) - a(n-1), n >=1} is unbounded, and that the 3rd difference sequence of (a(n)) consists entirely of 1's and 2's.
LINKS
EXAMPLE
b(0) = least not in {a(0), a(1)} = 3;
a(2) = 2*a(1) - a(0) + b(0) must exceed = 2*2 -1 + 3 = 6, so that b(0) = 3, b(1) = 4, b(2) = 5, and a(2) = 6.
MATHEMATICA
a = {1, 2}; b = {3, 4, 5};
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
Do[AppendTo[a, 2 Last[a] - a[[-2]] + b[[-3]]];
AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {120}]; a
(* Peter J. C. Moses, May 30 2018 *)
CROSSREFS
Sequence in context: A101586 A178080 A121968 * A161212 A256058 A294867
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 02 2018
STATUS
approved

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)