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 A297826 Solution (a(n)) of the near-complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)).  See Comments. 17
 1, 2, 7, 9, 11, 15, 18, 21, 22, 24, 28, 29, 33, 34, 40, 42, 43, 45, 51, 51, 53, 59, 59, 61, 63, 65, 69, 74, 76, 77, 79, 81, 83, 87, 90, 91, 93, 95, 97, 101, 104, 107, 110, 111, 113, 117, 118, 120, 122, 126, 131, 133, 136, 139, 140, 142, 146, 147, 153, 155 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The sequence (a(n)) generated by the equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 51.  If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures: (1) 0 <= a(k) - a(k-1) <= 6 for k>=1; (2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k-1) + d for infinitely many k. *** See A297830 for a guide to related sequences. LINKS Clark Kimberling, Table of n, a(n) for n = 0..10000 EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 7. Complement: (b(n)) = (3, 4, 5, 6, 8,10,12,13,14,16, ...) MATHEMATICA mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]); tbl = {}; a = 1; a = 2; b = 3; b = 4; a[n_] := a[n] = a*b[n - 1] - a*b[n - 2] + n; b[n_] := b[n] = mex[tbl = Join[{a[n], a[n - 1], b[n - 1]}, tbl], b[n - 1]]; Table[a[n], {n, 0, 300}]  (* A297826 *) Table[b[n], {n, 0, 300}]  (* A297997 *) (* Peter J. C. Moses, Jan 03 2017 *) CROSSREFS Cf. A297997, A297830. Sequence in context: A287359 A022424 A136498 * A288598 A277737 A082371 Adjacent sequences:  A297823 A297824 A297825 * A297827 A297828 A297829 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 04 2018 STATUS approved

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Last modified September 20 10:04 EDT 2020. Contains 337264 sequences. (Running on oeis4.)