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 A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)).  See Comments. 8
 1, 2, 10, 13, 16, 19, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 63, 68, 72, 74, 77, 80, 85, 89, 91, 94, 97, 102, 106, 108, 111, 114, 119, 123, 125, 128, 131, 136, 140, 142, 147, 149, 154, 156, 159, 162, 167, 169, 172, 177, 181, 183, 188, 190, 195, 197 (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. Conjectures:  a(n) - (2 +sqrt(2))*n < 4 for n >= 1.  Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, where (b(n)) is the increasing sequence of positive integers not in (a(n)): *** a(n) = a(1)*b(n) - a(0)*b(n-1) + n     (a(n)) = A297999; (b(n)) = A298110 a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n   (a(n)) = A298000; (b(n)) = A298111 a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n   (a(n)) = A298001; (b(n)) = A298112 a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n   (a(n)) = A298002; (b(n)) = A298113 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) = 10. Complement: (b(n)) = (3,4,5,6,8,9,11,12,14,15,17,18,20,...) MATHEMATICA a = 1; a = 2; b = 3; b = 4; b = 5; a[n_] := a[n] = a*b[n] - a*b[n - 1] + 2 n; j = 1; While[j < 100, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k Table[a[n], {n, 0, k}]  (* A298000 *) CROSSREFS Cf. A297826, A297836, A297837. Sequence in context: A177856 A296220 A297835 * A058216 A297998 A037386 Adjacent sequences:  A297997 A297998 A297999 * A298001 A298002 A298003 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 04 2018 STATUS approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)