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 A298113 Solution b( ) of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*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. 2
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 OFFSET 0,1 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A298002.  See A297830 for a guide to related sequences. Conjecture:  1/2 < a(n) - n*sqrt(2) < 3 for n >= 1. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. MATHEMATICA a = 1; a = 2; b = 3; b = 4; b = 5; a[n_] := a[n] = a*b[n] - a*b[n - 1] + 4 n; j = 1; While[j < 80000, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k u = Table[a[n], {n, 0, k}]; (* A298002 *) v = Table[b[n], {n, 0, k}]; (* A298113 *) Take[u, 50] Take[v, 50] CROSSREFS Cf. A297830, A298002. Sequence in context: A191838 A333214 A108473 * A026447 A194378 A283810 Adjacent sequences:  A298110 A298111 A298112 * A298114 A298115 A298116 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 09 2018 STATUS approved

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Last modified October 25 19:28 EDT 2020. Contains 338012 sequences. (Running on oeis4.)