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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

%I

%S 3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,20,21,22,23,25,26,27,28,30,31,

%T 32,33,35,36,37,38,40,41,42,43,45,46,47,48,50,51,52,53,55,56,57,58,59,

%U 60,62,63,64,66,67,68,69,71,72,73,74,76,77,78,79,80,81

%N 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.

%C 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.

%C Conjecture: 1/2 < a(n) - n*sqrt(2) < 3 for n >= 1.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%t a = 1; a = 2; b = 3; b = 4; b = 5;

%t a[n_] := a[n] = a*b[n] - a*b[n - 1] + 4 n;

%t j = 1; While[j < 80000, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k

%t u = Table[a[n], {n, 0, k}]; (* A298002 *)

%t v = Table[b[n], {n, 0, k}]; (* A298113 *)

%t Take[u, 50]

%t Take[v, 50]

%Y Cf. A297830, A298002.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Feb 09 2018

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Last modified November 25 19:43 EST 2020. Contains 338625 sequences. (Running on oeis4.)