A298111 - OEIS

%I #4 Feb 09 2018 11:10:24

%S 3,4,5,6,7,8,9,11,12,14,15,17,18,20,21,23,24,25,26,28,30,31,32,33,35,

%T 37,38,39,40,42,44,45,46,47,49,51,52,53,54,56,58,59,61,62,64,65,66,67,

%U 69,70,71,73,75,76,78,79,81,82,83,84,86,87,88,90,92,93

%N Solution b( ) 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.

%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A298000.  See A297830 for a guide to related sequences.

%C Conjecture:  1/5 < 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[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

%t a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 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}]; (* A298000 *)

%t v = Table[b[n], {n, 0, k}]; (* A298111 *)

%t Take[u, 50]

%t Take[v, 50]

%Y Cf. A297830, A298000.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Feb 09 2018