|
%I #6 Sep 30 2020 05:03:14
%S 1,2,9,12,15,18,21,26,28,33,35,40,42,47,49,54,56,59,62,67,71,73,76,79,
%T 84,88,90,93,96,101,105,107,110,113,118,122,124,127,130,135,139,141,
%U 146,148,153,155,158,161,166,168,171,176,180,182,187,189,194,196
%N Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 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.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Conjecture: a(n) - (2 +sqrt(2))*n < 3 for n >= 1.
%C Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
%C ***
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n (a(n)) = A297826; (b(n)) = A297997
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n (a(n)) = A297830; (b(n)) = A298003
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n (a(n)) = A297836; (b(n)) = A298004
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n (a(n)) = A297837; (b(n)) = A298005
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1 (a(n)) = A297831; (b(n)) = A298006
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2 (a(n)) = A297832; (b(n)) = A298007
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3 (a(n)) = A297833; (b(n)) = A298108
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4 (a(n)) = A297834; (b(n)) = A298109
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1 (a(n)) = A297835;
%C a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2) (a(n)) = A297998;
%C ***
%C For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800.
%H Clark Kimberling, <a href="/A297830/b297830.txt">Table of n, a(n) for n = 0..10000</a>
%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9.
%e Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...)
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n;
%t j = 1; While[j < 100, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
%t Table[a[n], {n, 0, k}] (* A297830 *)
%Y Cf. A297826, A297836, A297837.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Feb 04 2018
|
