cp's OEIS Frontend

A294170 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294170 #7 Feb 11 2018 10:54:47
%S A294170 1,2,12,26,53,97,171,292,490,813,1337,2187,3564,5794,9404,15247,24703,
%T A294170 40005,64766,104832,169662,274561,444294,718929,1163300,1882309,
%U A294170 3045692,4928087,7973868,12902047,20876010,33778155,54654266,88432525,143086898,231519533
%N A294170 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294170 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.
%H A294170 Clark Kimberling, <a href="/A294170/b294170.txt">Table of n, a(n) for n = 0..2000</a>
%H A294170 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A294170 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A294170 a(2) = a(0) + a(1) + b(2) + 4 = 12
%e A294170 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)
%t A294170 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A294170 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
%t A294170 j = 1; While[j < 16, k = a[j] - j - 1;
%t A294170  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A294170 Table[a[n], {n, 0, k}];  (* A294170 *)
%Y A294170 Cf. A296245, A296491, A296492.
%K A294170 nonn,easy
%O A294170 0,2
%A A294170 _Clark Kimberling_, Feb 10 2018