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A296849 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(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 A296849 #13 Jun 12 2018 10:20:56
%S A296849 1,2,10,28,73,182,446,1085,2628,6354,15350,37069,89504,216094,521710,
%T A296849 1259533,3040796,7341146,17723110,42787389,103297912,249383238,
%U A296849 602064414,1453512093,3509088629,8471689381,20452467422,49376624257,119205715969,287788056229
%N A296849 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(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 A296849 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 1 + sqrt(2) = A014176. [corrected by _Clark Kimberling_, Jun 09 2018]
%H A296849 Clark Kimberling, <a href="/A296849/b296849.txt">Table of n, a(n) for n = 0..1000</a>
%H A296849 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 A296849 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A296849 a(2) = 2*a(1) + a(0) + b(2) = 10
%e A296849 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
%t A296849 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296849 a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
%t A296849 j = 1; While[j < 7, k = a[j] - j - 1;
%t A296849 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296849 u = Table[a[n], {n, 0, k}]; (* A296849 *)
%t A296849 Table[b[n], {n, 0, 20}] (* complement *)
%t A296849 Take[u, 30]
%Y A296849 Cf. A014176, A296245, A296850, A296851.
%K A296849 nonn,easy
%O A296849 0,2
%A A296849 _Clark Kimberling_, Jan 12 2018