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Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
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%I #4 Nov 18 2017 09:05:25

%S 1,2,9,23,45,76,117,170,236,316,411,522,650,796,961,1146,1352,1580,

%T 1831,2106,2407,2735,3091,3476,3891,4337,4815,5326,5871,6451,7067,

%U 7720,8411,9141,9911,10722,11575,12471,13411,14396,15427,16506,17634,18812,20041

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

%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.

%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.

%e a(0) = 1, a(1) = 2, b(0) = 3

%e b(1) = 4 (least "new number")

%e a(2) = 2*a(1) - a(0) + b(1) + 2 = 9

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)

%e m

%t ex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 2; b[0] = 3;

%t a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 2;

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 18}] (* A294870 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A294860.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 16 2017