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 A296284 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 33

%I #4 Dec 13 2017 18:40:31

%S 1,2,9,23,52,105,199,360,639,1098,1857,3098,5123,8416,13763,22434,

%T 36485,59242,96087,155728,252255,408487,661292,1070377,1732317,

%U 2803394,4536465,7340669,11878002,19219599,31098591,50319244,81418955,131739387,213159600

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-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. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.

%H Clark Kimberling, <a href="/A296284/b296284.txt">Table of n, a(n) for n = 0..1000</a>

%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, b(1) = 4, b(2) = 5

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

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)

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

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

%t j = 1; While[j < 10, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t Table[a[n], {n, 0, k}]; (* A296284 *)

%t Table[b[n], {n, 0, 20}] (* complement *)

%Y Cf. A001622, A296245.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 13 2017

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