A296296 - OEIS

%I #4 Dec 14 2017 18:07:35

%S 2,3,15,36,79,155,288,513,889,1510,2529,4193,6914,11328,18494,30107,

%T 48921,79385,128702,208524,337706,546755,885033,1432409,2318114,

%U 3751248,6070142,9822227,15893265,25716449,41610734,67328268,108940186,176269708,285211220

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, 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="/A296296/b296296.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) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5

%e a(2) = a(0) + a(1) + 2*b(2) = 15

%e Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, ...)

%t a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;

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

%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}]; (* A296296 *)

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

%Y Cf. A001622, A296245.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Dec 14 2017