A294368 - OEIS

%I #14 Nov 01 2017 23:05:01

%S 1,3,11,23,45,81,141,239,399,660,1083,1769,2880,4679,7591,12304,19931,

%T 32273,52244,84559,136848,221454,358351,579856,938260,1518171,2456488,

%U 3974718,6431267,10406048,16837380,27243495,44080944,71324510,115405527,186730112

%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%C The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

%C Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences.

%H Robert Israel, <a href="/A294368/b294368.txt">Table of n, a(n) for n = 0..4775</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) = 3, b(0) = 2, b(1) = 4, so that

%e a(2)  = a(1) + a(0) + b(1) + 3 = 11;

%e b(2) is the first positive integer not already seen, namely 5.

%e Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)

%p A[0]:= 1: B[0]:= 2:

%p A[1]:= 3: B[1]:= 4:

%p Av:= {$5..200}:

%p for n from 2 to 100 do

%p   A[n]:= A[n-1]+A[n-2]+B[n-1]+n+1;

%p   B[n]:= min(Av minus {A[n]});

%p   Av:= Av minus {A[n],B[n]};

%p od:

%p seq(A[i],i=0..100); # _Robert Israel_, Oct 29 2017

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

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

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

%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, 40}]  (* A294368 *)

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

%Y Cf. A001622 (golden ratio), A293765.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Oct 29 2017

%E Example clarified by _Robert Israel_, Oct 29 2017