A091569 a(1) = 1; for n > 1, a(n) is the smallest positive integer not already used such that a(n)*a(n-1) + 1 is a perfect square.

1, 3, 5, 7, 9, 11, 13, 15, 8, 6, 4, 2, 12, 10, 36, 34, 32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 52, 50, 48, 35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 64, 62, 60, 40, 38, 148, 146, 144, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83

Offset: 1

Author

David Wasserman, Mar 04 2004

Keywords

• easy
• nonn

Comments

Does this sequence contain every positive integer? We could get an equally interesting sequence by choosing a(1) to be any other positive integer.

A sequence with the same condition but without the requirement for a(n) to be distinct would end up repeating (1,3) or (2,4), depending on the initial term. - Ivan Neretin, May 26 2015

Examples

			10 is followed by 36 because 10*36+1 = 19^2 and 8 and 12 were already used.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A083203.

Programs

MATLAB

A = zeros(1, 100); A(1) = 1; used = zeros(1, 1000); used(1) = 1; for i = 2:100; found = 0; k = 0; while found == 0; k = k + 1; if used(k) == 0; s = sqrt(k*A(i - 1) + 1); if s == floor(s); A(i) = k; used(k) = 1; found = 1; end; end; end; end; A

Maple

N:= 10^4: Used:= Vector(N,datatype=integer[4]):
a[1]:= 1: blocked:= false: Used[1]:= 1:
for n from 2 to 100 while not(blocked) do
    ndone:= false;
    if n = 2 then T:= [0]
      else T:= select(t -> t^2 mod a[n-1] = 1, [$0..a[n-1]-1])
      fi;
    for s from 0 while not (ndone) do
       for t in T while not (ndone) do
         x:= s * a[n-1] + t;
         if x <= 1 then next fi;
         y:= (x^2-1)/a[n-1];
         if y > N then blocked:= true; ndone:= true
         elif Used[y] = 0 then
            a[n]:= y;
            Used[y]:= 1;
            ndone:= true;
            print(n,y);
         fi
       od
    od
od:
seq(a[n],n=1..100); # Robert Israel, May 26 2015

Mathematica

a = {1}; Do[a = Join[a, Select[Complement[Range[(Max[a] + 1)*n], a], IntegerQ[Sqrt[#*a[[-1]] + 1]] &, 1]], {n, 2, 71}]; a (* Ivan Neretin, May 26 2015 *)