A268132 If a(n) is a square (and n > 1), then a(n+1) = a(n) + a(n-1), else a(n+1) is the smallest positive integer not occurring earlier.

1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 11, 12, 13, 14, 15, 16, 31, 18, 19, 20, 21, 22, 23, 24, 25, 49, 74, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 71, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 127, 65, 66, 67, 68, 69, 70, 72, 73, 75, 76, 77, 78, 79, 80, 81, 161

Offset: 1

Author

M. F. Hasler, Jan 26 2016

Keywords

• nonn

Comments

A variant of the sequence A267758 where the relation has to hold for prime numbers rather than for squares.

Conjectured to be a permutation of the positive integers (which could be enforced by definition). In case there would occur a duplicate, it must be of the form a(n+1) = a(n) + a(n-1) and equal to an earlier term a(m+1) of the same form, where furthermore the predecessor a(m-1) also is of that form, since otherwise a(m+1) would be smaller than this a(n+1). This seems extremely unlikely to happen, and maybe provably impossible.

Examples

			a(26) = 25 is a square, thus followed by a(26) + a(25) = 25 + 24 = 49 which is again a square, thus followed by 49 + 25 = 74. Where is the next occurrence of two subsequent squares?
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000

Programs

PARI

a(n,show=0,is=x->issquare(x),a=[1],L=0,U=[])={while(#aL+1,U=setunion(U,[a[#a]]),L++;while(#U&&U[1]<=L+1,U=U[^1];L++));a=concat(a,if(!is(a[#a])||#a<2,L+1,a[#a]+a[#a-1])));if(type(show)=="t_VEC",a,a[#a])}