A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.

2, 5, 7, 11, 12, 14, 16, 17, 18, 19, 20, 22, 25, 26, 30, 31, 34, 35, 37, 42, 46, 49, 52, 54, 59, 63, 64, 68, 72, 73, 77, 80, 81, 84, 85, 87, 92, 93, 94, 98, 100, 101, 108, 113, 115, 117, 118, 121, 122, 123, 125, 129, 130, 132, 133, 134, 141, 142, 143, 146, 149

Offset: 0

Author

Ctibor O. Zizka, Jan 21 2023

Keywords

• nonn

Comments

The sequence cannot contain any triangular numbers.

Examples

			a(0) = 2 by the definition of the sequence. The next number > a(0) is 3, but it is a triangular number, so we try 4, but 2 + 4 = 6 is a triangular number. Then we try 5; {5, 2 + 5} are not triangular numbers, thus a(1) = 5. a(2) cannot be 6, so we try 7; {7, 5 + 7, 2 + 5 + 7} are not triangular numbers, thus a(2) = 7.
		

Crossrefs

Cf. A000217, A084833, A332941.

Programs

Maple

q:= proc(n) option remember; issqr(8*n+1) end:
s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
a:= proc(n) option remember; local k; for k from 1+a(n-1) while
      ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
    end: a(-1):=-1:
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 21 2023

Mathematica

triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = a[n - 1] + 1, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 61, 0] (* Amiram Eldar, Jan 21 2023 *)

Extensions

More terms from Jon E. Schoenfield, Jan 21 2023