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

0,1

Comments

The sequence cannot contain any triangular numbers.

Links

Table of n, a(n) for n=0..60.

Example

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.

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 *)

Crossrefs

Cf. A000217, A084833, A332941.

Sequence in context: A155085 A077665 A025062 * A175034 A030498 A355774

Adjacent sequences: A360016 A360017 A360018 * A360020 A360021 A360022

Keyword

nonn

Author

Ctibor O. Zizka, Jan 21 2023

Extensions

More terms from Jon E. Schoenfield, Jan 21 2023

Status

approved