A359246 Lexicographically earliest sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.
2, 2, 5, 2, 2, 5, 2, 2, 5, 2, 2, 20, 2, 7, 2, 7, 2, 14, 9, 7, 2, 7, 2, 29, 2, 7, 2, 7, 2, 7, 2, 41, 9, 9, 16, 22, 2, 23, 7, 2, 7, 2, 7, 2, 7, 22, 9, 2, 7, 2, 7, 43, 9, 29, 2, 41, 9, 7, 2, 9, 5, 2, 7, 2, 22, 9, 9, 9, 25, 9, 29, 2, 7, 2, 7, 2, 32, 43, 65, 5, 2, 2
Offset: 0
Keywords
Examples
a(0) = 2 by the definition of the sequence. The next number >= 2 is 2; {2, 2 + 2} are not triangular numbers, thus a(1) = 2. Then we try 2; but 2 + 2 + 2 is a triangular number. We cannot try 3, which is a triangular number, so we try 4; but 4 + 2 is a triangular number, so we try 5; {5, 5 + 2, 5 + 2 + 2} are not triangular numbers, thus a(2) = 5.
Programs
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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 2 while ormap(q, [k+s(i, n-1)$i=0..n]) do od; k end: a(-1):=-1: seq(a(n), n=0..81); # Alois P. Heinz, Jan 21 2023 -
Mathematica
triQ[n_] := IntegerQ @ Sqrt[8*n + 1]; a[0] = 2; a[n_] := a[n] = Module[{k = 2, t = Accumulate @ Table[a[i], {i, n - 1, 0, -1}]}, While[triQ[k] || AnyTrue[t + k, triQ], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Jan 21 2023 *)
Extensions
More terms from Amiram Eldar, Jan 21 2023
Comments