cp's OEIS Frontend

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

%I A360019 #35 Jan 22 2023 17:48:05
%S A360019 2,5,7,11,12,14,16,17,18,19,20,22,25,26,30,31,34,35,37,42,46,49,52,54,
%T A360019 59,63,64,68,72,73,77,80,81,84,85,87,92,93,94,98,100,101,108,113,115,
%U A360019 117,118,121,122,123,125,129,130,132,133,134,141,142,143,146,149
%N A360019 Lexicographically earliest increasing sequence of positive numbers in which no nonempty subsequence of consecutive terms sums to a triangular number.
%C A360019 The sequence cannot contain any triangular numbers.
%e A360019 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.
%p A360019 q:= proc(n) option remember; issqr(8*n+1) end:
%p A360019 s:= proc(i, j) option remember; `if`(i>j, 0, a(j)+s(i, j-1)) end:
%p A360019 a:= proc(n) option remember; local k; for k from 1+a(n-1) while
%p A360019       ormap(q, [k+s(i, n-1)$i=0..n]) do od; k
%p A360019     end: a(-1):=-1:
%p A360019 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 21 2023
%t A360019 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 *)
%Y A360019 Cf. A000217, A084833, A332941.
%K A360019 nonn
%O A360019 0,1
%A A360019 _Ctibor O. Zizka_, Jan 21 2023
%E A360019 More terms from _Jon E. Schoenfield_, Jan 21 2023