A082184 The a(n)-th triangular number is the sum of the n-th triangular number and the smallest triangular number possible.

3, 6, 10, 6, 8, 28, 13, 10, 13, 18, 21, 16, 15, 26, 136, 21, 23, 40, 21, 23, 28, 38, 27, 31, 28, 28, 61, 36, 38, 496, 53, 36, 43, 36, 61, 46, 41, 44, 106, 51, 53, 91, 45, 49, 58, 78, 66, 52, 54, 53, 112, 66, 55, 58, 78, 62, 73, 98, 101, 76, 67, 106, 166, 66, 83, 142, 71

Offset: 2

Author

Ralf Stephan, Apr 06 2003

Keywords

• nonn

Comments

a(n) is triangular if n+1 is triangular. Conjectures: partial maxima of sequence are at index i with value from A068195 and also a(i) - A082183(i) = 1, where i is in A068194.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..65536
• J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Crossrefs

Cf. A000217, A080824, index of second term is in A082183.

Partial maxima have index in A068194.

Programs

Maple

a:= proc(n) local h, j; h:= n*(n+1); for j from n+1 do
      if issqr(1+4*(j*(j+1)-h)) then return j fi od
    end:
seq(a(n), n=2..70);  # Alois P. Heinz, Jul 31 2019

Mathematica

a[n_] := Module[{h = n(n+1), j}, For[j = n+1, True, j++, If[IntegerQ[ Sqrt[1 + 4 (j(j+1) - h)]], Return[j]]]];
a /@ Range[2, 70] (* Jean-François Alcover, Jun 05 2020, after Maple *)

PARI

for(n=2, 100, t=n*(n+1)/2; for(k=1, 10^9, u=t+k*(k+1)/2; v=floor(sqrt(2*u)); if(v*(v+1)/2==u, print1(v", "); break)))