A080824 -id:A080824 - cp's OEIS frontend

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

Ralf Stephan, Apr 06 2003

nonn

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.

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.

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

Partial maxima have index in A068194.

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

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 *)

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)))

A072522 Smallest number a(n)>a(n-1) such that T(a(n-1))+T(a(n))=T(m) for some m, a(1)=3; T(i) are the triangular numbers.

Original entry on oeis.org

3, 5, 6, 9, 13, 44, 58, 854, 896, 1194, 1253, 1469, 1628, 2170, 2403, 3205, 3366, 3395, 4370, 4454, 4676, 6234, 6545, 6873, 9165, 9624, 12833, 17110, 17450, 18524, 24012, 25025, 25652, 28242, 37657, 356750, 400894, 420938, 561250, 1347002

Offset: 1

Amarnath Murthy, Jul 31 2002

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..2401

Cf. A072521, A082183. Values of m are in A080824.

p=3; print1("3, "); for(n=2, 50, t=p*(p+1)/2; for(k=p, 10^9, u=t+k*(k+1)/2; v=floor(sqrt(2*u)); if(v*(v+1)/2==u, print1(k", "); p=k; break)))

Edited by Ralf Stephan, Mar 27 2003

cp's OEIS Frontend

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

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