A239071 Numbers k such that k+x+y is a triangular number (A000217), where x and y are the two triangular numbers nearest to k.

0, 2, 6, 11, 19, 39, 53, 84, 104, 122, 146, 195, 225, 285, 321, 352, 392, 434, 470, 516, 605, 657, 757, 815, 864, 926, 990, 1044, 1112, 1241, 1315, 1455, 1535, 1602, 1686, 1844, 1934, 2103, 2199, 2279, 2379, 2481, 2566, 2672, 2870, 2982, 3191, 3309, 3407, 3529

Offset: 1

Author

Alex Ratushnyak, Mar 10 2014

Keywords

• nonn

Comments

If k is a triangular number then y=k.

The sequence of terms that are triangular numbers begins: 0, 6, 990, 189420, 36709596, 7120958130, 1381422007290, 267988648725336, 51988415041636920, 10085484510081574110.

Those are the triangular numbers with indices from A011916. - Ivan Neretin, May 31 2015

Examples

			The two triangular numbers nearest to 11 are 10 and 15. Because 10+11+15=36 is a triangular number, 11 is in the sequence.
		

Crossrefs

Cf. A000217, A238489, A238599.

Programs

Mathematica

Module[{nn=3600,trnos},trnos=Accumulate[Range[100]];Join[{0},Select[ Range[ nn],OddQ[Sqrt[8(Total[Nearest[trnos,#,2]]+#) +1]]&]]] (* Harvey P. Dale, Dec 19 2020 *)

PARI

isok(k) = {my(x = k-1); while (! ispolygonal(x, 3), x--); my(y = k); while (! ispolygonal(y, 3), y++); ispolygonal(k+x+y, 3);} \\ Michel Marcus, May 31 2015

Python

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
def isTriang(x):
    x+=x
    r = isqrt(x)
    return r*(r+1)==x
print('0', end=', ')
for n in range(777):
    tn = n*(n+1)//2
    tn1 = (n+1)*(n+2)//2
    for t in range(tn+1, tn1+1):
        if isTriang(tn+t+tn1): print(str(t), end=',')