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

1,2

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

Links

Table of n, a(n) for n=1..50.

Example

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.

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 *)

Prog

(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=', ')
(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

Crossrefs

Cf. A000217, A238489, A238599.

Sequence in context: A048204 A259411 A058760 * A085573 A171516 A081691

Adjacent sequences: A239068 A239069 A239070 * A239072 A239073 A239074

Keyword

nonn

Author

Alex Ratushnyak, Mar 10 2014

Status

approved