A229134 Square numbers that are the sum of two non-consecutive triangular numbers.

16, 36, 81, 121, 196, 225, 256, 289, 361, 441, 484, 529, 576, 625, 676, 841, 900, 961, 1024, 1156, 1225, 1296, 1444, 1521, 1600, 1681, 1849, 1936, 2116, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3136, 3249, 3481, 3721, 4096, 4356, 4624, 4761, 4900, 5041, 5184

Offset

1,1

Comments

It is well known that tri(n) + tri(n+1) is always a square.

Sequence includes all terms of A001110 > 1. A number m is a term if and only if there exists k > 1 such that m >= tri(k) and 4m - k^2 + 1 is a perfect square. - Chai Wah Wu, Feb 25 2016

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

16 = 15+1, 81 = 78+3 = 66+15.

Mathematica

nn = 10000; mx = Floor[Sqrt[1 + 8 nn]/2]; tri = Table[n (n + 1)/2, {n, mx}]; t = {}; Do[s = tri[[i]] + tri[[j]]; If[s <= nn && IntegerQ[Sqrt[s]], AppendTo[t, s]], {i, mx - 2}, {j, i + 2, mx}]; t = Union[t] (* T. D. Noe, Sep 17 2013 *)

Prog

(JavaScript)
function isSquare(n) {
if (Math.sqrt(n)==Math.floor(Math.sqrt(n))) return true; else return false;
}
a=new Array();
ac=0;
for (i=0; i<100; i++)
for (j=i+2; j<100; j++)
if (isSquare(i*(i+1)/2+j*(j+1)/2)) a[ac++]=i*(i+1)/2+j*(j+1)/2;
a.sort(function(a, b) {return a-b; });
a=trimArray(a);
function trimArray(arr) {
var j, c=new Array(), i;
for (j=0; j<arr.length; j++) c[j]=arr[j];
c.sort(function(a, b) {return a-b; });
i=-1;
while(i++<c.length-1)
if (c[i]==c[i+1]) c.splice(i--, 1);
return c;
}
document.write(a+", ");
(Python)
from gmpy2 import is_square
A229134_list = []
for i in range(10**3):
    m, m2, j, k = 2, 4, 4*i**2+1, 2*i**2
    while k >= m2 + m:
        if is_square(j-m2):
            A229134_list.append(i**2)
            break
        m2 += 2*m+1
        m += 1 # Chai Wah Wu, Feb 25 2016

Crossrefs

Cf. A000217, A001110.

Sequence in context: A050775 A022040 A074985 * A069262 A076956 A075369

Adjacent sequences: A229131 A229132 A229133 * A229135 A229136 A229137

Keyword

nonn

Author

Jon Perry, Sep 15 2013

Extensions

Corrected and extended by T. D. Noe, Sep 17 2013

a(2) = 36 reinserted by Chai Wah Wu, Feb 27 2016

Status

approved