A229909 Triangular numbers t such that the following are three triangular numbers: x, y, x+y, where x and y are distances from t to the two nearest squares.

1, 2080, 8038045

Offset: 1

Author

Alex Ratushnyak, Dec 19 2013

Keywords

• nonn
• bref
• hard
• more

Comments

No more terms through 10^34. - Jon E. Schoenfield, Feb 09 2014

Examples

			2080 is in the sequence because the following three are triangular numbers:
2080-2025 = 55,
2116-2080 = 36,
55 + 36 = 91.
2025 = 45^2 and 2116 = 46^2 are the nearest to 2080 squares.
		

Crossrefs

Cf. A000217, A000290, A234143.

Programs

Mathematica

ttnQ[n_]:=Module[{s=Sqrt[n],x,y},x=If[IntegerQ[s],n-(s-1)^2,n- Floor[ s]^2];y=If[IntegerQ[s],(s+1)^2-n,Ceiling[s]^2-n];AllTrue[ {Sqrt[ 8x+1],Sqrt[8y+1],Sqrt[8(x+y)+1]},OddQ]]; Join[{1},Select[Accumulate[ Range[10000]],ttnQ]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 30 2015 *)

Python

import math
def isTriangular(a):
    a+=a
    sr = int(math.sqrt(a))
    return (a==sr*(sr+1))
for n in range(1, 1000000000):
    tn = int(n*(n+1)/2)  # = x+y = distance between squares
    if tn&1:
        k = tn>>1
        k*= k       # square below t
        a = int(math.sqrt(k*2))
        t = a*(a+1)/2
        if t <= k:
            a+=1
            t+=a
        ktn = k+tn   # square above t
        while t <= ktn:  # check if x and y are triangular:
            if isTriangular(t-k) and isTriangular(ktn-t):
                print(int(t))
            a+=1
            t+=a
    if (n&0xfffff)==0: print('.', end='')