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

1,2

Comments

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

Links

Table of n, a(n) for n=1..3.

Example

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.

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 *)

Prog

(Python)
import math
def isTriangular(a):
    a+=a
    sr = int(math.sqrt(a))
    return (a==sr*(sr+1))
for n in range(1, 1000000000L):
    tn = 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 t
            a+=1
            t+=a
    if (n&0xfffff)==0: print '.',

Crossrefs

Cf. A000217, A000290, A234143.

Sequence in context: A071235 A259414 A233088 * A323813 A323472 A224438

Adjacent sequences: A229906 A229907 A229908 * A229910 A229911 A229912

Keyword

nonn,bref,hard,more

Author

Alex Ratushnyak, Dec 19 2013

Status

approved