A234143 -id:A234143 - cp's OEIS frontend

A234141 Numbers k such that triangular(k) - m is a triangular number (A000217), where m is the largest square less than triangular(k).

1, 4, 5, 7, 10, 13, 16, 19, 21, 22, 25, 28, 40, 42, 59, 60, 64, 76, 85, 89, 93, 109, 110, 124, 127, 142, 144, 148, 161, 165, 178, 184, 195, 212, 229, 233, 246, 247, 263, 265, 268, 280, 297, 313, 314, 331, 346, 348, 349, 365, 373, 376, 382, 399, 416, 433, 445, 450

Offset: 1

Alex Ratushnyak, Dec 19 2013

nonn

The sequence of triangular(a(n)) begins: 1, 10, 15, 28, 55, 91, 136, 190, 231, 253, 325, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A234142, A234143.

Select[Range[450], IntegerQ@Sqrt[8 ((#^2 + #)/2 - (Ceiling@Sqrt[(#^2 + #)/2] - 1)^2) + 1] &] (* Ivan Neretin, May 29 2016 *)

A234142 Numbers k such that m - triangular(k) is a triangular number (A000217), where m is the least square above triangular(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 12, 15, 19, 20, 22, 25, 26, 29, 32, 33, 36, 40, 43, 47, 50, 52, 54, 57, 61, 64, 68, 70, 71, 73, 75, 78, 82, 85, 89, 90, 92, 96, 99, 103, 106, 110, 113, 114, 117, 120, 121, 122, 124, 127, 131, 134, 135, 141, 148, 152, 155, 172, 173, 188, 189, 196, 213

Offset: 1

Alex Ratushnyak, Dec 19 2013

nonn

The sequence of triangular(a(n)) begins: 1, 3, 6, 10, 15, 66, 78, 120, 190, 210, 253, 325, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A234141, A234143.

Select[Range[215], IntegerQ@Sqrt[8 ((Floor@Sqrt[(#^2 + #)/2] + 1)^2 - (#^2 + #)/2) + 1] &] (* Ivan Neretin, May 29 2016 *)

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.

Original entry on oeis.org

1, 2080, 8038045

Offset: 1

Alex Ratushnyak, Dec 19 2013

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

			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.

Cf. A000217, A000290, A234143.

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 *)

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='')

cp's OEIS Frontend

A234141 Numbers k such that triangular(k) - m is a triangular number (A000217), where m is the largest square less than triangular(k).

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A234142 Numbers k such that m - triangular(k) is a triangular number (A000217), where m is the least square above triangular(k).

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Mathematica

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.

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