A267077 -id:A267077 - cp's OEIS frontend

A267140 Least m>0 for which m + n^2 is a square and m + triangular(n) is a triangular number (A000217).

1, 35, 12, 72, 180, 336, 45, 792, 1092, 208, 1836, 2280, 112, 315, 3900, 4536, 644, 5952, 6732, 7560, 225, 715, 10332, 627, 12420, 13536, 924, 1575, 17172, 840, 396, 21240, 22692, 3267, 2565, 27336, 28980, 3392, 32412, 34200, 1881, 3795, 637, 1400, 1785, 45936, 2240

Offset: 0

Alex Ratushnyak, Jan 10 2016

nonn

For n>1, a(n) <= b(n), where b(n) = 24*n^2 - 60*n + 36 = A143698(n-1), because b(n) + n^2 = (5*n-6)^2, and b(n) + n*(n+1)/2 = (7*n-9)*(7*n-8)/2 = triangular(7*n-9).

			12 + 2^2 = 16 is a square, and 12 + 2*3/2 = 15 is a triangular number, and 12 is the least such integer, so a(2)=12.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A143698, A267077.

lmst[n_]:=Module[{m=1,n2=n^2,nt=(n(n+1))/2},While[ !IntegerQ[Sqrt[ m+n2]] || !OddQ[Sqrt[1+8(m+nt)]],m++];m]; Join[{1},Array[lmst,50]] (* Harvey P. Dale, Aug 15 2021 *)

a(n) = {m = 1; while (! (issquare(m+n^2) && ispolygonal(m+n*(n+1)/2, 3)), m++); m;} \\ Michel Marcus, Jan 11 2016

from math import sqrt
def A267140(n):
    u,r,k,m = 2*n+1,4*n*(n+1)+1,0,2*n+1
    while True:
        if int(sqrt(8*m+r))**2 == 8*m+r:
            return m
        k += 2
        m += u + k # Chai Wah Wu, Jan 13 2016

cp's OEIS Frontend

A267140 Least m>0 for which m + n^2 is a square and m + triangular(n) is a triangular number (A000217).

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