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

%I A267140 #14 Aug 15 2021 12:12:04
%S A267140 1,35,12,72,180,336,45,792,1092,208,1836,2280,112,315,3900,4536,644,
%T A267140 5952,6732,7560,225,715,10332,627,12420,13536,924,1575,17172,840,396,
%U A267140 21240,22692,3267,2565,27336,28980,3392,32412,34200,1881,3795,637,1400,1785,45936,2240
%N A267140 Least m>0 for which m + n^2 is a square and m + triangular(n) is a triangular number (A000217).
%C A267140 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).
%H A267140 Chai Wah Wu, <a href="/A267140/b267140.txt">Table of n, a(n) for n = 0..10000</a>
%e A267140 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.
%t A267140 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 *)
%o A267140 (PARI) a(n) = {m = 1; while (! (issquare(m+n^2) && ispolygonal(m+n*(n+1)/2, 3)), m++); m;} \\ _Michel Marcus_, Jan 11 2016
%o A267140 (Python)
%o A267140 from math import sqrt
%o A267140 def A267140(n):
%o A267140     u,r,k,m = 2*n+1,4*n*(n+1)+1,0,2*n+1
%o A267140     while True:
%o A267140         if int(sqrt(8*m+r))**2 == 8*m+r:
%o A267140             return m
%o A267140         k += 2
%o A267140         m += u + k # _Chai Wah Wu_, Jan 13 2016
%Y A267140 Cf. A000217, A000290, A143698, A267077.
%K A267140 nonn
%O A267140 0,2
%A A267140 _Alex Ratushnyak_, Jan 10 2016