|
%I #18 Oct 22 2021 23:57:10
%S 10,21,26519722651971,33388573338856,69954026995401,80863378086336
%N Triangular numbers obtained as the concatenation of n+1 and n.
%C There are only six terms less than 10^20.
%e 26519722651971 is the concatenation of 2651972 and 2651971 and a triangular number, because 26519722651971 = 7282818*7282819/2.
%t TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n+1], IntegerDigits[n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* _T. D. Noe_, Jun 18 2013 *)
%o (PARI)
%o concatint(a,b)=eval(concat(Str(a),Str(b)))
%o istriang(x)=issquare(8*x+1)
%o {for(n=1,10^7,a=concatint(n+1,n);if(istriang(a),print(a)))}
%o (Python)
%o from math import isqrt
%o def istri(n): t = 8*n+1; return isqrt(t)**2 == t
%o def afind(klimit, kstart=0):
%o strk = "0"
%o for k in range(kstart, klimit+1):
%o strkp1 = str(k+1)
%o t = int(strkp1 + strk)
%o if istri(t):
%o print(t, end=", ")
%o strk = strkp1
%o afind(81*10**5) # _Michael S. Branicky_, Oct 21 2021
%o (Python) # alternate version
%o def isconcat(n):
%o if n < 10: return False
%o s = str(n)
%o mid = (len(s)+1)//2
%o lft, rgt = int(s[:mid]), int(s[mid:])
%o return lft - 1 == rgt
%o def afind(tlimit, tstart=0):
%o for t in range(tstart, tlimit+1):
%o trit = t*(t+1)//2
%o if isconcat(trit):
%o print(trit, end=", ")
%o afind(13*10**6) # _Michael S. Branicky_, Oct 21 2021
%Y Cf. A003098, A068899, A226742, A226772, A226788.
%K nonn,base,more
%O 1,1
%A _Antonio Roldán_, Jun 18 2013
| 