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A226789 Triangular numbers obtained as the concatenation of n+1 and n. 3
10, 21, 26519722651971, 33388573338856, 69954026995401, 80863378086336 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
There are only six terms less than 10^20.
LINKS
EXAMPLE
26519722651971 is the concatenation of 2651972 and 2651971 and a triangular number, because 26519722651971 = 7282818*7282819/2.
MATHEMATICA
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 *)
PROG
(PARI)
concatint(a, b)=eval(concat(Str(a), Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1, 10^7, a=concatint(n+1, n); if(istriang(a), print(a)))}
(Python)
from math import isqrt
def istri(n): t = 8*n+1; return isqrt(t)**2 == t
def afind(klimit, kstart=0):
strk = "0"
for k in range(kstart, klimit+1):
strkp1 = str(k+1)
t = int(strkp1 + strk)
if istri(t):
print(t, end=", ")
strk = strkp1
afind(81*10**5) # Michael S. Branicky, Oct 21 2021
(Python) # alternate version
def isconcat(n):
if n < 10: return False
s = str(n)
mid = (len(s)+1)//2
lft, rgt = int(s[:mid]), int(s[mid:])
return lft - 1 == rgt
def afind(tlimit, tstart=0):
for t in range(tstart, tlimit+1):
trit = t*(t+1)//2
if isconcat(trit):
print(trit, end=", ")
afind(13*10**6) # Michael S. Branicky, Oct 21 2021
CROSSREFS
Sequence in context: A109326 A369120 A080454 * A110367 A104865 A063555
KEYWORD
nonn,base,more
AUTHOR
Antonio Roldán, Jun 18 2013
STATUS
approved

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Last modified June 16 08:45 EDT 2024. Contains 373424 sequences. (Running on oeis4.)