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A309828 Squares formed by concatenating k and 2*k+1.

%I A309828 #52 Mar 18 2023 16:27:32
%S A309828 25,49,1225,4489,112225,444889,11122225,44448889,816416329,1111222225,
%T A309828 1451229025,3832476649,4444488889,111112222225,444444888889,
%U A309828 10185602037121,11111122222225,44444448888889,46355849271169,997230019944601,1111111222222225,1231148024622961
%N A309828 Squares formed by concatenating k and 2*k+1.
%C A309828 The sequence is infinite. The squares of the form 66...67^2 = 4..48..89 are terms.
%C A309828 Another infinite family is the squares 33...35^2 = 1...122...25. - _Robert Israel_, Aug 20 2019
%D A309828 Ion Cucurezeanu, Perfect squares and cubes of integers, Ed. Gil, Zalău, (2007), ch. 4, p. 25, pr. 211, 212 (in Romanian).
%H A309828 Chai Wah Wu, <a href="/A309828/b309828.txt">Table of n, a(n) for n = 1..286</a>
%e A309828 5^2 = 25 = 2_(2 * 2 + 1);
%e A309828 7^2 = 49 = 4_(2 * 4 + 1);
%e A309828 35^2 = 1225 = 12_(2 * 12 + 1);
%e A309828 61907^2 = 3832476649 = 38324_(2 * 38324 + 1).
%p A309828 F:= proc(m) local x,X,A;
%p A309828   X:= [numtheory:-rootsunity(2,10^m+2)];
%p A309828   A:= map(x -> (x^2-1)/(10^m+2), X);
%p A309828   A:= sort(select(x -> 2*x+1>=10^(m-1) and 2*x+1<10^m, A));
%p A309828   op(map(x -> x*10^m+2*x+1, A))
%p A309828 end proc:
%p A309828 subsop(1=NULL, [seq(F(m),m=1..10)]); # _Robert Israel_, Aug 20 2019
%t A309828 Select[Array[FromDigits@ Flatten@ IntegerDigits[{#, 2 # + 1}] &, 10^5],
%t A309828 IntegerQ@ Sqrt@ # &] (* _Michael De Vlieger_, Aug 19 2019 *)
%o A309828 (Magma) [a:n in [1..30000000]|IsSquare(a) where a is 10^(#Intseq(2*n+1))*n+2*n+1];
%o A309828 (Python)
%o A309828 def Test(n):
%o A309828     s = str(n)
%o A309828     ps, ss = s[0:len(s)//2], s[len(s)//2:len(s)]
%o A309828     return int(ss) == 2*int(ps)+1 and s[len(s)//2] != "0"
%o A309828 n, a = 1, 4
%o A309828 while n < 23:
%o A309828     if Test(a*a):
%o A309828         print(n,a*a)
%o A309828         n = n+1
%o A309828     a = a+1 # _A.H.M. Smeets_, Aug 19 2019
%o A309828 (Python)
%o A309828 from itertools import count, islice
%o A309828 from sympy.ntheory.primetest import is_square
%o A309828 def A309828_gen(): # generator of terms
%o A309828     return filter(is_square,(int(str(k)+str((k<<1)+1)) for k in count(1)))
%o A309828 A309828_list = list(islice(A309828_gen(),20)) # _Chai Wah Wu_, Feb 20 2023
%Y A309828 Cf. A000290, A030466, A054215, A109344, A181719, A309808, A309809.
%K A309828 nonn,base
%O A309828 1,1
%A A309828 _Marius A. Burtea_, Aug 18 2019