cp's OEIS Frontend

A048612 Find smallest pair (x,y) such that x^2-y^2 = 11...1 (n times) = (10^n-1)/9; sequence gives value of y.

%I A048612 #24 Apr 05 2021 13:33:20
%S A048612 0,5,17,45,115,67,2205,2933,166667,44445,245795,6667,132683733,
%T A048612 4444445,2012917,23767083,2680575317,666667,555555555555555555,
%U A048612 83053525,3263104267,12488376483,5555555555555555555555,66666667,2952525627555
%N A048612 Find smallest pair (x,y) such that x^2-y^2 = 11...1 (n times) = (10^n-1)/9; sequence gives value of y.
%C A048612 Least solutions for 'Difference between two squares is a repunit of length n'.
%D A048612 David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, p. 119. ISBN 0-14-026149-4.
%H A048612 H. Havermann, <a href="http://chesswanks.com/pxp/RSD.html">Repunit Square Differences (gives many more terms)</a>
%F A048612 a(n) = (A033677((10^n-1)/9)-A033676((10^n-1)/9))/2. - _Chai Wah Wu_, Apr 05 2021
%e A048612 For n=2, 6^2 - 5^2 = 11.
%t A048612 s = Flatten[Table[r = (10^i - 1)/9; d = Divisors[r]; p = d[[Length[d]/2]]; Solve[{x - y == p, x + y == r/p}, {y, x}], {i, 2, 56}]]; Prepend[Cases[s, Rule[y, n_] -> n], 0]
%t A048612 Join[{0},Table[y/.Solve[{x>0,y>0,x^2-y^2==FromDigits[PadRight[{},n,1]]},{x,y},Integers][[1]],{n,2,30}]](* _Harvey P. Dale_, Jun 12 2018 *)
%o A048612 (Python)
%o A048612 from sympy import divisors
%o A048612 def A048612(n):
%o A048612     d = divisors((10**n-1)//9)
%o A048612     l = len(d)
%o A048612     return (d[l//2]-d[(l-1)//2])//2 # _Chai Wah Wu_, Apr 05 2021
%Y A048612 Cf. A048611, A000042, A002275, A033676, A033677.
%K A048612 nonn,nice
%O A048612 1,2
%A A048612 _Felice Russo_
%E A048612 Corrected and extended by _Patrick De Geest_, Jun 15 1999
%E A048612 More terms from _Hans Havermann_, Jul 02 2000
%E A048612 Offset corrected by _Chai Wah Wu_, Apr 05 2021