cp's OEIS Frontend

A383734 Numbers k such that 2+k and 2*k are squares.

%I A383734 #33 May 28 2025 00:04:14
%S A383734 2,98,3362,114242,3880898,131836322,4478554082,152139002498,
%T A383734 5168247530882,175568277047522,5964153172084898,202605639573839042,
%U A383734 6882627592338442562,233806732499933208098,7942546277405390632802,269812766699283348307202,9165691521498228451812098
%N A383734 Numbers k such that 2+k and 2*k are squares.
%C A383734 The limit of a(n+1)/a(n) is 33.97056... = 17+12*sqrt(2) = (3+2*sqrt(2))^2 (see A156164).
%H A383734 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NSWNumber.html">NSW numbers</a>.
%H A383734 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A383734 a(n) = (1/2) * ((3+2*sqrt(2))^(2*n-1) + (3-2*sqrt(2))^(1-2*n)) - 1.
%F A383734 a(n) = -2*sqrt(2)*sinh(n*log(17+12*sqrt(2))) + 3*cosh(n*log(17+12*sqrt(2))) - 1.
%F A383734 a(n) = 2*A002315(n-1)^2.
%F A383734 a(n) = A075870(n)^2 - 2.
%F A383734 a(n) = 34*a(n-1) - a(n-2) + 32.
%F A383734 G.f.: 2 * (1 + 14*x + x^2) / ((1 - x)*(1 - 34*x + x^2)). - _Stefano Spezia_, May 08 2025
%e A383734 98 is a term becouse 98+2=100 is a square and 98*2=196 is a square.
%t A383734 LinearRecurrence[{35, -35, 1}, {2, 98, 3362}, 20] (* _Amiram Eldar_, May 07 2025 *)
%o A383734 (Python)
%o A383734 from itertools import islice
%o A383734 def A383734_gen(): # generator of terms
%o A383734     x, y = 1, 7
%o A383734     while True:
%o A383734         yield 2*x**2
%o A383734         x, y = y, 6*y - x
%o A383734 A383734_list = list(islice(A383734_gen(), 100))
%Y A383734 Cf. A002315, A088165, A008843, A008844, A075870, A031396, A156164.
%Y A383734 Cf. A382209 (10+k and 10*k are squares).
%Y A383734 Cf. A245226 (m such that k+m and k*m are squares).
%K A383734 nonn,easy
%O A383734 1,1
%A A383734 _Emilio Martín_, May 07 2025