|
%I #23 Sep 08 2022 08:46:16
%S 2,3,7,10,18,23,35,42,58,67,87,98,122,135,163,178,210,227,263,282,322,
%T 343,387,410,458,483,535,562,618,647,707,738,802,835,903,938,1010,
%U 1047,1123,1162,1242,1283,1367,1410,1498,1543,1635,1682,1778,1827,1927,1978,2082,2135,2243,2298
%N Numbers n such that 3*n - 5 is a square.
%C Quasipolynomial of order 2 and degree 2. - _Charles R Greathouse IV_, Apr 12 2016
%C From _Ray Chandler_, Apr 13 2016: (Start)
%C Square roots of resulting squares gives A001651.
%C Sequence is the union of A141631 and A271740. (End)
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, -2, -1, 1).
%F G.f.: x*(2 + x + x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2). - _Ilya Gutkovskiy_, Apr 12 2016
%F a(n) = (3/2)*n^2 + O(n). - _Charles R Greathouse IV_, Apr 12 2016
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5. - _Wesley Ivan Hurt_, Apr 13 2016
%e a(3) = 7 because 3*7 - 5 = 16 = 4^2.
%p A271713:=n->`if`(issqr(3*n-5), n, NULL): seq(A271713(n), n=1..5000); # _Wesley Ivan Hurt_, Apr 13 2016
%t Select[Range@ 2300, IntegerQ@ Sqrt[3 # - 5] &] (* _Michael De Vlieger_, Apr 12 2016 *)
%o (Magma) [ n: n in [0..2500] | IsSquare(3*n - 5)];
%o (PARI) is(n)=issquare(3*n-5) \\ _Charles R Greathouse IV_, Apr 12 2016
%o (PARI) a(n)=((n\2*3-(-1)^n)^2+5)/3 \\ _Charles R Greathouse IV_, Apr 12 2016
%o (Python)
%o from __future__ import division
%o A271713_list = [(n**2+5)//3 for n in range(10**6) if not (n**2+5) % 3] # _Chai Wah Wu_, Apr 13 2016
%Y Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), this sequence (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6).
%Y Cf. A001651, A141631, A271740.
%K nonn,easy
%O 1,1
%A _Juri-Stepan Gerasimov_, Apr 12 2016
|
