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A271723 Numbers k such that 3*k - 8 is a square.

%I A271723 #34 Sep 08 2022 08:46:16
%S A271723 3,4,8,11,19,24,36,43,59,68,88,99,123,136,164,179,211,228,264,283,323,
%T A271723 344,388,411,459,484,536,563,619,648,708,739,803,836,904,939,1011,
%U A271723 1048,1124,1163,1243,1284,1368,1411,1499,1544,1636,1683,1779,1828,1928,1979,2083,2136,2244,2299
%N A271723 Numbers k such that 3*k - 8 is a square.
%C A271723 Square roots of resulting squares gives A001651. - _Ray Chandler_, Apr 14 2016
%H A271723 Robert Israel, <a href="/A271723/b271723.txt">Table of n, a(n) for n = 1..10000</a>
%H A271723 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A271723 From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
%F A271723 G.f.: x*(3 + x - 2*x^2 + x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
%F A271723 a(n) = (6*(n - 1)*n - (2*n - 1)*(-1)^n + 23)/8. (End)
%e A271723 a(1) = 3 because 3*3 - 8 = 1^2.
%p A271723 seq(seq(((3*m+k)^2+8)/3, k=1..2),m=0..50); # _Robert Israel_, Dec 05 2016
%t A271723 Select[Range@ 2400, IntegerQ@ Sqrt[3 # - 8] &] (* _Bruno Berselli_, Apr 14 2016 *)
%t A271723 LinearRecurrence[{1,2,-2,-1,1},{3,4,8,11,19},60] (* _Harvey P. Dale_, Oct 02 2020 *)
%o A271723 (Magma) [n: n in [1..2400] | IsSquare(3*n-8)];
%o A271723 (Python) from gmpy2 import is_square
%o A271723 [n for n in range(3000) if is_square(3*n-8)] # _Bruno Berselli_, Dec 05 2016
%o A271723 (Python) [(6*(n-1)*n-(2*n-1)*(-1)**n+23)/8 for n in range(1, 60)] # _Bruno Berselli_, Dec 05 2016
%Y A271723 Cf. A001651.
%Y A271723 Cf. numbers n such that 3*n + k is a square: this sequence (k=-8), A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6), A271741 (k=7), A067725 (k=9).
%K A271723 nonn,easy
%O A271723 1,1
%A A271723 _Juri-Stepan Gerasimov_, Apr 13 2016