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Integers m such that 7*m + 1 is a square.
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%I #31 Mar 15 2022 04:33:39

%S 0,5,9,24,32,57,69,104,120,165,185,240,264,329,357,432,464,549,585,

%T 680,720,825,869,984,1032,1157,1209,1344,1400,1545,1605,1760,1824,

%U 1989,2057,2232,2304,2489,2565,2760,2840,3045,3129,3344,3432,3657,3749,3984,4080

%N Integers m such that 7*m + 1 is a square.

%C Numbers of the form m*(7*m + 2) for m = 0, -1, 1, -2, 2, -3, 3, ... - _Bruno Berselli_, Feb 26 2018

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(2*k) = k*(7*k + 2), a(2*k + 1) = 7*k^2 + 12*k + 5.

%F a(n) = n^2 + n + 3*ceiling(n/2)^2. - _Gary Detlefs_, Feb 23 2010

%F G.f.: -x*(5*x^2 + 4*x + 5)/((x - 1)^3*(x + 1)^2). - _Colin Barker_, Oct 24 2012

%F Sum_{n>=1} 1/a(n) = 7/4 - cot(2*Pi/7)*Pi/2. - _Amiram Eldar_, Mar 15 2022

%p seq(n^2+n+3*ceil(n/2)^2, n=0..48); # _Gary Detlefs_, Feb 23 2010

%t f[n_]:=IntegerQ[Sqrt[1+7*n]]; Select[Range[0,8! ],f[ # ]&] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)

%o (Magma) [n^2+n+3*Ceiling(n/2)^2: n in [0..50]]; // _Wesley Ivan Hurt_, Jul 07 2014

%Y Cf. A054000, A056220, A000217, A087475, A028347, A062717, A036666, A002378, A001082, A046092, A005563.

%K nonn,easy

%O 0,2

%A _Mohamed Bouhamida_, Nov 08 2007

%E More terms from _Max Alekseyev_, Nov 13 2009

%E Better definition from _Max Alekseyev_, Oct 24 2012