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A281445 Nonnegative k for which (2*k^2 + 1)/11 is an integer.

%I A281445 #54 Nov 24 2024 01:52:05
%S A281445 4,7,15,18,26,29,37,40,48,51,59,62,70,73,81,84,92,95,103,106,114,117,
%T A281445 125,128,136,139,147,150,158,161,169,172,180,183,191,194,202,205,213,
%U A281445 216,224,227,235,238,246,249,257,260,268,271,279,282,290,293,301,304,312,315
%N A281445 Nonnegative k for which (2*k^2 + 1)/11 is an integer.
%C A281445 For prime d < 11, (2*k^2 + 1)/d can provide integers when d = 3 (A186424).
%C A281445 Corresponding values of (2*k^2 + 1)/11 are listed in A179088.
%C A281445 All k == 4 or 7 (mod 11). - _Robert Israel_, Apr 25 2017
%H A281445 Bruno Berselli, <a href="/A281445/b281445.txt">Table of n, a(n) for n = 1..1000</a>
%H A281445 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A281445 O.g.f.: x*(4 + 3*x + 4*x^2)/((1 + x)*(1 - x)^2).
%F A281445 E.g.f.: 4 - 5*exp(-x)/4 - 11*(1 - 2*x)*exp(x)/4.
%F A281445 a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
%F A281445 a(n) = (22*n - 5*(-1)^n - 11)/4. Therefore: a(2*h) = 11*h - 4, a(2*h+1) = 11*h + 4.
%F A281445 If h>0,
%F A281445 h*a(n) + (6*h - 5*(-1)^h - 11)/4 = a(h*n) for odd n; otherwise:
%F A281445 h*a(n) + 4*(h - 1) = a(h*n). Some special cases:
%F A281445 h=2: 2*a(n) - 1 = a(2*n) for odd n, 2*a(n) +  4 = a(2*n) for even n;
%F A281445 h=3: 3*a(n) + 3 = a(3*n) for odd n, 3*a(n) +  8 = a(3*n) for even n;
%F A281445 h=4: 4*a(n) + 2 = a(4*n) for odd n, 4*a(n) + 12 = a(4*n) for even n;
%F A281445 h=5: 5*a(n) + 6 = a(5*n) for odd n, 5*a(n) + 16 = a(5*n) for even n, and so on.
%F A281445 Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/22)*Pi/11. - _Amiram Eldar_, Feb 27 2023
%F A281445 From _Amiram Eldar_, Nov 23 2024: (Start)
%F A281445 Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/22)*sec(3*Pi/22).
%F A281445 Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(3*Pi/22). (End)
%p A281445 seq(seq(11*i+j,j=[4,7]),i=0..50); # _Robert Israel_, Apr 25 2017
%t A281445 Select[Range[400], IntegerQ[(2*#^2 + 1)/11] &]
%o A281445 (Sage) [k for k in range(400) if ((2*k^2+1)/11).is_integer()]
%o A281445 (Magma) &cat [[11*n+4, 11*n+7]: n in [0..30]];
%Y A281445 Cf. A179088.
%Y A281445 Cf. A001651 (nonnegative k for which (2*k^2 + 1)/3 is an integer).
%K A281445 nonn,easy
%O A281445 1,1
%A A281445 _Bruno Berselli_, Apr 13 2017