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A156638 Numbers k such that k^2 + 2 == 0 (mod 9). 8

%I #56 Jan 31 2023 17:46:06

%S 4,5,13,14,22,23,31,32,40,41,49,50,58,59,67,68,76,77,85,86,94,95,103,

%T 104,112,113,121,122,130,131,139,140,148,149,157,158,166,167,175,176,

%U 184,185,193,194,202,203,211,212,220,221,229,230,238,239,247,248,256

%N Numbers k such that k^2 + 2 == 0 (mod 9).

%C From _Artur Jasinski_, Apr 30 2010: (Start)

%C Numbers congruent to 4 or 5 mod 9.

%C Numbers which are not the sum of 3 cubes.

%C Complement to A060464. (End)

%C Numbers k such that A010888(k^2) = 7. - _V.J. Pohjola_, Aug 18 2012

%D Henri Cohen, Number Theory Volume I: Tools and Diophantine Equations. Springer Verlag (2007) p. 380. - _Artur Jasinski_, Apr 30 2010

%H Vincenzo Librandi, <a href="/A156638/b156638.txt">Table of n, a(n) for n = 1..1000</a>

%H Andrew Sutherland, <a href="https://drive.google.com/file/d/1qzD__dviONTqHQH7DBFmsQ0MdCa7ePRg/view">Sums of three cubes</a>, Slides of a talk given May 07 2020 on the Number Theory Web.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F For n > 2, a(n) = a(n-2) + 9.

%F G.f.: x*(4*x^2 + x + 4)/(x^3 - x^2 - x + 1). - _Alexander R. Povolotsky_, Feb 15 2009

%F From _R. J. Mathar_, Feb 19 2009: (Start)

%F a(n) = a(n-1) + a(n-2) - a(n-3), n>3.

%F a(n) = 9*n/2 - 9/4 - 7*(-1)^n/4.

%F G.f.: x*(4 + x + 4*x^2)/((1 + x)*(1 - x)^2). (End)

%F a(n) = -a(-n+1). - _Bruno Berselli_, Jan 08 2012

%F E.g.f.: 4 + ((18*x - 9)*exp(x) - 7*exp(-x))/4. - _David Lovler_, Aug 21 2022

%F Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/18)*Pi/9. - _Amiram Eldar_, Sep 26 2022

%p A156638:=n->9*n/2 - 9/4 - 7*(-1)^n/4: seq(A156638(n), n=1..80); # _Wesley Ivan Hurt_, Aug 16 2015

%t LinearRecurrence[{1, 1, -1}, {4, 5, 13}, 50] (* _Vincenzo Librandi_, Mar 01 2012 *)

%t Flatten[Table[9n - {5, 4}, {n, 30}]] (* _Alonso del Arte_, Aug 09 2015 *)

%t Select[Range[300],PowerMod[#,2,9]==7&] (* _Harvey P. Dale_, Jan 31 2023 *)

%o (Magma) [9*n/2 - 9/4 - 7*(-1)^n/4 : n in [1..80]]; // _Wesley Ivan Hurt_, Aug 16 2015

%o (PARI) a(n) = (18*n - 9 - 7*(-1)^n)/4 \\ _David Lovler_, Aug 21 2022

%Y Cf. A060464, A010888, A334521, A334522.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Feb 12 2009

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