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A226782 If n == 0 (mod 2) then a(n) = 0, otherwise a(n) = 4^(-1) in Z/nZ*.

%I A226782 #30 Mar 14 2023 11:41:59
%S A226782 0,0,1,0,4,0,2,0,7,0,3,0,10,0,4,0,13,0,5,0,16,0,6,0,19,0,7,0,22,0,8,0,
%T A226782 25,0,9,0,28,0,10,0,31,0,11,0,34,0,12,0,37,0,13,0,40,0,14,0,43,0,15,0,
%U A226782 46,0,16,0,49,0,17
%N A226782 If n == 0 (mod 2) then a(n) = 0, otherwise a(n) = 4^(-1) in Z/nZ*.
%H A226782 Charles R Greathouse IV, <a href="/A226782/b226782.txt">Table of n, a(n) for n = 1..10000</a>
%H A226782 Bertrand Teguia Tabuguia and Wolfram Koepf, <a href="https://arxiv.org/abs/2207.01031">FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences</a>, arXiv:2207.01031 [cs.SC], 2022.
%H A226782 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).
%F A226782 From _Colin Barker_, Jun 20 2013: (Start)
%F A226782 G.f.: -x^3*(x^6 - 4*x^2 - 1) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ).
%F A226782 a(2n+1) = A225126(n+1). (End)
%p A226782 A226782 := proc(n)
%p A226782     local x ,a,m;
%p A226782     a := 4 ;
%p A226782     m := 2 ;
%p A226782     if n mod m = 0 or n = 1 then
%p A226782         0;
%p A226782     else
%p A226782         msolve(x*a=1,n) ;
%p A226782         op(%) ;
%p A226782         op(2,%) ;
%p A226782     end if;
%p A226782 end proc: # _R. J. Mathar_, Jun 28 2013
%t A226782 Inv[a_, mod_] := Which[mod == 1, 0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]]; Table[Inv[4, n], {n, 1, 122}]
%t A226782 (* Second program: *)
%t A226782 Table[If[EvenQ[n], 0, ModularInverse[4, n], 0], {n, 1, 100}] (* _Jean-François Alcover_, Mar 14 2023 *)
%o A226782 (PARI) a(n)=if(n%2,lift(Mod(1, n)/4),0) \\ _Charles R Greathouse IV_, Jun 18 2013
%Y A226782 Cf. A092092, A226783, A226784, A226785, A226786, A226787.
%K A226782 nonn,easy
%O A226782 1,5
%A A226782 _José María Grau Ribas_, Jun 18 2013