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A262672 Expansion of (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).

%I A262672 #49 May 26 2023 12:45:31
%S A262672 3,2,2,1,4,3,3,2,5,4,4,3,6,5,5,4,7,6,6,5,8,7,7,6,9,8,8,7,10,9,9,8,11,
%T A262672 10,10,9,12,11,11,10,13,12,12,11,14,13,13,12,15,14,14,13,16,15,15,14,
%U A262672 17,16,16,15,18,17,17,16,19,18,18,17,20,19,19,18,21
%N A262672 Expansion of (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).
%C A262672 From _Altug Alkan_, Sep 29 2015: (Start)
%C A262672 Sequence can be defined as a composition of arithmetic sequences and it can be generated by the equations: a(4k) = k+3, a(4k+3) = k+1, a(4k+1) = a(4k+2) = k+2, with k >= 0.
%C A262672 Obviously, minimum value of sequence is a(3), which is also unique value that has no repetition in sequence. (End)
%H A262672 G. C. Greubel, <a href="/A262672/b262672.txt">Table of n, a(n) for n = 0..400</a>
%H A262672 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A262672 G.f.: (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).
%F A262672 a(n) = a(n-1) + a(n-4) - a(n-5), for n>4
%F A262672 a(n) = ( 2*n + 13+5*(-1)^n - 6*(-1)^((2*n + 3 + (-1)^n)/4) )/8.
%F A262672 a(n) = 3*A002265(n+4) - A002265(n+3) - A002265(n+1).
%F A262672 a(2n) = A028242(n+4), a(2n+1) = A028242(n+2).
%F A262672 4*a(n) = |A118402(n+8)| + 3*i^(n*(n-1)), where i=sqrt(-1). [_Bruno Berselli_, Sep 29 2015]
%F A262672 E.g.f.: (1/8)*(2*x*exp(x) + 5*exp(-x) + 13*exp(x) + 6*sin(x) + 6*cos(x)). - _G. C. Greubel_, Sep 29 2015
%F A262672 a(n) = -a(-13 - n) for all n in Z. - _Michael Somos_, Oct 02 2015
%e A262672 G.f. = 3 + 2*x + 2*x^2 + x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
%p A262672 A262672:=n->(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8: seq(A262672(n), n=0..100);
%t A262672 Table[(2n + 13 + 5 (-1)^n - 6 (-1)^((2n + 3 + (-1)^n)/4))/8, {n, 0, 100}]
%t A262672 LinearRecurrence[{1, 0, 0, 1, -1}, {3, 2, 2, 1, 4}, 100] (* _Vincenzo Librandi_, Sep 29 2015 *)
%t A262672 CoefficientList[Series[(3-x-x^3)/((x-1)^2(1+x+x^2+x^3)),{x,0,100}],x] (* _Harvey P. Dale_, May 26 2023 *)
%o A262672 (Magma) [(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n) div 4))/8 : n in [0..100]];
%o A262672 (Magma) &cat[[3+n,2+n,2+n,1+n]: n in [0..20]]; // _Bruno Berselli_, Sep 29 2015
%o A262672 (Magma) I:=[3,2,2,1,4]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..100]]; // _Vincenzo Librandi_, Sep 29 2015
%o A262672 (PARI) a(n) = (2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8;
%o A262672 vector(80, n, a(n-1)) \\ _Altug Alkan_, Sep 29 2015
%o A262672 (PARI) {a(n) = my(k = n\4); [ 3, 2, 2, 1][n%4 + 1] + k}; /* _Michael Somos_, Oct 02 2015 */
%Y A262672 Row 2 of the example in A238340 (read from bottom).
%Y A262672 Cf. A002265, A028242, A118402.
%K A262672 nonn,easy
%O A262672 0,1
%A A262672 _Wesley Ivan Hurt_, Sep 26 2015
%E A262672 Edited by _Bruno Berselli_, Sep 30 2015