This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156174 #39 Jan 05 2025 19:51:39
%S A156174 1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,
%T A156174 -1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,
%U A156174 -1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,1,-1,1,-1,0
%N A156174 Period 5: repeat [1,-1,1,-1,0].
%C A156174 C(n) := a(n+4) appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + B(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), B(n) = A080891(n) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116(n+5). - _Wolfdieter Lang_, Feb 26 2014
%C A156174 With offset 1 this is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 1, z = 1. - _Michael Somos_, Oct 17 2018
%D A156174 Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).
%H A156174 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1).
%H A156174 C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%F A156174 G.f.: (1+x^2)/(1 + x + x^2 + x^3 + x^4).
%F A156174 Sum_{i=0..n} a(i) = A198517(n). - _Bruno Berselli_, Nov 02 2011
%F A156174 From _Wesley Ivan Hurt_, May 31 2015: (Start)
%F A156174 a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 4.
%F A156174 a(n) = Sum_{i=0..3} A011558(n+2+i)*(-1)^i. (End)
%F A156174 Euler transform of length 5 sequence [-1, 1, 0, -1, 1]. - _Michael Somos_, Jun 17 2015
%F A156174 G.f.: (1-x)*(1-x^4)/((1-x^2)*(1-x^5)). - _Michael Somos_, Jun 17 2015
%F A156174 a(n) = -a(-2-n) = a(n+5) for all n in Z. - _Michael Somos_, Jun 17 2015
%F A156174 a(n) = (2/5) * (cos(4*(n-2)*Pi/5) + cos(2*n*Pi/5) + cos(4*n*Pi/5) - cos(2*(n-3)*Pi/5) - cos(4*(n-3)*Pi/5) - cos(2*(n-1)*Pi/5) - cos(4*(n-1)*Pi/5) - cos((2*n+1)*Pi/5)). - _Wesley Ivan Hurt_, Sep 26 2018
%F A156174 a(n) = (-1)^n * A099443(n). - _Michael Somos_, Oct 17 2018
%F A156174 a(5*n) = a(5*n + 2) = 1, a(5*n + 1) = a(5*n + 3) = -1, a(5*n + 4) = 0 for all n in Z. - _Michael Somos_, Nov 27 2019
%e A156174 G.f. = 1 - x + x^2 - x^3 + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 + ...
%p A156174 A156174:=n->[1,-1,1,-1,0][(n mod 5)+1]: seq(A156174(n), n=0..100); # _Wesley Ivan Hurt_, May 31 2015
%t A156174 CoefficientList[Series[(1 + x^2)/(1 + x + x^2 + x^3 + x^4), {x, 0, 100}], x] (* _Wesley Ivan Hurt_, May 31 2015 *)
%t A156174 a[ n_] := { -1, 1, -1, 0, 1}[[Mod[n, 5, 1]]]; (* _Michael Somos_, Jun 17 2015 *)
%t A156174 a[ n_] := (-1)^Mod[n, 5] Sign @ Mod[n + 1, 5]; (* _Michael Somos_, Jun 17 2015 *)
%o A156174 (PARI) a(n)=[1,-1,1,-1,0][n%5+1] \\ _Charles R Greathouse IV_, Oct 28 2011
%o A156174 (PARI) {a(n) = (-1)^(n%5) * sign((n+1)%5)}; /* _Michael Somos_, Jun 17 2015 */
%Y A156174 Cf. A010874, A011558 (this read mod 2), A099443, A198517.
%K A156174 sign,easy
%O A156174 0,1
%A A156174 _N. J. A. Sloane_, Nov 06 2009