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A335298 a(n) is the squared distance between the points P(n) and P(0) on a plane, n >= 0, such that the distance between P(n) and P(n+1) is n+1 and, going from P(n) to P(n+2), a 90-degree left turn is taken in P(n+1).

%I A335298 #82 Oct 10 2022 10:20:35
%S A335298 0,1,5,8,8,13,25,32,32,41,61,72,72,85,113,128,128,145,181,200,200,221,
%T A335298 265,288,288,313,365,392,392,421,481,512,512,545,613,648,648,685,761,
%U A335298 800,800,841,925,968,968,1013,1105,1152,1152,1201,1301,1352,1352,1405,1513
%N A335298 a(n) is the squared distance between the points P(n) and P(0) on a plane, n >= 0, such that the distance between P(n) and P(n+1) is n+1 and, going from P(n) to P(n+2), a 90-degree left turn is taken in P(n+1).
%C A335298 P(n) is a corner on a spiral like this:
%C A335298     * * * * * * * * * * * *
%C A335298                           *
%C A335298         * * * * * * * *   *
%C A335298         *             *   *
%C A335298         *   * * * *   *   *
%C A335298         *   *     *   *   *
%C A335298         *   *   * *   *   *
%C A335298         *   *         *   *
%C A335298         *   * * * * * *   *
%C A335298         *                 *
%C A335298         * * * * * * * * * *
%C A335298 If we interpret the pointer from P(0) to P(n) as a complex number z(n), the description of the spiral is short because a 90-degree left turn is a multiplication by i (imaginary unit) and the distance of P(n) from P(0) is abs(z(n))^2, see formula 1.
%H A335298 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-5,7,-7,5,-3,1).
%F A335298 a(n) = abs(z(n))^2 with
%F A335298   1) z(n) = z(n-1)+n*i^(n-1), z(0)=0.    (recursive)
%F A335298   2) z(n) = i/2*(n*i^(n+1)-(n+1)*i^n+1). (explicit)
%F A335298 Without complex numbers for k >= 0:
%F A335298   a(4*k) = 8*k^2,
%F A335298   a(4*k+1) = 8*k^2+4*k+1,
%F A335298   a(4*k+2) = 8*k^2+12*k+5,
%F A335298   a(4*k+3) = 8*(k+1)^2.
%F A335298 From _Stefano Spezia_, Jun 28 2020: (Start)
%F A335298 G.f.: x*(1 + 2*x - 2*x^2 + 2*x^3 + x^4)/((1 - x)^3*(1 + x^2)^2).
%F A335298 a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n > 6. (End)
%e A335298   n  n*i^(n-1)  z(n)  a(n)
%e A335298 ------------------------------------
%e A335298   0     0        0     0
%e A335298   1     1        1     1
%e A335298   2     2i      1+2i   5 = 1^2 + 2^2
%e A335298   3    -3      -2+2i   8 = 2^2 + 2^2
%e A335298   4    -4i     -2-2i   8
%e A335298   5     5       3-2i  13 = 3^2 + 2^2
%e A335298   6     6i      3+4i  25 = 3^2 + 4^2
%t A335298 z[0]=0; z[n_]:=z[n-1]+n*I^(n-1); a[n_]:=z[n]*Conjugate[z[n]]; Array[a,55,0] (* _Stefano Spezia_, Jun 28 2020 *)
%Y A335298 Cf. A000982, A174344, A268038, A274923, A336336.
%K A335298 nonn
%O A335298 0,3
%A A335298 _Gerhard Kirchner_, Jun 28 2020