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).

0, 1, 5, 8, 8, 13, 25, 32, 32, 41, 61, 72, 72, 85, 113, 128, 128, 145, 181, 200, 200, 221, 265, 288, 288, 313, 365, 392, 392, 421, 481, 512, 512, 545, 613, 648, 648, 685, 761, 800, 800, 841, 925, 968, 968, 1013, 1105, 1152, 1152, 1201, 1301, 1352, 1352, 1405, 1513

Offset: 0

Author

Gerhard Kirchner, Jun 28 2020

Keywords

• nonn

Comments

P(n) is a corner on a spiral like this:

* * * * * * * * * * * *

*

* * * * * * * * *

* * *

* * * * * * *

* * * * *

* * * * * *

* * * *

* * * * * * * *

* *

* * * * * * * * * *

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.

Examples

			  n  n*i^(n-1)  z(n)  a(n)
------------------------------------
  0     0        0     0
  1     1        1     1
  2     2i      1+2i   5 = 1^2 + 2^2
  3    -3      -2+2i   8 = 2^2 + 2^2
  4    -4i     -2-2i   8
  5     5       3-2i  13 = 3^2 + 2^2
  6     6i      3+4i  25 = 3^2 + 4^2
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Crossrefs

Cf. A000982, A174344, A268038, A274923, A336336.

Programs

Mathematica

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 *)

Formula

a(n) = abs(z(n))^2 with

1) z(n) = z(n-1)+n*i^(n-1), z(0)=0. (recursive)

2) z(n) = i/2*(n*i^(n+1)-(n+1)*i^n+1). (explicit)

Without complex numbers for k >= 0:

a(4*k) = 8*k^2,

a(4*k+1) = 8*k^2+4*k+1,

a(4*k+2) = 8*k^2+12*k+5,

a(4*k+3) = 8*(k+1)^2.

From Stefano Spezia, Jun 28 2020: (Start)

G.f.: x*(1 + 2*x - 2*x^2 + 2*x^3 + x^4)/((1 - x)^3*(1 + x^2)^2).

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)