A335298 - OEIS

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

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OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..54.

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

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.