A356465 - OEIS

A356465

The number of unit squares enclosed by the rectangular spiral of which the n-th side has length prime(n).

0, 2, 6, 12, 27, 59, 113, 179, 257, 359, 497, 747, 963, 1227, 1577, 1799, 2081, 2611, 3223, 3663, 4167, 4817, 5231, 5847, 6657, 7527, 8801, 9869, 10439, 11057, 11699, 12425, 14675, 16817, 18027, 19139, 20855, 22595, 23803, 25711, 27321, 29011, 31063, 32495

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OFFSET

0,2

COMMENTS

The pictures in the links show how the spiral is constructed. The first segment is the small black rectangle in the center, of which the left lower corner is at the origin (0,0). It represents prime(1) = 2 (its width) and is given a height of one. The first part of the boundary of the spiral is the line between (0,0) and (2,0). Prime(2) = 3 yields the next part of the boundary, the line connecting (2,0) and (2,3). The next primes determine how many unit steps the boundary of the spiral goes left, down, right, up, etc.

LINKS

Table of n, a(n) for n=0..43.

Bob Andriesse, Graphical representation of the first 16 segments of the spiral with grid.

Bob Andriesse, Graphical representation of the first 44 segments of the spiral without grid.

FORMULA

a(n) = a(n-1) +(prime(n) - prime(n-2) + prime(n-4))*(prime(n-1) - prime(n-3)) for n > 4.

MATHEMATICA

a[4]:=27; a[n_]:=a[n]=a[n-1]+(Prime[n]-Prime[n-2]+Prime[n-4])(Prime[n-1]-Prime[n-3]); Join[{0, 2, 6, 12, 27}, Table[a[n], {n, 5, 45}]] (* Stefano Spezia, Aug 09 2022 *)

PROG

(Python)

from sympy import prime as p

a = [0, 2, 6, 12, 27] #first 4 area values

area = 27

for n in range(5, 44+1):

darea = (p(n) - p(n-2) + p(n-4)) * (p(n-1) - p(n-3))

area += darea

a.append(area)

print('a(n)=', a)

CROSSREFS

Cf. A000040.