A336336 - OEIS

A336336

Squared distance from start of a point moving in a square spiral.

0, 1, 2, 1, 2, 1, 2, 1, 2, 5, 4, 5, 8, 5, 4, 5, 8, 5, 4, 5, 8, 5, 4, 5, 8, 13, 10, 9, 10, 13, 18, 13, 10, 9, 10, 13, 18, 13, 10, 9, 10, 13, 18, 13, 10, 9, 10, 13, 18, 25, 20, 17, 16, 17, 20, 25, 32, 25, 20, 17, 16, 17, 20, 25, 32, 25, 20, 17, 16, 17, 20, 25, 32

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The terms corresponding to the corner points of the spiral with a(k-1) < a(k) > a(k+1), i.e., 2, 2, 2, 5, 8, 8, 8, 13, 18, 18, 18, ... are given by the sequence A001105(1) repeated 3 times, (A001105(1)+A001105(2))/2, A001105(2) repeated 3 times, (A001105(2)+A001105(3))/2, A001105(3) repeated 3 times, ... .

These numbers are the norms of the Gaussian integers discussed in A345436. - N. J. A. Sloane, Jun 25 2021

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Sketch showing first 49 cells (numbered in black, starting at 0) of the square spiral and the corresponding squared distances from the origin (red). (This illustration assumes the sequence has offset 0.)

FORMULA

a(n) = A174344(n)^2 + A268038(n)^2 = A174344(n)^2 + A274923(n)^2.

PROG

(PARI) A336336(m)={my(v=vectorsmall(m)); for(Lstart=0, 1, my(L=Lstart, d=1, n=0); for(r=1, oo, d=-d; my(k=floor(r/2)*d); for(j=1, L++, n++; if(n<=m, v[n]+=k*k)); forstep(j=k-d, -floor((r+1)/2)*d+d, -d, n++; if(n<=m, v[n]+=j*j)); if(n>m, break))); v};

A336336(73)

CROSSREFS

Cf. A001105, A174344, A268038, A274923, A335298.