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

Offset: 1

Author

Hugo Pfoertner, Jul 18 2020

Keywords

• nonn

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.)

Crossrefs

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

See also A345436.

Programs

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)

Formula

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