A345440 Represent the ring R = {x+y*sqrt(-2): x, y rational integers} by the cells centered at the points (x,y) of a square grid, as in A345437; number the cells of the grid along a counterclockwise square spiral, with the cells at (0,0) and (1,0) numbered 0, 1; then a(n) is the norm x^2+2*y^2 of the element of R occupying the n-th cell.

0, 1, 3, 2, 3, 1, 3, 2, 3, 6, 4, 6, 12, 9, 8, 9, 12, 6, 4, 6, 12, 9, 8, 9, 12, 17, 11, 9, 11, 17, 27, 22, 19, 18, 19, 22, 27, 17, 11, 9, 11, 17, 27, 22, 19, 18, 19, 22, 27, 34, 24, 18, 16, 18, 24, 34, 48, 41, 36, 33, 32, 33, 36, 41, 48, 34, 24, 18, 16, 18, 24

Offset: 0

Author

N. J. A. Sloane, Jun 25 2021

Keywords

• nonn

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, PARI program for A345440
• N. J. A. Sloane, Illustration of initial terms [An enlargement of Figure 3 of Wichmann (2019), showing the numbering of the initial cells of the square spiral. The origin is black, the two units +-1 are red, and the primes are blue.]
• N. J. A. Sloane, Illustration of initial terms. [The cell numbers are black, their norms are red.]
• Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019.

Crossrefs

Cf. A174344, A274923, A336336, A345435, A345437, A345439.

Programs

PARI

See Links section.

Formula

a(n) = A174344(n+1)^2 + 2*A274923(n+1)^2. - Rémy Sigrist, Jun 26 2021

Extensions

More terms from Rémy Sigrist, Jun 26 2021