A053416 Circle numbers (version 4): a(n)= number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in a circle of diameter n, centered at (0,0).
1, 1, 7, 7, 19, 19, 37, 43, 61, 73, 91, 109, 127, 151, 187, 199, 241, 253, 301, 313, 367, 397, 439, 475, 517, 571, 613, 661, 721, 757, 823, 859, 931, 979, 1045, 1111, 1165, 1237, 1303, 1381, 1459, 1519, 1615, 1663, 1765, 1813, 1921, 1993, 2083, 2173, 2263
Offset: 0
Links
- H. v. Eitzen, Table of n, a(n) for n = 0..1000
- Kival Ngaokrajang, Illustration of initial terms
- Index entries for sequences related to A2 = hexagonal = triangular lattice
Crossrefs
Programs
-
Maple
A053416 := proc(d) local a,j,imin,imax ; a := 0 ; for j from -floor(d/sqrt(3)) do if j^2*3 > d^2 and j> 0 then break ; end if; imin := ceil((-j-sqrt(d^2-3*j^2))/2) ; imax := floor((-j+sqrt(d^2-3*j^2))/2) ; a := a+imax-imin+1 ; end do: a ; end proc: seq(A053416(d),d=0..30) ; # R. J. Mathar, Nov 22 2022
-
Mathematica
a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) <= n^2], {i, -n, n}, {j, -n, n}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 06 2013, updated Apr 08 2022 to correct a discrepancy wrt b-file noticed by Georg Fischer *)
Formula
a(n)/(n/2)^2->Pi*2/sqrt(3).
a(n) >= A053458(n). - R. J. Mathar, Nov 22 2022
a(2*n) = A308685(n). - R. J. Mathar, Nov 22 2022
Extensions
Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
Comments