A128127 The number of regular pentagons found by constructing n equally-spaced points on each side of the pentagon and drawing lines parallel to the pentagon side.

1, 3, 9, 21, 37, 59

Offset

0,2

Comments

A similar pattern of construction to A000330 (dividing a square), A002717 (dividing a triangle), dividing a hexagon and any other polygon in a similar fashion (sequences pending).

Use 1 midpoint (resp. 2 points) on each side placed to divide each side into 2 (resp. 3) equally-sized segments or so on, do the same construction for every side of the pentagon so that each side is equally divided in the same way. Connect all such points to each other with lines that are parallel to at least 1 side of the polygon.

Links

Table of n, a(n) for n=0..5.

Michel Marcus, Figure with 2 points on each side

Noah Priluck, On Counting Regular Polygons Formed by Special Families of Parallel Lines, Geombinatorics Quarterly, Vol XVII (4), 2008, pp. 166-171. (note there is no document to download).

Noah Priluck, On Counting Regular Polygons Formed by Special Families of Parallel Lines

Formula

a(n) = (10*n^2 - 4*n + 5 -(-1)^n)/4 (conjectural).

Example

With 0 point, there is only 1 pentagon, the original one. With 1 point (a midpoint on each side), 3 regular pentagons are found. With two points, 9 regular pentagons are found in total.

Prog

(PARI)
ldraw(w, vx, vy, np, with, ia, ib, jb, ja) = {if (with, kdeb = 0; kend = np, kdeb = 1; kend = np-1; ); for (k=kdeb, kend, plotmove(w, vx[ia]+k*(vx[ib]-vx[ia])/np, vy[ia]+k*(vy[ib]-vy[ia])/np); plotlines(w, vx[ja]+k*(vx[jb]-vx[ja])/np, vy[ja]+k*(vy[jb]-vy[ja])/np); ); }
modnv(i, nv) = {i =  i % nv; if (i == 0, i = nv); return (i); }
poly(nv, np, with) = {w = 2; s = plothsizes(); plotinit(w, s[1]-1, s[2]-1); plotscale(w, 0, 1000, 0, 1000); xc = 500; yc = 500; vx = vector(nv, i, xc + 500*sin(i*2*Pi/nv)); vy = vector(nv, i, yc + 500*cos(i*2*Pi/nv)); plotlines(w , vx, vy, 1); plotmove(w, vx[nv], vy[nv]); plotlines(w, vx[1], vy[1]); np++; for (ia=1, nv, ia =  modnv(ia, nv); ib =  modnv(ia+1, nv); for (ja=1, nv, ja = modnv(ja, nv); if (ja != ia, jb = modnv(ja+1, nv); ldraw(w, vx, vy, np, with, ia, ib, ja, jb); ); ); ); plotdraw([w, 0, 0]); return(0); } \\ use poly(5, n, 0) to get figure with n points \\ Michel Marcus, Jul 09 2013

Crossrefs

Cf. A128153 (same construction but with pentagon vertices also connected).

Sequence in context: A287057 A048780 A009864 * A363242 A341433 A014857

Adjacent sequences: A128124 A128125 A128126 * A128128 A128129 A128130

Keyword

more,nonn

Author

Noah Priluck (npriluck(AT)gmail.com), May 02 2007

Extensions

Edited by Michel Marcus, Jul 09 2013

a(4) and a(5) from Michel Marcus, Jul 21 2013

Status

approved