A337031 - OEIS

A337031

Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2h x 2h x 2h where the walk starts at the center of one of the box's faces.

5, 17, 5, 52, 21, 5, 148, 89, 21, 5, 400, 357, 93, 21, 5, 1060, 1424, 405, 93, 21, 5, 2700, 5484, 1789, 409, 93, 21, 5, 6720, 20960, 7705, 1849, 409, 93, 21, 5, 15760, 78412, 33048, 8257, 1853, 409, 93, 21, 5, 36248, 292168, 139032, 37097, 8329, 1853, 409, 93, 21, 5 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Table of n, a(n) for n=1..55.`
	FORMULA	`For n <= h, T(h,n) = A116904(n).` `Row 1 = T(1,n) = A337033(n).` `For n >= (2h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the 2h X 2h X 2h box.`
	EXAMPLE	`T(1,2) = 17. Taking the first step right,left,forward or backward hits the box's edge after which the walks has three directions for the second step, giving 4*3 = 12 walks in all. A first step upward can be followed by a second step in five directions. The total number of 2-step walks is therefore 12+5 = 17.` `.` `The table begins:` `.` `5 17 52 148 400 1060 2700 6720 15760 36248 77856 163296 312760...` `5 21 89 357 1424 5484 20960 78412 292168 1072272 3919000 14145220 50832492...` `5 21 93 405 1789 7705 33048 139032 583256 2422480 10053452 41415564 170419680...` `5 21 93 409 1849 8257 37097 164533 728808 3194636 13978148 60739156 263711448...` `5 21 93 409 1853 8329 37877 171117 776065 3496769 15758504 70593984 315942684...` `5 21 93 409 1853 8333 37961 172165 786089 3577129 16326745 74257917 337994448...` `5 21 93 409 1853 8333 37965 172261 787445 3591637 16455441 75254865 344977177...` `5 21 93 409 1853 8333 37965 172265 787553 3593341 16475617 75451269 346633713...` `5 21 93 409 1853 8333 37965 172265 787557 3593461 16477709 75478437 346921841...` `5 21 93 409 1853 8333 37965 172265 787557 3593465 16477841 75480957 346957465...` `5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481101 346960453...` `5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481105 346960609...` `5 21 93 409 1853 8333 37965 172265 787557 3593465 16477845 75481105 346960613...`
	CROSSREFS	`Cf. A116904 (h->infinity), A337033 (h=1), A337023 (start at center of box), A336862 (start at middle of edge), A337035 (start at corner of box), A001412.` `Sequence in context: A340706 A093558 A170866 * A125636 A355658 A156323` `Adjacent sequences: A337028 A337029 A337030 * A337032 A337033 A337034`
	KEYWORD	`nonn,walk,tabl`
	AUTHOR	`Scott R. Shannon, Aug 12 2020`
	STATUS	`approved`