A105373 Square array by antidiagonals of number of straight lines with n points in a k-dimensional hypercube with n points on each edge.

1, 1, 1, 1, 6, 1, 1, 28, 8, 1, 1, 120, 49, 10, 1, 1, 496, 272, 76, 12, 1, 1, 2016, 1441, 520, 109, 14, 1, 1, 8128, 7448, 3376, 888, 148, 16, 1, 1, 32640, 37969, 21280, 6841, 1400, 193, 18, 1, 1, 130816, 192032, 131776, 51012, 12496, 2080, 244, 20, 1, 1, 523776

Offset

1,5

Links

Table of n, a(n) for n=1..57.

Formula

T(1, k)=1. For n>1: T(n, k) = ((n+2)^k-n^k)/2 = (n+2)*T(n, k-1)+n^(k-1) = A102728(k, n+1).

Example

Rows start:
  1,  1,   1,   1,    1,     1, ...;
  1,  6,  28, 120,  496,  2016, ...;
  1,  8,  49, 272, 1441,  7448, ...;
  1, 10,  76, 520, 3376, 21280, ...;
  1, 12, 109, 888, 6841, 51012, ...;
  etc.
T(5,3)=109 because in a 5 X 5 X 5 cube there are 25 columns, 25 linear rows in one direction, 25 linear rows in another direction, 5 short diagonals in each of 6 directions and 4 long diagonals; and 3*25 + 6*5 + 4 = 109.

Crossrefs

See A102728. Rows essentially include A000012, A006516, A005059, A016149 or A081199, A016161 or A081200, A016170 or A081201, A016178 or A081202 etc. Columns essentially include A000012, A005843, A056107, A105373.

Sequence in context: A394704 A166960 A155908 * A296548 A201461 A340475

Adjacent sequences: A105370 A105371 A105372 * A105374 A105375 A105376

Keyword

nonn,tabl

Author

Henry Bottomley, Apr 02 2005

Status

approved