A187162 T(n,k) = Number of n-step self-avoiding walks on a k X k X k cube summed over all starting positions.
1, 8, 0, 27, 24, 0, 64, 108, 48, 0, 125, 288, 342, 96, 0, 216, 600, 1056, 1104, 144, 0, 343, 1080, 2370, 3984, 3240, 240, 0, 512, 1764, 4464, 9612, 14256, 9504, 192, 0, 729, 2688, 7518, 18888, 37470, 51504, 25344, 144, 0, 1000, 3888, 11712, 32712, 77184, 148224
Offset: 1
Examples
Solution for n=9 3X3X3 0 0 0 9 0 0 8 0 0 0 2 1 6 0 0 7 0 0 0 3 0 5 4 0 0 0 0 Table starts 1 8 27 64 125 216 343 512 729 1000 0 24 108 288 600 1080 1764 2688 3888 5400 0 48 342 1056 2370 4464 7518 11712 17226 24240 0 96 1104 3984 9612 18888 32712 51984 77604 110472 0 144 3240 14256 37470 77184 137754 223536 338886 488160 0 240 9504 51504 148224 320328 588924 975216 1500408 2185704 0 192 25344 177120 568248 1298016 2466510 4175136 6525450 9619008 0 144 67824 608928 2188608 5299056 10416624 18026640 28617228 42676728 0 0 167016 2013360 8227752 21274896 43422072 76964016 124223214 187527168 0 0 414912 6654048 30938640 85654320 181790352 330218544 541990896 828222216
Links
- R. H. Hardin, Table of n, a(n) for n = 1..241
Formula
a(1,k) = k^3
a(2,k) = 6*k^3 - 6*k^2
a(3,k) = 30*k^3 - 60*k^2 + 24*k for k>1
a(4,k) = 150*k^3 - 426*k^2 + 312*k - 48 for k>2
a(5,k) = 726*k^3 - 2640*k^2 + 2688*k - 720 for k>3
a(6,k) = 3534*k^3 - 15366*k^2 + 19536*k - 7056 for k>4
a(7,k) = 16926*k^3 - 85380*k^2 + 128832*k - 57312 for k>5
a(8,k) = 81390*k^3 - 463074*k^2 + 801216*k - 418032 for k>6
a(9,k) = 387966*k^3 - 2452704*k^2 + 4766544*k - 2833872 for k>7
a(10,k) = 1853886*k^3 - 12825630*k^2 + 27515184*k - 18252624 for k>8
["Empirical" removed by Andrey Zabolotskiy, Feb 28 2022]