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Row 5 of A186861.

From David A. Corneth, Sep 04 2023: (Start)

Proof of a(n) = 2336*n^2 - 10456*n + 11160 for n > 3.

For any walk we can find the surrounding rectangle it fits in.

For example, the walk

0 1 2

0 3 5

0 4 0

has width 2 and height 3.

So it fits max(0, (5 - 2 + 1))*max(0, (5 - 3 + 1)) times in a 5 X 5 grid. This way we can set up a matrix m for all possible walks where element m(r, k) is the number of walks with dimensions (r, k).

That matrix is as follows:

[0 0 0 0 2]

[0 0 160 192 60]

[0 160 568 312 72]

[0 192 312 120 24]

[2 60 72 24 4]

To find a(n) by iterating over this matrix we can compute Sum_{r=1..min(n, 5)} Sum_{k=1..min(n, 5)} m(r, k)*(n - r + 1)*(n - k + 1). This is the sum of 25 quadratics and gives the stated quadratic which completes the proof. (End)

From J. Volkmar Schmidt, Oct 25 2023 (Start)

Proof of formula for a(n) follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 4-step walks is:

[0 0 0 2]

[0 24 80 28]

[0 80 80 20]

[2 28 20 4] (End)

From J. Volkmar Schmidt, Oct 24 2023 (Start)

Proof of a(n) follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 8-step walks is:

[0 0 0 0 0 0 0 2]

[0 0 0 416 3264 4224 2304 508]

[0 0 2384 26004 38120 26164 10080 1764]

[0 416 26004 67424 53320 26480 8460 1328]

[0 3264 38120 53320 32032 13428 3816 560]

[0 4224 26164 26480 13428 4952 1260 172]

[0 2304 10080 8460 3816 1260 288 36]

[2 508 1764 1328 560 172 36 4]

(End)

From J. Volkmar Schmidt, Oct 18 2023: (Start)

Proof of a(n) = 14576*n^2 - 77924*n + 99292 for n>4 follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 6-step walks is:

[0 0 0 0 0 2]

[0 0 96 576 448 124]

[0 96 1856 2272 1064 224]

[0 576 2272 1552 560 104]

[0 448 1064 560 168 28]

[2 124 224 104 28 4]

(End)

From J. Volkmar Schmidt, Oct 24 2023 (Start)

Proof of a(n) follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 7-step walks is:

[0 0 0 0 0 0 2]

[0 0 0 800 1600 1024 252]

[0 0 2984 9780 7904 3360 644]

[0 800 9780 12416 6636 2272 388]

[0 1600 7904 6636 2912 864 136]

[0 1024 3360 2272 864 224 32]

[2 252 644 388 136 32 4]

(End)

Proof of the formula follows proof scheme from David A. Corneth for A186864.

Distribution matrix of surrounding rectangles for 9-step walks is:

[0 0 0 0 0 0 0 0 2]

[0 0 0 0 3584 10496 10752 5120 1020]

[0 0 784 43856 129100 136320 83208 29160 4680]

[0 0 43856 258424 318816 215096 99984 29680 4296]

[0 3584 129100 318816 262816 142888 57376 15400 2100]

[0 10496 136320 215096 142888 67688 24288 5960 768]

[0 10752 83208 99984 57376 24288 7864 1760 212]

[0 5120 29160 29680 15400 5960 1760 360 40]

[2 1020 4680 4296 2100 768 212 40 4]