A low-resolution example of the undefinable aspect of a Pascalloid is a progressively generated number crystal based on a five cornered polygon. This seems at first to be only the logical way to generate a 5-sided number crystal or Pascalloid, but there are other ways. Yet at what resolution? 8X9 seems a close guesstimate.

Here is my example of a 5-sided Pascalloid (or 5-variable / corners). It is superficial, accuracy is only at an unreachable infinite resolution, and by then there would be no room to display it. Terms collapsing to unity in this case is very important, because otherwise there seems to be no way to sort them out. Deriving fractional exponents for (A+B+C+D+E)N is possible algorithmically but not identical at all to the terms derived from that expansion.

There is a divergence of polynomial accuracy with respect to the five variable algebraic expansion beginning at N=4. There are six instances of the number 24, whereas in the expansion of (a+b+c+d+e) N there are only five terms with that coefficient. In this Pascalloid at N=4, there are two instances of the number 18, whereas in the expansion there are turns out to be no number 18. Though this seems to be an error, it is accurate to the algorithm, and because our original pentagonal algorithm takes place on an eight by nine cell grid, it very far from a precise pentagonal algorithm. The imprecision pointed out here results in those "polynomial overlap" errors starting in the following Pascalloid at N=4. This calculation is still continued up to N=8.

1

total = 50 = 1


1
1
1
1
1

total = 51 = 5



1
2
1 2
2 2
2 2 1
2 2
1 2
2
1

total = 52 = 25



1
3
3 3
1 3 6
6 3 3 3
3 6 6 3
6 6 6 1
3 6 6 3
6 3 3 3
1 3 6
3 3
3
1

total = 53 = 125



1
4
6 4
4 4 12
1 12 12 4 6
12 4 12 12 12
4 18 24 6 12 4
16 12 12 24 12 4
6 24 24 12 6 12 1
16 12 12 24 12 4
4 18 24 6 12 4
12 4 12 12 12
1 12 12 4 6
4 4 12
6 4
4
1

total = 54 = 625



1
5
10 5
10 5 20
5 20 30 5 10
1 30 20 20 20 30
20 5 40 60 30 20 10
5 50 60 10 20 60 30 20
35 20 60 90 60 10 10 30 5
10 70 80 30 30 60 60 20 5
40 30 60 120 60 20 30 20 1
10 70 80 30 30 60 60 20 5
35 20 60 90 60 10 10 30 5
5 50 60 10 20 60 30 20
20 5 40 60 30 20 10
1 30 20 20 20 30
5 20 30 5 10
10 5 20
10 5
5
1

total = 55 = 3,125



1
6
15
6
20
6
30
15
30
60
6
15
6
60
60
30
30
60
1
60
30
75
120
90
30
20
30
6
120
180
60
30
120
60
60
6
120
120
15
120
240
180
15
60
60
15
66
30
200
300
180
60
20
120
180
60
30
15
180
210
60
150
360
270
120
60
15
60
6
90
60
240
420
240
60
60
180
180
120
30
6
20
210
240
90
180
360
360
20
120
30
90
30
1
90
60
240
420
240
60
60
180
180
120
30
6
15
180
210
60
150
360
270
120
60
15
60
6
66
30
200
300
180
60
20
120
180
60
30
6
120
120
15
120
240
180
15
60
60
15
30
6
120
180
60
30
120
60
60
1
60
30
75
120
90
30
20
6
60
60
30
30
60
15
30
60
6
15
20
6
30
15
6
6
1

total = 56 = 15,625



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1
7
21 7
35 7 42
420 105 42
182 105 770 1260 735 420 140 1050 1260 35 630 210 210 21 105 7
35 546 630 210 770 1680 1470 140 560 420 105 630 420 210 42 7
210 140 840 1470 840 420 210 1260 1260 840 140 210 42 210 42 1
35 546 630 210 770 1680 1470 140 560 420 105 630 420 210 42 7
182 105 770 1260 735 420 140 1050 1260 35 630 210 210 21 105 7
21 420 462 105 630 1400 1050 105 420 420 35 420 420 105 42
112 42 560 840 420 315 35 840 840 420 105 105 140 21
7 252 210 21 455 840 630 105 140 210 420 140 105
42 7 315 420 105 210 525 420 21 210 105 35
1 105 42 245 420 210 42 210 105 140
7 140 105 126 210 210 42 35
21 105 140 42 42 105
35 42 105 7 21
35 7 42
21 7
7
1

total = 57 = 78,125



1
8
28 8
56 8 56
70 56 168 8 28
56 168 280 56 56 168
28 280 280 196 336 420 56 56
8 280 168 448 840 560 56 336 168 280
1 168 56 700 1120 420 336 1008 840 28 560 168 70
56 8 728 840 168 896 1960 1680 168 560 336 840 280 280
8 476 336 28 1400 2520 1680 588 280 1680 2100 1120 168 420 280 56
176 56 1400 2016 840 1400 56 3640 3360 168 1680 840 280 840 1120 280 168
28 896 896 168 2170 4480 3360 840 1120 2520 70 3360 56 2240 840 420 168 280 28
336 168 2128 3360 1848 1960 280 5040 5600 280 2800 840 840 1680 56 1120 840 168 56
56 1288 1456 420 2800 6160 5040 980 1960 3360 280 4200 3360 280 1260 336 560 28 168 8
448 280 2576 4368 2520 2240 560 6160 6720 280 3920 1120 1120 1680 168 1680 840 336 56 8
70 1456 1680 560 3080 6720 5880 1120 2240 3360 420 5040 70 3360 1680 560 336 56 420 56 1
448 280 2576 4368 2520 2240 560 6160 6720 280 3920 1120 1120 1680 168 1680 840 336 56 8
56 1288 1456 420 2800 6160 5040 980 1960 3360 280 4200 3360 280 1260 336 560 28 168 8
336 168 2128 3360 1848 1960 280 5040 5600 280 2800 840 840 1680 56 1120 840 168 56
28 896 896 168 2170 4480 3360 840 1120 2520 70 3360 56 2240 840 420 168 280 28
176 56 1400 2016 840 1400 56 3640 3360 168 1680 840 280 840 1120 280 168
8 476 336 28 1400 2520 1680 588 280 1680 2100 1120 168 420 280 56
56 8 728 840 168 896 1960 1680 168 560 336 840 280 280
1 168 56 700 1120 420 336 1008 840 28 560 168 70
8 280 168 448 840 560 56 336 168 280
28 280 280 196 336 420 56 56
56 168 280 56 56 168
70 56 168 8 28
56 8 56
28 8
8
1

total = 58 = 390,625

...

35 42 105 7 21 21 105 140 42 42 105 7 140 105 126 210 210 42 35 1 105 42 245 420 210 42 210 105 140 42 7 315 420 105 210 525 420 21 210 105 35 7 252 210 21 455 840 630 105 140 210 420 140 105 112 42 560 840 420 315 35 840 840 420 105 105 140 21 21 420 462 105 630 1400 1050 105 420 420 35