User:Gangleri/tests/sandbox/squares

Counting[edit]

counting sequences[edit]

sequence details

patterns			$\beth (3)$	$\beth (2,1)$
→ and ↓	→	↓		→	↓

→→→↓↓↓	x,x,x	x,x,x	1	1	1	1

→→↓↓↓→	x,x x	x,x,x	1	2	1	2
→↓↓↓→→	x,x x	x,x,x	1	2	1	2

→→↓↓→↓	x,x x	x,x x	1	2	2	4
→↓↓→→↓	x,x x	x,x x	1	2	2	4
→→↓→↓↓	x,x x	x,x x	1	2	2	4
→↓→→↓↓	x,x x	x,x x	1	2	2	4

→↓↓→↓→	x x x	x,x x	1	3	2	6
→↓→↓↓→	x x x	x,x x	1	3	2	6

→↓→↓→↓	x x x	x x x	1	3	3	9

			10			42

			20			84

			$\beth (3)$	$\beth (2,1)$

patterns			$\beth (4)$	$\beth (3,1)$			$\beth (2,2)$			$\beth (2,1,1)$
→ and ↓	→	↓		→	↓		→	↓		→	↓

→→→→↓↓↓↓	x,x,x,x	x,x,x,x	1	1	1	1	1	1	1	1	1	1

→→→↓↓↓↓→	x,x,x x	x,x,x,x	1	2	1	2	2	1	2	3	1	3
→→↓↓↓↓→→	x,x x,x	x,x,x,x	1	2	1	2	3	1	3	4	1	4
→↓↓↓↓→→→	x,x,x x	x,x,x,x	1	2	1	2	2	1	2	3	1	3

→→→↓↓↓→↓	x,x,x x	x,x,x x	1	2	2	4	2	2	4	3	3	9
→→↓↓↓→→↓	x,x x,x	x,x,x x	1	2	2	4	3	2	6	4	3	12
→↓↓↓→→→↓	x,x,x x	x,x,x x	1	2	2	4	2	2	4	3	3	9
→→→↓↓→↓↓	x,x,x x	x,x x,x	1	2	2	4	2	3	6	3	4	12
→→↓↓→→↓↓	x,x x,x	x,x x,x	1	2	2	4	3	3	9	4	4	16
→↓↓→→→↓↓	x,x,x x	x,x x,x	1	2	2	4	2	3	6	3	4	12
→→→↓→↓↓↓	x,x,x x	x,x,x x	1	2	2	4	2	2	4	3	3	9
→→↓→→↓↓↓	x,x x,x	x,x,x x	1	2	2	4	3	2	6	4	3	12
→↓→→→↓↓↓	x,x,x x	x,x,x x	1	2	2	4	2	2	4	3	3	9

→→↓↓↓→↓→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21
→↓↓↓→→↓→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21
→↓↓↓→↓→→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21
→→↓↓→↓↓→	x,x x x	x,x x,x	1	4	2	8	4	3	12	7	4	28
→↓↓→→↓↓→	x,x x x	x,x x,x	1	4	2	8	4	3	12	7	4	28
→↓↓→↓↓→→	x,x x x	x,x x,x	1	4	2	8	4	3	12	7	4	28
→→↓→↓↓↓→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21
→↓→→↓↓↓→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21
→↓→↓↓↓→→	x,x x x	x,x,x x	1	4	2	8	4	2	8	7	3	21

→→↓↓→↓→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓↓→→↓→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓↓→↓→→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→→↓→↓↓→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓→→↓↓→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓→↓↓→→↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→→↓→↓→↓↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓→→↓→↓↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49
→↓→↓→→↓↓	x,x x x	x,x x x	1	4	4	16	4	4	16	7	7	49

→↓↓→↓→↓→	x x x x	x,x x x	1	4	4	16	6	4	24	12	7	84
→↓→↓↓→↓→	x x x x	x,x x x	1	4	4	16	6	4	24	12	7	84
→↓→↓→↓↓→	x x x x	x,x x x	1	4	4	16	6	4	24	12	7	84

→↓→↓→↓→↓	x x x x	x x x x	1	4	4	16	6	6	36	12	12	144

			35			323			393			1158

			70			646			786			2316

			$\beth (4)$	$\beth (3,1)$			$\beth (2,2)$			$\beth (2,1,1)$

patterns			$\beth (5)$	$\beth (4,1)$			$\beth (3,2)$			$\beth (3,1,1)$			$\beth (2,2,1)$			$\beth (2,1,1,1)$
→ and ↓	→	↓		→	↓		→	↓		→	↓		→	↓		→	↓

→→→→→↓↓↓↓↓	x,x,x,x,x	x,x,x,x,x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

→→→→↓↓↓↓↓→	x,x,x,x x	x,x,x,x,x	1	2	1	2	2	1	2	3	1	3	3	1	3	4	1	4
→→→↓↓↓↓↓→→	x,x,x x,x	x,x,x,x,x	1	2	1	2	3	1	3	4	1	4	5	1	5	7	1	7
→→↓↓↓↓↓→→→	x,x,x x,x	x,x,x,x,x	1	2	1	2	3	1	3	4	1	4	5	1	5	7	1	7
→↓↓↓↓↓→→→→	x,x,x,x x	x,x,x,x,x	1	2	1	2	2	1	2	3	1	3	3	1	3	4	1	4

→→→→↓↓↓↓→↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16
→→→↓↓↓↓→→↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→→↓↓↓↓→→→↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→↓↓↓↓→→→→↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16
→→→→↓↓↓→↓↓	x,x,x,x x	x,x,x x,x	1	2	2	4	2	3	6	3	4	12	3	5	15	4	7	28
→→→↓↓↓→→↓↓	x,x,x x,x	x,x,x x,x	1	2	2	4	3	3	9	4	4	16	5	5	25	7	7	49
→→↓↓↓→→→↓↓	x,x,x x,x	x,x,x x,x	1	2	2	4	3	3	9	4	4	16	5	5	25	7	7	49
→↓↓↓→→→→↓↓	x,x,x,x x	x,x,x x,x	1	2	2	4	2	3	6	3	4	12	3	5	15	4	7	28
→→→→↓↓→↓↓↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16
→→→↓↓→→↓↓↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→→↓↓→→→↓↓↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→↓↓→→→→↓↓↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16
→→→→↓→↓↓↓↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16
→→→↓→→↓↓↓↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→→↓→→→↓↓↓↓	x,x,x x,x	x,x,x,x x	1	2	2	4	3	2	6	4	3	12	5	3	15	7	4	28
→↓→→→→↓↓↓↓	x,x,x,x x	x,x,x,x x	1	2	2	4	2	2	4	3	3	9	3	3	9	4	4	16

→→→↓↓↓↓→↓→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52
→→↓↓↓↓→→↓→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→→↓↓↓↓→↓→→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→↓↓↓↓→→→↓→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52
→↓↓↓↓→→↓→→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→↓↓↓↓→↓→→→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52
→→→↓↓↓→↓↓→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→→↓↓↓→→↓↓→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→→↓↓↓→↓↓→→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→↓↓↓→→→↓↓→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→↓↓↓→→↓↓→→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→↓↓↓→↓↓→→→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→→→↓↓→↓↓↓→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→→↓↓→→↓↓↓→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→→↓↓→↓↓↓→→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→↓↓→→→↓↓↓→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→↓↓→→↓↓↓→→	x,x x,x x	x,x,x x,x	1	3	2	6	5	3	15	8	4	32	11	5	55	18	7	126
→↓↓→↓↓↓→→→	x,x,x x x	x,x,x x,x	1	3	2	6	4	3	12	7	4	28	8	5	40	13	7	91
→→→↓→↓↓↓↓→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52
→→↓→→↓↓↓↓→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→→↓→↓↓↓↓→→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→↓→→→↓↓↓↓→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52
→↓→→↓↓↓↓→→	x,x x,x x	x,x,x,x x	1	3	2	6	5	2	10	8	3	24	11	3	33	18	4	72
→↓→↓↓↓↓→→→	x,x,x x x	x,x,x,x x	1	3	2	6	4	2	8	7	3	21	8	3	24	13	4	52

→→→↓↓↓→↓→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→→↓↓↓→→↓→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→→↓↓↓→↓→→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓↓↓→→→↓→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→↓↓↓→→↓→→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓↓↓→↓→→→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→→→↓↓→↓↓→↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→↓↓→→↓↓→↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→→↓↓→↓↓→→↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓↓→→→↓↓→↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→↓↓→→↓↓→→↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓↓→↓↓→→→↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→→↓↓→↓→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→↓↓→→↓→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→→↓↓→↓→→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓↓→→→↓→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→↓↓→→↓→→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓↓→↓→→→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→→↓→↓↓↓→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→→↓→→↓↓↓→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→→↓→↓↓↓→→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓→→→↓↓↓→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→↓→→↓↓↓→→↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓→↓↓↓→→→↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→→→↓→↓↓→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→↓→→↓↓→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→→↓→↓↓→→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓→→→↓↓→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→↓→→↓↓→→↓↓	x,x x,x x	x,x x,x x	1	3	3	9	5	5	25	8	8	64	11	11	121	18	18	324
→↓→↓↓→→→↓↓	x,x,x x x	x,x x,x x	1	3	3	9	4	5	20	7	8	56	8	11	88	13	18	234
→→→↓→↓→↓↓↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→→↓→→↓→↓↓↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→→↓→↓→→↓↓↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓→→→↓→↓↓↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169
→↓→→↓→→↓↓↓	x,x x,x x	x,x,x x x	1	3	3	9	5	4	20	8	7	56	11	8	88	18	13	234
→↓→↓→→→↓↓↓	x,x,x x x	x,x,x x x	1	3	3	9	4	4	16	7	7	49	8	8	64	13	13	169

→→↓↓↓→↓→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓↓↓→→↓→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓↓↓→↓→→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓↓↓→↓→↓→→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→→↓↓→↓↓→↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→→↓↓→↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→↓↓→→↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→↓↓→↓→→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→→↓→↓↓↓→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→→↓↓↓→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→↓↓↓→→↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→↓↓↓→↓→→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→→↓↓→↓→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→→↓→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→↓→→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓↓→↓→↓↓→→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→→↓→↓↓→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓→→↓↓→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓→↓↓→→↓↓→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→↓→↓↓→↓↓→→	x,x x x x	x,x x,x x	1	4	3	12	7	5	35	13	8	104	15	11	165	33	18	594
→→↓→↓→↓↓↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→→↓→↓↓↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→↓→→↓↓↓→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429
→↓→↓→↓↓↓→→	x,x x x x	x,x,x x x	1	4	3	12	7	4	28	13	7	91	15	8	120	33	13	429

→→↓↓→↓→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓↓→→↓→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓↓→↓→→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓↓→↓→↓→→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→→↓→↓↓→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→→↓↓→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓↓→→↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓↓→↓→→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→→↓→↓→↓↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→→↓→↓↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓→→↓↓→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓→↓↓→→↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→→↓→↓→↓→↓↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→→↓→↓→↓↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓→→↓→↓↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089
→↓→↓→↓→→↓↓	x,x x x x	x,x x x x	1	4	4	16	7	7	49	13	13	169	15	15	225	33	33	1089

→↓↓→↓→↓→↓→	x x x x x	x,x x x x	1	5	4	20	10	7	70	20	13	260	30	15	450	60	33	1980
→↓→↓↓→↓→↓→	x x x x x	x,x x x x	1	5	4	20	10	7	70	20	13	260	30	15	450	60	33	1980
→↓→↓→↓↓→↓→	x x x x x	x,x x x x	1	5	4	20	10	7	70	20	13	260	30	15	450	60	33	1980
→↓→↓→↓→↓↓→	x x x x x	x,x x x x	1	5	4	20	10	7	70	20	13	260	30	15	450	60	33	1980

→↓→↓→↓→↓→↓	x x x x x	x x x x x	1	5	5	25	10	10	100	20	20	400	30	30	900	60	60	3600

			126			1190			3020			9336			14122			52356

			252			2380			6040			18672			28244			104712

			$\beth (5)$	$\beth (4,1)$			$\beth (3,2)$			$\beth (3,1,1)$			$\beth (2,2,1)$			$\beth (2,1,1,1)$

counting subsequences[edit]

subsequence details

Let $p,{\mbox{ }}q,{\mbox{ }}r$ be some prime numbers where $2<p<q<r$ .

$\beth (2,1)$	x,x,x	1	x,x x	2	x x x	3
	2,2,p		2,2 p 2,p 2		2 2 p 2 p 2 p 2 2

$\beth (3,1)$	x,x,x,x	1	x,x,x x	2	x,x x,x	2	x,x x x	3	x x x x	4
	2,2,2,p		2,2,2 p 2,2,p 2		2,2 2,p 2,p 2,2		2,2 2 p 2,2 p 2 2,p 2 2		2 2 2 p 2 2 p 2 2 p 2 2 p 2 2 2
$\beth (2,2)$	x,x,x,x	1	x,x,x x	2	x,x x,x	3	x,x x x	4	x x x x	6
	2,2,p,p		2,2,p p 2,p,p 2		2,2 p,p 2,p 2,p p,p 2,2		2,2 p p 2,p 2 p 2,p p 2 p,p 2 2		2 2 p p 2 p 2 p 2 p p 2 p 2 2 p p 2 p 2 p p 2 2
$\beth (2,1,1)$	x,x,x,x	1	x,x,x x	3	x,x x,x	4	x,x x x	7	x x x x	12
	2,2,p,q		2,2,p q 2,2,q p 2,p,q 2		2,2 p,q 2,p 2,q 2,q 2,p p,q 2,2		2,2 p q 2,2 q p 2,p 2 q 2,p q 2 2,q 2 p 2,q p 2 p,q 2 2		2 2 p q 2 2 q p 2 p 2 q 2 p q 2 2 q 2 p 2 q p 2 p 2 2 q p 2 q 2 p q 2 2 q 2 2 p q 2 p 2 q p 2 2

$\beth (4,1)$	x,x,x,x,x	1	x,x,x,x x	2	x,x,x x,x	2	x,x,x x x	3	x,x x,x x	3	x,x x x x	4	x x x x x	5
	2,2,2,2,p		2,2,2,2 p 2,2,2,p 2		2,2,2 2,p 2,2,p 2,2		2,2,2 2 p 2,2,2 p 2 2,2,p 2 2		2,2 2,2 p 2,2 2,p 2 2,p 2,2 2		2,2 2 2 p 2,2 2 p 2 2,2 p 2 2 2,p 2 2 2		2 2 2 2 p 2 2 2 p 2 2 2 p 2 2 2 p 2 2 2 p 2 2 2 2
$\beth (3,2)$	x,x,x,x,x	1	x,x,x,x x	2	x,x,x x,x	3	x,x,x x x	4	x,x x,x x	5	x,x x x x	7	x x x x x	10
	2,2,2,p,p		2,2,2,p p 2,2,p,p 2		2,2,2 p,p 2,2,p 2,p 2,p,p 2,2		2,2,2 p p 2,2,p 2 p 2,2,p p 2 2,p,p 2 2		2,2 2,p p 2,2 p,p 2 2,p 2,2 p 2,p 2,p 2 p,p 2,2 2		2,2 2 p p 2,2 p 2 p 2,2 p p 2 2,p 2 2 p 2,p 2 p 2 2,p p 2 2 p,p 2 2 2		2 2 2 p p 2 2 p 2 p 2 2 p p 2 2 p 2 2 p 2 p 2 p 2 2 p p 2 2 p 2 2 2 p p 2 2 p 2 p 2 p 2 2 p p 2 2 2
$\beth (3,1,1)$	x,x,x,x,x	1	x,x,x,x x	3	x,x,x x,x	4	x,x,x x x	7	x,x x,x x	8	x,x x x x	13	x x x x x	20
	2,2,2,p,q		2,2,2,p q 2,2,2,q p 2,2,p,q 2		2,2,2 p,q 2,2,p 2,q 2,2,q 2,p 2,p,q 2,2		2,2,2 p q 2,2,2 q p 2,2,p 2 q 2,2,p q 2 2,2,q 2 p 2,2,q p 2 2,p,q 2 2		2,2 2,p q 2,2 2,q p 2,2 p,q 2 2,p 2,2 q 2,p 2,q 2 2,q 2,2 p 2,q 2,p 2 p,q 2,2 2		2,2 2 p q 2,2 2 q p 2,2 p 2 q 2,2 p q 2 2,2 q 2 p 2,2 q p 2 2,p 2 2 q 2,p 2 q 2 2,p q 2 2 2,q 2 2 p 2,q 2 p 2 2,q p 2 2 p,q 2 2 2		2 2 2 p q 2 2 2 q p 2 2 p 2 q 2 2 p q 2 2 2 q 2 p 2 2 q p 2 2 p 2 2 q 2 p 2 q 2 2 p q 2 2 2 q 2 2 p 2 q 2 p 2 2 q p 2 2 p 2 2 2 q p 2 2 q 2 p 2 q 2 2 p q 2 2 2 q 2 2 2 p q 2 2 p 2 q 2 p 2 2 q p 2 2 2
$\beth (2,2,1)$	x,x,x,x,x	1	x,x,x,x x	3	x,x,x x,x	5	x,x,x x x	8	x,x x,x x	11	x,x x x x	15	x x x x x	30
	2,2,p,p,q		2,2,p,p q 2,2,p,q p 2,p,p,q 2		2,2,p p,q 2,2,q p,p 2,p,p 2,q 2,p,q 2,p p,p,q 2,2		2,2,p p q 2,2,p q p 2,2,q p p 2,p,p 2 q 2,p,p q 2 2,p,q 2 p 2,p,q p 2 p,p,q 2 2		2,2 p,p q 2,2 p,q p 2,p 2,p q 2,p 2,q p 2,p p,q 2 2,q 2,p p 2,q p,p 2 p,p 2,2 q p,p 2,q 2 p,q 2,2 p p,q 2,p 2		2,2 p p q 2,2 p q p 2,2 q p p 2,p 2 p q 2,p 2 q p 2,p p 2 q 2,p p q 2 2,p q 2 p 2,p q p 2 2,q 2 p p 2,q p 2 p 2,q p p 2 p,q 2 2 p p,q 2 p 2 p,q p 2 2		2 2 p p q 2 2 p q p 2 2 q p p 2 p 2 p q 2 p 2 q p 2 p p 2 q 2 p p q 2 2 p q 2 p 2 p q p 2 2 q 2 p p 2 q p 2 p 2 q p p 2 p 2 2 p q p 2 2 q p p 2 p 2 q p 2 p q 2 p 2 q 2 p p 2 q p 2 p p 2 2 q p p 2 q 2 p p q 2 2 p q 2 2 p p q 2 p 2 p q p 2 2 q 2 2 p p q 2 p 2 p q 2 p p 2 q p 2 2 p q p 2 p 2 q p p 2 2
$\beth (2,1,1,1)$	x,x,x,x,x	1	x,x,x,x x	4	x,x,x x,x	7	x,x,x x x	13	x,x x,x x	18	x,x x x x	33	x x x x x	60
	2,2,p,q,r		2,2,p,q r 2,2,p,r q 2,2,q,r p 2,p,q,r 2		2,2,p q,r 2,2,q p,r 2,2,r p,q 2,p,q 2,r 2,p,r 2,q 2,q,r 2,p p,q,r 2,2		2,2,p q r 2,2,p r q 2,2,q p r 2,2,q r p 2,2,r p q 2,2,r q p 2,p,q 2 r 2,p,q r 2 2,p,r 2 q 2,p,r q 2 2,q,r 2 p 2,q,r p 2 p,q,r 2 2		2,2 p,q r 2,2 p,r q 2,2 q,r p 2,p 2,q r 2,p 2,r q 2,p q,r 2 2,q 2,p r 2,q 2,r p 2,q p,r 2 2,r 2,p q 2,r 2,q p 2,r p,q 2 p,q 2,2 r p,q 2,r 2 p,r 2,2 q p,r 2,q 2 q,r 2,2 p q,r 2,p 2		2,2 p q r 2,2 p r q 2,2 q p r 2,2 q r p 2,2 r p q 2,2 r q p 2,p 2 q r 2,p 2 r q 2,p q 2 r 2,p q r 2 2,p r 2 q 2,p r q 2 2,q 2 p r 2,q 2 r p 2,q p 2 r 2,q p r 2 2,q r 2 p 2,q r p 2 2,r 2 p q 2,r 2 q p 2,r p 2 q 2,r p q 2 2,r q 2 p 2,r q p 2 p,q 2 2 r p,q 2 r 2 p,q r 2 2 p,r 2 2 q p,r 2 q 2 p,r q 2 2 q,r 2 2 p q,r 2 p 2 q,r p 2 2		2 2 p q r 2 2 p r q 2 2 q p r 2 2 q r p 2 2 r p q 2 2 r q p 2 p 2 q r 2 p 2 r q 2 p q 2 r 2 p q r 2 2 p r 2 q 2 p r q 2 2 q 2 p r 2 q 2 r p 2 q p 2 r 2 q p r 2 2 q r 2 p 2 q r p 2 2 r 2 p q 2 r 2 q p 2 r p 2 q 2 r p q 2 2 r q 2 p 2 r q p 2 p 2 2 q r p 2 2 r q p 2 q 2 r p 2 q r 2 p 2 r 2 q p 2 r q 2 p q 2 2 r p q 2 r 2 p q r 2 2 p r 2 2 q p r 2 q 2 p r q 2 2 q 2 2 p r q 2 2 r p q 2 p 2 r q 2 p r 2 q 2 r 2 p q 2 r p 2 q p 2 2 r q p 2 r 2 q p r 2 2 q r 2 2 p q r 2 p 2 q r p 2 2 r 2 2 p q r 2 2 q p r 2 p 2 q r 2 p q 2 r 2 q 2 p r 2 q p 2 r p 2 2 q r p 2 q 2 r p q 2 2 r q 2 2 p r q 2 p 2 r q p 2 2