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Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123.....12)*
+10
1
27, 25, 13, 24, 19, 24, 25, 13, 23, 24, 13, 12, 37, 25, 13, 24, 25, 24, 25
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...20)*
+10
1
83, 80, 41, 21, 27, 80, 81, 40, 31, 40, 81, 80, 55, 41, 21, 80, 99, 40, 21
Array read by antidiagonals: A(n,k) (n,k >= 2) is the base-n state complexity of the partitioned finite deterministic automaton (PFDA) for the periodic sequence (123..k)*.
+10
1
3, 6, 2, 7, 4, 3, 20, 8, 3, 2, 13, 20, 5, 6, 3, 21, 7, 10, 4, 4, 2, 15, 42, 7, 6, 9, 3, 3, 54, 16, 21, 12, 5, 8, 6, 2, 41, 13, 13, 42, 7, 20, 5, 4, 3, 110, 40, 27, 16, 14, 6, 20, 4, 3, 2, 27, 55, 21, 54, 23, 8, 13, 10, 9, 6, 3, 156, 25, 55, 11
COMMENTS
Rows are ultimately periodic.
FORMULA
A(n,n^k) = Sum_{i=0..k} n^i.
A(n+1,n) = n.
It also appears that A(n-1,n) = 2n.
EXAMPLE
Array begins:
3 2 3 2 3
6 4 3 6 4
7 8 5 4 9 ...
20 20 10 6 5
13 7 7 12 7
...
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123....13)*
+10
0
156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13, 156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13, 156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13, 156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13, 156, 39, 78, 52, 156, 156, 52, 39
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,1).
FORMULA
G.f.: x^2*(156 + 39*x + 78*x^2 + 52*x^3 + 156*x^4 + 156*x^5 + 52*x^6 + 39*x^7 + 78*x^8 + 156*x^9 + 26*x^10 + 14*x^11 + 13*x^12)/(1 - x^13).
MATHEMATICA
CoefficientList[Series[(156 + 39 x + 78 x^2 + 52 x^3 + 156 x^4 + 156 x^5 + 52 x^6 + 39 x^7 + 78 x^8 + 156 x^9 + 26 x^10 + 14 x^11 + 13 x^12)/(1 - x^13), {x, 0, 60}], x]
PadRight[{}, 120, {156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13}] (* Harvey P. Dale, Mar 19 2021 *)
PROG
(Magma) &cat[[156, 39, 78, 52, 156, 156, 52, 39, 78, 156, 26, 14, 13]: n in [0..10]];
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...14)*
+10
0
43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14, 43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14, 43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14, 43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14, 43, 84, 43, 84, 29, 15, 15, 42
COMMENTS
Period 14, repeat [43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14].
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1).
FORMULA
G.f.: -x^2*(43+84*x+43*x^2+84*x^3+29*x^4+15*x^5+15*x^6+42*x^7+85*x^8+42*x^9+85
*x^10+28*x^11+15*x^12+14*x^13) / ( (x-1)*(1+x^6+x^5+x^4+x^3+x^2+x)*(1+x)*(1-x+
x^2-x^3+x^4-x^5+x^6) ).
MATHEMATICA
CoefficientList[Series[(43 + 84 x + 43 x^2 + 84 x^3 + 29 x^4 + 15 x^5 + 15 x^6 + 42 x^7 + 85 x^8 + 42 x^9 + 85 x^10 + 28 x^11 + 15 x^12 + 14 x^13)/(1 - x^14), {x, 0, 60}], x]
PROG
(Magma) &cat[[43, 84, 43, 84, 29, 15, 15, 42, 85, 42, 85, 28, 15, 14]: n in [0..10]];
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...15)*
+10
0
60, 61, 30, 31, 16, 60, 60, 31, 16, 30, 61, 60, 30, 16, 15, 60, 61, 30, 31
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...16)*
+10
0
31, 64, 21, 64, 55, 32, 25, 32, 59, 64, 29, 64, 63, 32, 17, 16, 65, 64, 33
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...17)*
+10
0
136, 272, 68, 272, 272, 272, 136, 136, 272, 272, 272, 68, 272, 136, 34, 18, 17, 136, 272
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...18)*
+10
0
109, 22, 55, 108, 25, 54, 37, 19, 19, 108, 31, 54, 109, 34, 55, 36, 19, 18, 109
Base-n state complexity of partitioned deterministic finite automaton (PDFA) for the periodic sequence (123...19)*
+10
0
342, 342, 171, 171, 171, 57, 114, 171, 342, 57, 114, 342, 342, 342, 171, 171, 38, 20, 19
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