[seqfan] Re: Max Locations after Smoothing - recurrences

Ron Hardin rhhardin at att.net
Thu Jan 24 21:40:29 CET 2013


The idea started based on a long-ago wondering how fast you could get zero 
crossings of a band-limited function.  It seemed that there was no limit but the 
energy in the crossings would have to go exponentially small.

Here increasing k amounts to supplying additional energy for fast peaks, if that 
idea is right.

Anyway...

Some higher level 1,1 convolution kernels of less recurrence simplicity


/tmp/cxx
T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima 
of a random length-n 0..k array extended with zeros and convolved with 1,2,1

Table starts
...2...2...2....2....2....2....2....2....2....2....2...2...2...2...2...2...2..2..2..2

...3...3...3....3....3....3....3....3....3....3....3...3...3...3...3...3...3..3..3..3

...4...4...4....4....4....4....4....4....4....4....4...4...4...4...4...4...4..4..4..4

...6...6...7....8....8....8....8....8....8....8....8...8...8...8...8...8...8..8..8..8

...9..10..11...12...12...12...12...12...12...12...12..12..12..12..12..12..12.12.12.12

..13..15..17...19...19...19...19...20...21...21...21..21..21..21..21..21..21.21.21.21

..19..24..27...31...31...31...31...32...33...33...33..33..33..33..33..33..33.33.33.33

..28..38..42...48...48...49...49...51...53...53...53..53..53..53..54..55..55.55.55...

..41..59..66...79...79...80...80...83...86...86...86..86..86..86..87..88..88.88......

..60..92.104..126..126..128..128..132..136..137..138.138.138.138.140.142.142.........

..88.144.163..200..201..207..207..215..224..224..224.224.224.224.227.230.............

.129.224.256..322..323..334..334..346..360..360..360.360.360.360.....................

.189.350.402..513..516..536..536..558..584..584..584.585.585.........................

.277.547.631..823..827..866..866..900..943..943..943.945.............................

.406.854.991.1318.1324.1394.1394.1449.1520.1522.1524.................................


Empirical for column k:
k=1: a(n)=a(n-1)+a(n-3)
k=2: a(n)=a(n-1)+a(n-3)+a(n-6)+a(n-8)
k=3: a(n)=a(n-1)+a(n-3)+a(n-5)
k=4: a(n)=a(n-1)+a(n-3)+2*a(n-5)-a(n-6)
k=5: a(n)=a(n-1)+a(n-3)+a(n-5)+a(n-8)+a(n-10)+2*a(n-12)-a(n-13)
k=6: a(n)=a(n-1)+a(n-3)+a(n-5)+2*a(n-7)-a(n-8)-a(n-14)+a(n-15)
k=7: 
a(n)=a(n-1)+a(n-3)+a(n-5)+3*a(n-7)-2*a(n-8)-a(n-10)-a(n-12)-2*a(n-14)+a(n-15)

/tmp/cyb
T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima 
of a random length-n 0..k array extended with zeros and convolved with 1,3,3,1

Table starts
...2...2...2...2...2..2..2..2..2..2.2.2.2.2
...3...3...3...3...3..3..3..3..3..3.3.3.3..
...4...4...4...4...4..4..4..4..4..4.4.4....
...6...6...7...7...7..7..7..7..7..7.7......
...9..10..11..11..11.11.11.11.11.11........
..13..15..16..16..17.17.17.17.17...........
..19..23..26..26..28.28.28.28..............
..28..36..41..42..44.44.44.................
..41..55..62..65..70.70....................
..60..84..99.105.113.......................
..88.129.156.168...........................
.129.198.242...............................
.189.303...................................
.277.......................................

Empirical for column k:
k=1: a(n)=a(n-1)+a(n-3)
k=2: a(n)=a(n-1)+a(n-3)+a(n-7)+a(n-9)
k=3: a(n)=a(n-1)+2*a(n-3)-a(n-4)+a(n-8)-a(n-9)+a(n-11)-a(n-12)-2*a(n-14)+a(n-15)
k=4: a(n)=a(n-1)+a(n-2)+2*a(n-3)-3*a(n-4)-2*a(n-5)+ [omitted terms] 
-3*a(n-48)-a(n-49)+a(n-51)

/tmp/cyc
T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima 
of a random length-n 0..k array extended with zeros and convolved with 1,4,6,4,1

Table starts
...2...2...2...2...2..2..2..2..2..2.2.2.2.2
...3...3...3...3...3..3..3..3..3..3.3.3.3..
...4...4...4...4...4..4..4..4..4..4.4.4....
...6...6...6...6...6..7..7..7..7..7.8......
..10..10..10..10..10.11.11.11.11.11........
..15..15..15..15..15.16.16.16.16...........
..22..22..22..24..25.27.27.27..............
..35..36..36..39..40.43.43.................
..54..56..56..60..61.65....................
..81..84..84..96.100.......................
.125.133.135.154...........................
.193.208.211...............................
.294.318...................................
.450.......................................

Empirical for column k:
k=1: a(n)=a(n-1)+2*a(n-3)-a(n-4)-a(n-6)+a(n-7)
k=2: a(n)=a(n-1)+3*a(n-3)-2*a(n-4)-3*a(n-6)+2*a(n-7)+a(n-9)
k=3: a(n)=a(n-1)+a(n-3)+a(n-6)+3*a(n-9)-2*a(n-10)- [omitted terms] 
+3*a(n-28)-2*a(n-30)+a(n-31)


 rhhardin at mindspring.com
rhhardin at att.net (either)



----- Original Message ----

> (Other  convolution kernels don't seem to come out so simple.)



More information about the SeqFan mailing list