# [seqfan] Re: Guess the formula T(n,k)=?

Susanne Wienand susanne.wienand at gmail.com
Tue Feb 17 23:05:43 CET 2015

```Hello Ron,

The terms seem to be related to A000930, Narayana's cow sequence.
If decreased by 22 and then divided by 3, the terms become

.1/3....0....3....7....13....22..35..54..82..123..183..271..400..589..866..1272..1867..2739

..0.....2....5....9....15....24..37..56..84..125..185..273..402..591..868..1274..1869..2741

..3.....5....8...12....18....27..40..59..87..128..188..276..405..594..871..1277..1872..2744

..7.....9...12...16....22....31..44..63..91..132..192..280..409..598..875..1281..1876..2748

.13....15...18...22....28....37..50..69..97..138..198..286..415..604..881..1287..1882..2754

.22....24...27...31....37....46..59..78.106..147..207..295..424..613..890..1296..1891..2763

.35....37...40...44....50....59..72..91.119..160..220..308..437..626..903..1309..1904..2776

.54....56...59...63....69....78..91.110.138..179..239..327..456..645..922..1328..1923..2795

.82....84...87...91....97...106.119.138.166..207..267..355..484..673..950..1356..1951..2823

123...125..128..132...138...147.160.179.207..248..308..396..525..714..991..1397..1992..2864

Most differences (except upper left corner) along the columns and the rows
seem to match with A000930:
*2*, *3*, *4*, *6*, *9*, *13*, *19*, *28*, 41, 60, 88, 129, 189, 277, 406,
595, 872, 1278

T(2,2) - T(2,1) = 2 - 0 = 2
T(2,3) - T(2,2) = 5 - 2 = 3
T(2,4) - T(2,3) = 9 - 5 = 4
...

I hope this can help to find the formula.

Regards
Susanne

2015-02-16 17:22 GMT+01:00 Ron Hardin <rhhardin at att.net>:

> This is likely to be T(n,k)=f(n)+f(k) with f() having a 2^() flavor.
>
>
> Every row, column and the diagonal satisfy the same recurrence.
>
> The solutions seem to pick every 3rd row or column for 1's and sometimes
> add cross rows or columns of 1's.
>
>
> /tmp/eyg
> T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with no 3x3 subblock diagonal sum
> 0 and no antidiagonal sum 0 and no row sum 2 and no column sum 2
>
> Table starts
>
> ..23..22..31..43..61..88.127.184.268.391.571..835.1222.1789.2620.3838.5623.8239
>
> ..22..28..37..49..67..94.133.190.274.397.577..841.1228.1795.2626.3844.5629.8245
>
> ..31..37..46..58..76.103.142.199.283.406.586..850.1237.1804.2635.3853.5638.8254
>
> ..43..49..58..70..88.115.154.211.295.418.598..862.1249.1816.2647.3865.5650.8266
>
> ..61..67..76..88.106.133.172.229.313.436.616..880.1267.1834.2665.3883.5668.8284
>
> ..88..94.103.115.133.160.199.256.340.463.643..907.1294.1861.2692.3910.5695.8311
>
> .127.133.142.154.172.199.238.295.379.502.682..946.1333.1900.2731.3949.5734.8350
>
> .184.190.199.211.229.256.295.352.436.559.739.1003.1390.1957.2788.4006.5791.8407
>
> .268.274.283.295.313.340.379.436.520.643.823.1087.1474.2041.2872.4090.5875.8491
>
> .391.397.406.418.436.463.502.559.643.766.946.1210.1597.2164.2995.4213.5998.8614
>
> Empirical for diagonal:
> a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4) for n>5
> Empirical for column k:
> k=1: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4) for n>5
> k=2: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
> k=3: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
> k=4: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
> k=5: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
> k=6: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
> k=7: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
>
> Two solutions for n=8 k=8
> ..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
> ..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....1..1..1..1..1..1..1..1..1..1
> ..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....1..1..1..1..1..1..1..1..1..1
> ..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
> ..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
>
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
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>
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>
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