# [seqfan] Re: Closed form from recurrence? Isosceles right triangles on a grid

Ron Hardin rhhardin at att.net
Fri Apr 29 12:30:24 CEST 2011

```Curiously, the same final recurrence
a(n)=4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6)

works for the diagonal.

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

----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: seqfan at list.seqfan.eu
> Sent: Thu, April 28, 2011 6:01:50 PM
> Subject: [seqfan] Closed form from recurrence? Isosceles right triangles on a
>grid
>
> T(n,k)=Number of isosceles right triangles on a (n+1)X(k+1) grid
>
> Table  starts
> ..4..10..16..22...28...34...40...46...52...58...64...70...76....82....88....94
> .10..28..50..74...98..122..146..170..194..218..242..266..290...314...338...362
> .16..50..96.150..208..268..328..388..448..508..568..628..688...748...808...868
> .22..74.150.244..350..464..582..702..822..942.1062.1182.1302..1422..1542..1662
> .28..98.208.350..516..700..896.1100.1308.1518.1728.1938.2148..2358..2568..2778
> .34.122.268.464..700..968.1260.1570.1892.2222.2556.2892.3228..3564..3900..4236
> .40.146.328.582..896.1260.1664.2100.2560.3038.3528.4026.4528..5032..5536..6040
> .46.170.388.702.1100.1570.2100.2680.3300.3952.4628.5322.6028..6742..7460..8180
> .52.194.448.822.1308.1892.2560.3300.4100.4950.5840.6762.7708..8672..9648.10632
> .58.218.508.942.1518.2222.3038.3952.4950.6020.7150.8330.9550.10802.12078.13372
>
>
> Empirical:  a(n) = k*(k+1)*(k+2)*n + b(k) for n>2*k-2
> k=1: a(n) = 6*n - 2
> k=2:  a(n) = 24*n - 22 for n>2
> k=3: a(n) = 60*n - 92 for n>4
> k=4: a(n) =  120*n - 258 for n>6
> k=5: a(n) = 210*n - 582 for n>8
> k=6: a(n) =  336*n - 1140 for n>10
> k=7: a(n) = 504*n - 2024 for n>12
> k=8: a(n) =  720*n - 3340 for n>14
> k=9: a(n) = 990*n - 5210 for n>16
> k=10: a(n) =  1320*n - 7770 for n>18
> k=11: a(n) = 1716*n - 11172 for n>20
> k=12:  a(n) = 2184*n - 15582 for n>22
> k=13: a(n) = 2730*n - 21182 for  n>24
> k=14: a(n) = 3360*n - 28168 for n>26
>
> This is  k*(k+1)*(k+2)*n - b(k) for some b(k), and if b(k) is known then T(n,k)
>
> is  known at least outside the k,2*k-2 boundary.
>
> Computing the value of  T(2*k-1,k)=T(k,2*k-1) to get at b(k)
>
> Number of isosceles right triangles  on a 2nX(n+1) grid
> a(n)=4 50 208 582 1308 2556 4528 7460 11620 17310 24864  34650 47068 62552
>81568
>
> 104616 132228 164970 203440 248270
>
> Empirical:  a(n)=4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6)
>
> The question is what's a  closed form for a(n).
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
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>
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>

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