# [seqfan] Re: No isosceles triangles in a square grid, continued

Ron Hardin rhhardin at att.net
Tue Apr 26 13:29:25 CEST 2016

```Having 177 terms now,  an order 72 linear recurrence for column 3 turns up

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2)
k=2: a(n) = a(n-1) +3*a(n-2) -2*a(n-3) -3*a(n-4) +a(n-5) +a(n-6)
k=3: a(n) = 6*a(n-1) -15*a(n-2) +26*a(n-3) -30*a(n-4) -6*a(n-5) +71*a(n-6) -144*a(n-7) +168*a(n-8) +94*a(n-9) -469*a(n-10) +652*a(n-11) -378*a(n-12) -1158*a(n-13) +2908*a(n-14) -3158*a(n-15) +1005*a(n-16) +4842*a(n-17) -10378*a(n-18) +8764*a(n-19) +1617*a(n-20) -16040*a(n-21) +24082*a(n-22) -14472*a(n-23) -15317*a(n-24) +45188*a(n-25) -49946*a(n-26) +26510*a(n-27) +22275*a(n-28) -82222*a(n-29) +110930*a(n-30) -74226*a(n-31) -14272*a(n-32) +113348*a(n-33) -178337*a(n-34) +141136*a(n-35) +7266*a(n-36) -149432*a(n-37) +185895*a(n-38) -129140*a(n-39) +22038*a(n-40) +101372*a(n-41) -166288*a(n-42) +124584*a(n-43) -34773*a(n-44) -55018*a(n-45) +132927*a(n-46) -129638*a(n-47) +12757*a(n-48) +101176*a(n-49) -91095*a(n-50) +8640*a(n-51) +38603*a(n-52) -32838*a(n-53) +3964*a(n-54) +15224*a(n-55) -10065*a(n-56) -5948*a(n-57) +8968*a(n-58) +140*a(n-59) -4048*a(n-60) +2312*a(n-61) -236*a(n-62) -772*a(n-63) +628*a(n-64) -48*a(n-65) -252*a(n-66) -32*a(n-67) +108*a(n-68) -8*a(n-69) +12*a(n-70) +16*a(n-71) -16*a(n-72)

The small coefficients at each end perhaps indicate another (-1)^n  type formula underneath.
rhhardin at mindspring.com rhhardin at att.net (either)

From: Chris <cgribble263 at btinternet.com>
To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
Sent: Monday, April 25, 2016 5:53 PM
Subject: [seqfan] Re: No isosceles triangles in a square grid, continued

Also, for k = 2, a(n) = a(n-1) + a(n-2) + (n+1)(3-(-1)^(n+1))/4

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Ron Hardin
Sent: Sunday, April 24, 2016 23:42
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: No isosceles triangles in a square grid, continued

T(n,k)=Number of nXk 0..1 arrays with exactly n+k-2 having value 1 and no three 1s forming an isosceles right triangle

..1....2.....3......4......5......6......7......8......9
..2....6....10.....21.....34.....62....100....171....276
..3...10....24.....24....107....236....499...1228...2753
..4...21....24.....60....210....637...1840...5792..18556
..5...34...107....210....768...1898...8211..37402.192579
..6...62...236....637...1898...7468..26052.138476.831738
..7..100...499...1840...8211..26052.131056.648178.......
..8..171..1228...5792..37402.138476.648178..............
..9..276..2753..18556.192579.831738.....................
.10..458..6292..54034.635086............................
.11..740.14751.160246...................................
.12.1211.34824..........................................
.13.1958................................................
.14.....................................................
So far only cols 1 and 2 have recurrences Empirical for column k:
k=1: a(n)=2*a(n-1)-a(n-2)
k=2: a(n)=a(n-1)+3*a(n-2)-2*a(n-3)-3*a(n-4)+a(n-5)+a(n-6)
Column 3 does not look promising for a recurrence, after 94 terms.
work not double-checked! rhhardin at mindspring.com rhhardin at att.net (either)

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