# [seqfan] Re: A269254

Don Reble djr at nk.ca
Sun Oct 22 23:15:40 CEST 2017

```> could a(110) be -1?

It could!
Trisect sequence-110 to get:

s(0) -1 = -1 * 1
s(3) 12209 = 29 * 421
s(6) 16246150031 = 3191 * 5091241
s(9) 21618264461738561 = 350981 * 61593831181
s(12) 28766775971285404815839 = 38604719 * 745162164534481
s(15) 38279085781688731361830743569 = 4246168109 * 9014971804944317941
s(18) 50936831077090977385276030140145391
= 467039887271 * 109063128151054193913721
s(21) 67780113009314371791483566295225436698401
= 51370141431701 * 1319445715356481833023876701
s(24) 90192962978053418280696346184682355821321113279
= 5650248517599839 * 15962654155319589064868666412961

So far, this subsequence is the product of a Lucas sequence
[t(n+2) = 110 t(n+1) - t(n)] and an integer sequence.

s(1) 1 = 1 * 1
s(4) 1342879 = 139 * 9661
s(7) 1786928798929 = 15289 * 116876761
s(10) 2377812544869509551 = 1681651 * 1413975042901
s(13) 3164083819079723345430241 = 184966321 * 17106269952137521
s(16) 4210351415532437651518789281919 = 20344613659 * 206951652466984684141
s(19) 5602588318103384725927427610425725489
= 2237722536169 * 2503701074439310756598281
s(22) 7455196197246420601834317660713681347165711
= 246129134364931 * 30289775391615129066341317381
s(25) 9920405923784291913924768096855946930622570930881
= 27071967057606241 * 366445700184058757005286501075041

... product of a Lucas sequence [t(n+2) = 110 t(n+1) - t(n)]
and an integer sequence.

s(2) 111 = 111 * 1
s(5) 147704481 = 111 * 1330671
s(8) 196545921732159 = 111 * 1770683979569
s(11) 261537761671184312049 = 111 * 2356196051091750559
s(14) 348020453322798282592510671 = 111 * 3135319399304489032364961
s(17) 463100376622786452935704990267521
= 111 * 4172075465070148224645990903311
s(20) 616233778160295228874631761116689658399
= 111 * 5551655659101758818690376226276483409
s(23) 820003801584096951829983459112209722751529809
= 111 * 7387421635892765331801652784794682186950719
s(26) 1091154458653294057113443794307969480012661481283631
= 111 * 9830220348227874388409403552324049369483436768321

... multiples of 111. (But there's another factorization.)

--
Don Reble  djr at nk.ca

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