# [seqfan] Re: Number of solutions of a*b + c*d + ... + y*z = 0 (mod n)

Ron Hardin rhhardin at att.net
Wed Sep 8 14:34:04 CEST 2010

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----- Original Message ----
> From: Max Alekseyev <maxale at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Tue, September 7, 2010 5:02:29 AM
> Subject: [seqfan] Re: Number of solutions of a*b + c*d + ... + y*z = 0 (mod n)
>
> On Mon, Sep 6, 2010 at 2:28 PM, Ron Hardin <rhhardin at att.net> wrote:
>
> > Zeros  make the other half of the product free, so you'd be counting
> > 0*0 0*1  0*2 0*3 ... as distinct
>
> I don't see a problem with that.
>
> >  Instead, to pick up zeros, adding in nonzero results from fewer products
>seems
> > more natural.
>
> Perhaps, both variants should be represented  in the  OEIS.

The four versions (1..n-1 & mod=0 ; 1..n-1 & mod=1; 0..n-1 & mod=0; 0..n-1 &
mod=1)
come out

0,0,0,0,1,0,0,0,2,1,0,1,5,6,0,0,0,3,14,13,2,0,1,7,34,42,23,0,0,0,12,62,
145,119,36,3,0,1,9,118,402,522,253,62,3,0,0,15,198,995,1960,1518,526,78,
4,0,1,22,327,2294,6518,7587,4041,944,123,0,0,0,18,499,4856,19467,32893
T(n,k)=number of distinct solutions of sum{i=1..k}(x(2i-1)*x(2i)) = 0 (mod n),
with x() only in 1..n-1
Table starts
.0...0....0.....0......0.......0........0........0.........0..........0
.0...1....0.....1......0.......1........0........1.........0..........1
.0...2....5.....3......7......12........9.......15........22.........18
.1...6...14....34.....62.....118......198......327.......499........756
.0..13...42...145....402.....995.....2294.....4856......9730......18478
.2..23..119...522...1960....6518....19467....53455....136491.....327338
.0..36..253..1518...7587...32893...126864...444003...1430727....4292145
.3..62..526..4041..25470..139369...674922..2948305..11783122...43576141
.3..78..944..9150..73146..499659..2997717.16112289..78770456..354466005
.4.123.1702.19970.193036.1599064.11615754.75389116.443607864.2394458182

0,0,1,0,0,2,0,1,1,2,0,0,3,4,3,0,1,7,14,9,2,0,0,5,30,46,19,4,0,1,9,64,
142,106,31,4,0,0,15,112,400,502,254,42,4,0,1,12,198,1004,1914,1519,494,
75,3,0,0,18,318,2282,6404,7589,3828,939,91,6,0,1,26,502,4868,19300
T(n,k)=number of distinct solutions of sum{i=1..k}(x(2i-1)*x(2i)) = 1 (mod n),
with x() only in 1..n-1
Table starts
.0..0....0.....0......0.......0........0........0.........0..........0
.1..0....1.....0......1.......0........1........0.........1..........0
.2..1....3.....7......5.......9.......15.......12........18.........26
.2..4...14....30.....64.....112......198......318.......502........744
.3..9...46...142....400....1004.....2282.....4868......9721......18475
.2.19..106...502...1914....6404....19300....53133....135902.....326446
.4.31..254..1519...7589...32890...126861...444019...1430707....4292149
.4.42..494..3828..24902..137528...670058..2935676..11752870...43506488
.4.75..939..9145..73134..499641..2997685.16112283..78770466..354466019
.3.91.1528.18966.188315.1579113.11539243.75118525.442714845.2391684440

1,1,2,1,4,3,1,6,8,5,1,9,21,20,5,1,12,45,68,28,8,1,16,87,205,142,56,7,1,
20,159,549,620,355,64,11,1,25,270,1336,2338,1957,589,122,12,1,30,435,
3000,7770,9383,4507,1250,150,14,1,36,676,6302,23282,39768,28783,11171
T(n,k)=number of distinct solutions of sum{i=1..k}(x(2i-1)*x(2i)) = 0 (mod n),
with x() in 0..n-1
Table starts
..1...1....1.....1......1.......1........1.........1..........1...........1
..2...4....6.....9.....12......16.......20........25.........30..........36
..3...8...21....45.....87.....159......270.......435........676........1011
..5..20...68...205....549....1336.....3000......6302......12502.......23616
..5..28..142...620...2338....7770....23282.....63988.....163480......392306
..8..56..355..1957...9383...39768...151473....526152....1687310.....5046199
..7..64..589..4507..28783..158242...768541...3362296...13449091....49761514
.11.122.1250.11171..85956..576277..3422978..18294610...89142751...400327073
.12.150.1946.22146.213618.1770957.12879078..83656074..492505059..2659219164
.14.218.3372.45350.518268.5103172.44111410.340334735.2376231744.15184890632

0,0,1,0,2,2,0,4,7,2,0,6,18,12,3,0,9,42,50,24,2,0,12,84,166,136,31,4,0,
16,153,474,612,262,59,4,0,20,264,1200,2325,1649,583,74,4,0,25,429,2768,
7752,8468,4501,974,111,3,0,30,666,5920,23256,37264,28778,9772,1794,121
T(n,k)=number of distinct solutions of sum{i=1..k}(x(2i-1)*x(2i)) = 1 (mod n),
with x() in 0..n-1
Table starts
.0...0....0.....0......0.......0........0.........0..........0...........0
.1...2....4.....6......9......12.......16........20.........25..........30
.2...7...18....42.....84.....153......264.......429........666........1001
.2..12...50...166....474....1200.....2768......5920......11900.......22696
.3..24..136...612...2325....7752....23256.....63954.....163438......392250
.2..31..262..1649...8468...37264...145098....510927....1652886.....4972016
.4..59..583..4501..28778..158239...768537...3362288...13449080....49761505
.4..74..974..9772..79670..550860..3329262..17975676...88130750...397307654
.4.111.1794.21631.212049.1766580.12867751..83628636..492442470..2659084116
.3.121.2603.39911.484180.4912624.43149935.335900991.2357340087.15109820870

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