A263060 -id:A263060 - cp's OEIS frontend

A263053 Number of (n+1) X 2 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

2, 2, 10, 10, 42, 42, 170, 170, 682, 682, 2730, 2730, 10922, 10922, 43690, 43690, 174762, 174762, 699050, 699050, 2796202, 2796202, 11184810, 11184810, 44739242, 44739242, 178956970, 178956970, 715827882, 715827882, 2863311530, 2863311530

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Each row must be either 01 or 10. The two columns are therefore binary complements with sum 2^k-1, where k = n + 1 is the number of rows. If k is even then 2^k-1 is divisible by 3 and the number of solutions is 2*(2^k-1)/3. If k is odd then 2^k-1 == 1 (mod 3) and the number of solutions is (2^k-2)/3. - Andrew Howroyd, Feb 03 2022

			All solutions for n=4:
  0 1   0 1   1 0   1 0   1 0   0 1   1 0   1 0   0 1   0 1
  0 1   0 1   0 1   1 0   0 1   1 0   1 0   0 1   1 0   1 0
  1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1   0 1
  0 1   1 0   0 1   0 1   1 0   1 0   1 0   0 1   1 0   0 1
  1 0   0 1   1 0   1 0   1 0   0 1   0 1   0 1   1 0   0 1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 1 of A263060.

Cf. A052992.

[int(2**n - 2/3 -((-2)**n)/3) for n in range(1,40)] # Pascal Bisson, Feb 03 2022

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
From Colin Barker, Jan 01 2019: (Start)
G.f.: 2*x / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 2^n - 2/3 - (-2)^n/3.
(End)
a(n) = 2*A052992(n). - Pascal Bisson, Feb 03 2022

A263052 Number of (n+1)X(n+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

2, 33, 1666, 498597, 439260642, 1825928130193, 26380510465997186, 1607516667474074649381, 373842629115554331580791202, 353418497958270475251697468005873

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Diagonal of A263060.

			Some solutions for n=3
..1..0..1..1....1..0..0..0....0..1..0..1....1..0..1..1....1..0..1..1
..0..1..0..0....0..1..1..1....0..0..1..0....0..0..0..1....0..1..0..1
..1..0..1..1....1..0..1..0....0..1..0..1....1..0..1..1....0..1..1..1
..0..0..1..0....1..0..1..1....1..1..1..0....1..1..1..0....0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..11

Cf. A263060.

A263054 Number of (n+1) X (2+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

2, 33, 142, 895, 4314, 22921, 113486, 577071, 2877562, 14455993, 72225582, 361607935, 1807659674, 9041669481, 45205690126, 226052092111, 1130241870522, 5651375017753, 28256745002222, 141284885366175, 706423516399834

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

			Some solutions for n=4:
..1..0..1....0..0..1....1..0..0....1..0..0....1..0..1....1..0..1....1..0..0
..1..1..1....1..0..0....0..1..0....0..1..0....1..0..1....0..0..1....1..0..1
..1..1..1....1..1..1....1..1..1....0..1..0....0..0..1....1..1..1....0..1..0
..1..1..1....0..0..1....1..0..0....0..1..0....0..1..0....1..0..0....0..0..1
..1..0..1....1..1..1....0..1..0....1..0..1....1..0..0....0..0..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 2 of A263060.

Empirical: a(n) = 5*a(n-1) + 12*a(n-2) - 60*a(n-3) - 39*a(n-4) + 195*a(n-5) + 28*a(n-6) - 140*a(n-7).

Empirical g.f.: x*(2 + 23*x - 47*x^2 - 91*x^3 + 193*x^4 + 28*x^5 - 140*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 5*x)*(1 - 7*x^2)). - Colin Barker, Jan 01 2019

A263055 Number of (n+1)X(3+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

10, 142, 1666, 18390, 188370, 1941702, 19499266, 196698070, 1968558130, 19732383462, 197355588066, 1974792933750, 19748600169490, 197518001324422, 1975194815632066, 19752765325169430, 197527991815422450

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Column 3 of A263060.

			Some solutions for n=4
..1..0..1..0....1..1..1..0....1..1..0..1....1..1..0..1....1..0..1..1
..1..1..0..1....0..1..1..1....0..0..1..0....0..0..0..1....0..1..0..1
..1..0..1..0....0..1..1..1....0..0..0..1....1..0..1..1....0..0..1..0
..0..0..1..0....0..0..0..1....1..0..1..0....1..0..0..0....1..1..0..1
..1..0..0..0....1..0..1..0....1..0..0..0....1..0..1..1....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A263060.

Empirical: a(n) = 11*a(n-1) +35*a(n-2) -495*a(n-3) -114*a(n-4) +6204*a(n-5) -4040*a(n-6) -17600*a(n-7) +16000*a(n-8)

A263056 Number of (n+1)X(4+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

10, 895, 18390, 498597, 10416690, 232738767, 4880746710, 104100946101, 2185333961490, 46071984907935, 967423105276470, 20335923462641157, 427044510774483570, 8970161111918918127, 188372284635598141590

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Column 4 of A263060.

			Some solutions for n=3
..1..0..0..1..1....0..0..0..1..0....0..0..1..1..1....0..0..0..0..1
..0..0..1..0..0....1..1..0..0..1....1..1..0..0..1....1..0..1..0..0
..1..1..0..1..0....1..1..1..0..0....0..1..0..1..1....0..1..1..1..0
..0..0..0..1..0....1..1..0..0..1....0..1..0..1..0....1..0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A263060.

Empirical: a(n) = 21*a(n-1) +189*a(n-2) -3969*a(n-3) -10719*a(n-4) +225099*a(n-5) +250803*a(n-6) -5266863*a(n-7) -2527200*a(n-8) +53071200*a(n-9) +9062928*a(n-10) -190321488*a(n-11)

A263057 Number of (n+1)X(5+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

42, 4314, 188370, 10416690, 439260642, 19763462754, 830550961170, 35491321238130, 1490799705490242, 62885673930539394, 2641262874276464370, 111053568381384471570, 4664277162998698135842, 195952869163413914931234

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Column 5 of A263060.

			Some solutions for n=3
..0..1..0..0..1..1....0..1..1..0..1..0....0..1..1..0..0..1....0..1..1..0..0..1
..1..0..0..0..0..0....1..0..1..1..1..0....1..0..1..0..0..0....1..0..0..0..1..0
..1..1..0..1..1..1....1..0..0..0..1..1....0..1..0..1..1..1....1..1..1..1..1..0
..1..0..1..0..0..1....1..0..1..1..0..0....0..1..1..0..0..1....1..1..1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A263060.

Empirical: a(n) = 42*a(n-1) +711*a(n-2) -29862*a(n-3) -141183*a(n-4) +5929686*a(n-5) +10349613*a(n-6) -434683746*a(n-7) -267400116*a(n-8) +11230804872*a(n-9) +1666598976*a(n-10) -69997156992*a(n-11)

A263058 Number of (n+1)X(6+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

42, 22921, 1941702, 232738767, 19763462754, 1825928130193, 155157177242886, 13464303796343791, 1144373527121975682, 97776065732064546321, 8310794250378741921702, 707331447947977777389487

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Column 6 of A263060.

			Some solutions for n=2
..1..0..0..1..1..0..0....1..0..0..0..0..1..1....0..0..1..1..1..0..1
..1..0..0..0..1..1..1....0..1..0..1..0..0..0....1..0..1..1..0..0..0
..1..1..1..1..1..0..0....0..0..1..0..1..0..0....0..1..1..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A263060.

Empirical: a(n) = 85*a(n-1) +3026*a(n-2) -257210*a(n-3) -2725780*a(n-4) +231691300*a(n-5) +1042604066*a(n-6) -88621345610*a(n-7) -184834243867*a(n-8) +15710910728695*a(n-9) +16148556399500*a(n-10) -1372627293957500*a(n-11) -690707626945552*a(n-12) +58710148290371920*a(n-13) +13366198928883008*a(n-14) -1136126908955055680*a(n-15) -92117546962355200*a(n-16) +7829991491800192000*a(n-17)

A263059 Number of (n+1)X(7+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

Original entry on oeis.org

170, 113486, 19499266, 4880746710, 830550961170, 155157177242886, 26380510465997186, 4591006241105432150, 780492175300947773170, 133462112418824638334886, 22688696181791779887035106

Offset: 1

R. H. Hardin, Oct 08 2015

nonn

Column 7 of A263060.

			Some solutions for n=2
..1..0..1..1..1..1..1..1....1..0..0..0..1..1..1..1....1..1..0..1..0..1..1..0
..1..0..1..0..0..0..0..1....0..0..1..1..0..0..1..0....0..1..0..0..1..1..0..0
..1..1..1..1..1..1..0..1....1..1..0..0..0..0..1..0....0..1..1..0..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A263060.

Empirical: a(n) = 170*a(n-1) +12088*a(n-2) -2054960*a(n-3) -43463646*a(n-4) +7388819820*a(n-5) +66398454660*a(n-6) -11287737292200*a(n-7) -47263888640505*a(n-8) +8034861068885850*a(n-9) +16948827007298316*a(n-10) -2881300591240713720*a(n-11) -3142452853334937648*a(n-12) +534216985066939400160*a(n-13) +298488023933669046336*a(n-14) -50742964068723737877120*a(n-15) -13934074238604397209600*a(n-16) +2368792620562747525632000*a(n-17) +287178122770920663040000*a(n-18) -48820280871056512716800000*a(n-19) -2061386929701535744000000*a(n-20) +350435778049261076480000000*a(n-21)

cp's OEIS Frontend

A263053 Number of (n+1) X 2 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263052 Number of (n+1)X(n+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263054 Number of (n+1) X (2+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263055 Number of (n+1)X(3+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263056 Number of (n+1)X(4+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263057 Number of (n+1)X(5+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263058 Number of (n+1)X(6+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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A263059 Number of (n+1)X(7+1) 0..1 arrays with each row and column not divisible by 3, read as a binary number with top and left being the most significant bits.

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