A202471 -id:A202471 - cp's OEIS frontend

A050974 Number of binary arrangements on n X n array without three adjacent 1's in a row or column.

1, 2, 16, 265, 16561, 3157010, 1828904402, 3323590649777, 18691199385898465, 325778072452564800064, 17617718915229579206450786, 2954164381835835259001326344913, 1536134628973698280539373190731911729, 2477137610106747308204461168746042225266836, 12387488188151269567355592399321080831513078632498, 192102098800681202990688566451981906679020804069237862571, 9238409697848267958752630399467598421213391733838644131510525089

Offset: 0

Eric W. Weisstein

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.

Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, 01-Matrix.

Any connected three 1's gives A067968.

Cf. A006506. Diagonal of A202471.

t[m_] := t[m] = Map[ArrayReshape[#, {m, m}] &, Tuples[{0, 1}, m^2]]; a[m_] := a[m] = Count[Table[AnyTrue[Flatten[{Table[Equal[1, t[m][[n, a, b]], t[m][[n, a, b + 1]], t[m][[n, a, b + 2]]], {a, 1, m}, {b, 1, m - 2}], Table[Equal[1, t[m][[n, a, b]], t[m][[n, a + 1, b]], t[m][[n, a + 2, b]]], {a, 1, m - 2}, {b, 1, m}]}], TrueQ], {n, 1, 2^(m^2)}], False]; (* Robert P. P. McKone, Jan 04 2022 *)

More terms from R. H. Hardin, Feb 02 2002

a(0)=1 prepended and a(13)-a(16) from Peter J. Taylor, Sep 26 2024

A202464 Number of (n+2) X 3 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

265, 1573, 9253, 54085, 317179, 1858993, 10894297, 63850777, 374215201, 2193190681, 12853833343, 75333557377, 441513844561, 2587618272973, 15165477047269, 88881616706701, 520916142332515, 3052978074328417

Offset: 1

R. H. Hardin, Dec 19 2011

nonn

Column 1 of A202471.

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

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

Cf. A202471.

Empirical: a(n) = 4*a(n-1) +8*a(n-2) +18*a(n-3) -6*a(n-4) +2*a(n-5) -a(n-6).

Empirical g.f.: x*(265 + 513*x + 841*x^2 - 281*x^3 + 91*x^4 - 49*x^5) / (1 - 4*x - 8*x^2 - 18*x^3 + 6*x^4 - 2*x^5 + x^6). - Colin Barker, Jun 01 2018

A202465 Number of (n+2)X4 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

1573, 16561, 172197, 1774281, 18374521, 190112777, 1966463785, 20345296193, 210482281689, 2177526620233, 22527699867037, 233060360137857, 2411126165020333, 24944324990455409, 258061639015145685

Offset: 1

R. H. Hardin Dec 19 2011

nonn

Column 2 of A202471

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

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

Empirical: a(n) = 4*a(n-1) +36*a(n-2) +272*a(n-3) +350*a(n-4) +392*a(n-5) -3016*a(n-6) -164*a(n-7) -5522*a(n-8) +11427*a(n-9) -4608*a(n-10) +10140*a(n-11) -11817*a(n-12) +4108*a(n-13) -2432*a(n-14) +1176*a(n-15) -166*a(n-16) +124*a(n-17) -50*a(n-18) +4*a(n-19) -2*a(n-20) +a(n-21)

A202466 Number of (n+2)X5 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

9253, 172197, 3157010, 57262144, 1044865367, 19045287529, 347030298362, 6325139804762, 115276893817821, 2100921552416129, 38289856014452302, 697839603278397490, 12718251853368355369, 231792595465614884173

Offset: 1

R. H. Hardin Dec 19 2011

nonn

Column 3 of A202471

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

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

Empirical: a(n) = 5*a(n-1) +111*a(n-2) +2014*a(n-3) +6557*a(n-4) +16985*a(n-5) -300395*a(n-6) -493842*a(n-7) -3268704*a(n-8) +24359757*a(n-9) -12644117*a(n-10) +245975711*a(n-11) -1125487262*a(n-12) +1522045563*a(n-13) -8087124133*a(n-14) +26777225059*a(n-15) -37534618180*a(n-16) +126433919538*a(n-17) -328489679534*a(n-18) +391815137702*a(n-19) -986011308148*a(n-20) +2122690461724*a(n-21) -1946484646192*a(n-22) +3965682502748*a(n-23) -7576159630469*a(n-24) +5031247471425*a(n-25) -9030564869781*a(n-26) +16603266615558*a(n-27) -7654100893731*a(n-28) +12605001997031*a(n-29) -23568791472338*a(n-30) +7284025623281*a(n-31) -11592415733849*a(n-32) +22376915552667*a(n-33) -4108143304538*a(n-34) +7534067585378*a(n-35) -14586401994989*a(n-36) +851438613707*a(n-37) -3651106111783*a(n-38) +6680505527134*a(n-39) +516890844951*a(n-40) +1330578342773*a(n-41) -2180316021478*a(n-42) -457849189081*a(n-43) -350426435259*a(n-44) +504531345207*a(n-45) +156767801632*a(n-46) +63587131764*a(n-47) -80543670710*a(n-48) -29202619778*a(n-49) -7801427038*a(n-50) +8520515392*a(n-51) +3153222478*a(n-52) +630256472*a(n-53) -563860143*a(n-54) -184231091*a(n-55) -26381427*a(n-56) +24372940*a(n-57) +5813919*a(n-58) +360687*a(n-59) -727057*a(n-60) -95650*a(n-61) +8776*a(n-62) +14851*a(n-63) +693*a(n-64) -335*a(n-65) -184*a(n-66) -a(n-67) +3*a(n-68) +a(n-69)

A202467 Number of (n+2)X6 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

54085, 1774281, 57262144, 1828904402, 58746383721, 1885066867201, 60472625443366, 1940390570621922, 62257900228006925, 1997548362431307729, 64092086253488512734, 2056411827878286085746, 65980528433359436484485

Offset: 1

R. H. Hardin Dec 19 2011

nonn

Column 4 of A202471

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

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

A202468 Number of (n+2)X7 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

317179, 18374521, 1044865367, 58746383721, 3323590649777, 187820723381421, 10610932604054547, 599616560193259281, 33881738532796270907, 1914496303093962715299, 108180387260313623312899

Offset: 1

R. H. Hardin Dec 19 2011

nonn

Column 5 of A202471

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

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

A202469 Number of (n+2) X 8 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

1858993, 190112777, 19045287529, 1885066867201, 187820723381421, 18691199385898465, 1859521153822030023, 185045901525934158257, 18413176084400451943067, 1832207303424558956985631

Offset: 1

R. H. Hardin, Dec 19 2011

nonn

Column 6 of A202471.

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

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

Cf. A202471.

A202470 Number of (n+2)X9 binary arrays with no more than two of any consecutive three bits set in any row or column.

Original entry on oeis.org

10894297, 1966463785, 347030298362, 60472625443366, 10610932604054547, 1859521153822030023, 325778072452564800064, 57089473505402700427612, 10003712847692227326534941, 1752925427213891417645669229

Offset: 1

R. H. Hardin Dec 19 2011

nonn

Column 7 of A202471

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

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

cp's OEIS Frontend

A050974 Number of binary arrangements on n X n array without three adjacent 1's in a row or column.

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A202464 Number of (n+2) X 3 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202465 Number of (n+2)X4 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202466 Number of (n+2)X5 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202467 Number of (n+2)X6 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202468 Number of (n+2)X7 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202469 Number of (n+2) X 8 binary arrays with no more than two of any consecutive three bits set in any row or column.

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A202470 Number of (n+2)X9 binary arrays with no more than two of any consecutive three bits set in any row or column.

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