A295275 -id:A295275 - cp's OEIS frontend

A282990 Number of nX2 0..1 arrays with no 1 equal to more than one of its horizontal and vertical neighbors.

4, 11, 33, 98, 291, 865, 2570, 7637, 22693, 67432, 200373, 595405, 1769236, 5257255, 15621845, 46420050, 137936399, 409875693, 1217938738, 3619084505, 10754048825, 31955475472, 94955158681, 282157659273, 838426745764, 2491370993875

Offset: 1

R. H. Hardin, Feb 26 2017

nonn

Column 2 of A282996, A295275.

			Some solutions for n=5
..0..0. .1..1. .1..0. .0..0. .1..1. .1..0. .0..1. .1..0. .1..0. .0..0
..0..1. .0..0. .1..0. .1..0. .0..0. .0..0. .1..0. .1..0. .0..0. .0..1
..1..0. .0..0. .0..1. .0..0. .0..0. .0..1. .0..0. .0..0. .0..0. .0..0
..0..1. .0..0. .1..0. .1..0. .0..0. .0..0. .0..0. .0..0. .0..1. .0..0
..0..0. .0..0. .0..0. .1..0. .1..1. .0..1. .0..0. .1..1. .1..0. .1..1

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

Cf. A282996, A295275.

Empirical: a(n) = 2*a(n-1) +3*a(n-2) -a(n-4).

Empirical: G.f.: -x*(-4-3*x+x^2+x^3)/(1-2*x-3*x^2+x^4) . - R. J. Mathar, Mar 02 2017

A295269 Number of n X n 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1's.

Original entry on oeis.org

2, 11, 164, 6659, 748740, 230659662, 196091375073, 458510348625561, 2951683183977839155, 52303382279027148884728, 2551191530898428081696382880, 342537007989863347885673784022387

Offset: 1

R. H. Hardin, Nov 19 2017

nonn

Diagonal of A295275.

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

R. H. Hardin, Table of n, a(n) for n = 1..14

Cf. A295275.

A295270 Number of n X 3 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

Original entry on oeis.org

7, 33, 164, 811, 4035, 19997, 99245, 492401, 2443097, 12121712, 60143345, 298407987, 1480586061, 7346099129, 36448521869, 180843564461, 897276298340, 4451940313371, 22088817679653, 109596228179271, 543774384192739

Offset: 1

R. H. Hardin, Nov 19 2017

nonn

Column 3 of A295275.

			Some solutions for n=7:
  0 1 0    1 0 0    0 0 0    1 0 0    0 1 0    0 1 1    0 0 0
  1 1 1    0 0 1    0 0 1    0 1 0    1 0 0    1 0 0    1 0 0
  0 1 0    0 0 0    0 0 0    1 0 0    0 0 1    1 0 0    0 0 1
  1 0 0    1 0 0    1 0 1    1 0 0    0 0 1    0 0 0    0 0 0
  0 0 1    0 1 0    0 0 0    0 0 0    1 1 0    1 0 0    0 1 0
  1 0 0    1 0 0    0 1 0    0 0 0    0 0 0    0 0 1    1 0 0
  0 1 1    0 1 1    0 0 0    0 0 0    1 0 1    1 0 0    0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of formula
Index entries for linear recurrences with constant coefficients, signature (2, 10, 20, 17, -1, -9, -12, -1, -1, 1).

Cf. A295275.

Empirical: a(n) = 2*a(n-1) + 10*a(n-2) + 20*a(n-3) + 17*a(n-4) - a(n-5) - 9*a(n-6) - 12*a(n-7) - a(n-8) - a(n-9) + a(n-10).

Empirical formula is true: see link. - Robert Israel, Nov 19 2017

A295271 Number of n X 4 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

Original entry on oeis.org

13, 98, 811, 6659, 54773, 449827, 3697742, 30386185, 249719021, 2052217315, 16865303569, 138600705864, 1139033988179, 9360690397437, 76927054505407, 632193927094967, 5195430458657358, 42696546916624813, 350884326612255131, 2883601123790880675, 23697711212919539231, 194750068617952556626, 1600474783664067960559, 13152855612946330563815, 108091431705620580695441

Offset: 1

R. H. Hardin, Nov 19 2017

Column 4 of A295275.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of formula
Index entries for linear recurrences with constant coefficients, signature (4,27,63,14,-110,-43,-45,89,-150,89,-126,72,-30,8,-2,1).

Cf. A295275.

R:= [13, 98, 811, 6659, 54773, 449827, 3697742, 30386185, 249719021, 2052217315, 16865303569, 138600705864, 1139033988179, 9360690397437, 76927054505407, 632193927094967]:
rec:= a(n) = 4*a(n-1) +27*a(n-2) +63*a(n-3) +14*a(n-4) -110*a(n-5) -43*a(n-6) -45*a(n-7) +89*a(n-8) -150*a(n-9) +89*a(n-10) -126*a(n-11) +72*a(n-12) -30*a(n-13) +8*a(n-14) -2*a(n-15) +a(n-16):
f:= gfun:-rectoproc({rec,seq(a(i)=R[i],i=1..16)},a(n),remember):
map(f, [$1..40]); # Robert Israel, Nov 19 2017

a = DifferenceRoot[Function[{a, n},
{-a[n] + 2 a[n+1] - 8 a[n+2] + 30 a[n+3] - 72 a[n+4] + 126 a[n+5] - 89 a[n+6] + 150 a[n+7] - 89 a[n+8] +
   45 a[n+9] + 43 a[n+10] + 110 a[n+11] - 14 a[n+12] - 63 a[n+13] - 27 a[n+14] - 4 a[n+15] + a[n+16] == 0,
   a[1] == 13, a[2] == 98, a[3] == 811, a[4] == 6659,
   a[5] == 54773, a[6] == 449827, a[7] == 3697742, a[8] == 30386185,
   a[9] == 249719021, a[10] == 2052217315, a[11] == 16865303569, a[12] == 138600705864,
   a[13] == 1139033988179, a[14] == 9360690397437, a[15] == 76927054505407, a[16] == 632193927094967}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 27 2022, after Robert Israel *)

Empirical: a(n) = 4*a(n-1) +27*a(n-2) +63*a(n-3) +14*a(n-4) -110*a(n-5) -43*a(n-6) -45*a(n-7) +89*a(n-8) -150*a(n-9) +89*a(n-10) -126*a(n-11) +72*a(n-12) -30*a(n-13) +8*a(n-14) -2*a(n-15) +a(n-16).
From Robert Israel, Nov 19 2017: (Start)
Empirical formula confirmed (see link).
G.f.: (13*x + 46*x^2 + 68*x^3 - 50*x^4 - 116*x^5 - 93*x^6 + 31*x^7 - 28*x^8 - 60*x^9 + 2*x^10 - 73*x^11 + 51*x^12 - 22*x^13 + 7*x^14 - x^15 + x^16) / (1 - 4*x - 27*x^2 - 63*x^3 - 14*x^4 + 110*x^5 + 43*x^6 + 45*x^7 - 89*x^8 + 150*x^9 - 89*x^10 + 126*x^11 - 72*x^12 + 30*x^13 - 8*x^14 + 2*x^15 - x^16).
(End)

A295272 Number of nX5 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

Original entry on oeis.org

24, 291, 4035, 54773, 748740, 10203847, 139221499, 1898916677, 25902054180, 353312300646, 4819296560043, 65736776328102, 896671271822835, 12230890096966674, 166833355609514988, 2275661718111926504

Offset: 1

R. H. Hardin, Nov 19 2017

nonn

Column 5 of A295275.

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

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

Cf. A295275.

Empirical: a(n) = 5*a(n-1) +84*a(n-2) +421*a(n-3) +716*a(n-4) -1544*a(n-5) -7895*a(n-6) -11069*a(n-7) +23659*a(n-8) +46894*a(n-9) -28597*a(n-10) -25649*a(n-11) -112201*a(n-12) +282924*a(n-13) -604659*a(n-14) +1353567*a(n-15) -2132282*a(n-16) +2688487*a(n-17) -2990791*a(n-18) +2796525*a(n-19) -2113112*a(n-20) +1443409*a(n-21) -944144*a(n-22) +464982*a(n-23) -217099*a(n-24) +125161*a(n-25) -1805*a(n-26) -32202*a(n-27) +26125*a(n-28) -18772*a(n-29) -1914*a(n-30) -838*a(n-31) +1685*a(n-32) +2105*a(n-33) -329*a(n-34) -147*a(n-35) -176*a(n-36) +64*a(n-37) +7*a(n-38) +a(n-39) -a(n-40)

A295273 Number of nX6 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

Original entry on oeis.org

44, 865, 19997, 449827, 10203847, 230659662, 5221177246, 118139247475, 2673342943199, 60493860184777, 1368887903616187, 30975935104352291, 700940389558441876, 15861259863124103232, 358917220084448007191

Offset: 1

R. H. Hardin, Nov 19 2017

nonn

Column 6 of A295275.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 78

Cf. A295275.

Empirical recurrence of order 78 (see link above)

A295274 Number of nX7 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

Original entry on oeis.org

81, 2570, 99245, 3697742, 139221499, 5221177246, 196091375073, 7361533108317, 276386201580189, 10376697508977992, 389585088134351369, 14626667590533685932, 549146940474831592906, 20617295057949494189434

Offset: 1

R. H. Hardin, Nov 19 2017

nonn

Column 7 of A295275.

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

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

Cf. A295275.

cp's OEIS Frontend

A282990 Number of nX2 0..1 arrays with no 1 equal to more than one of its horizontal and vertical neighbors.

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A295269 Number of n X n 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1's.

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A295270 Number of n X 3 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

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A295271 Number of n X 4 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

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A295272 Number of nX5 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

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A295273 Number of nX6 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

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A295274 Number of nX7 0..1 arrays with each 1 horizontally or vertically adjacent to 0, 1 or 4 1s.

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