A268457 -id:A268457 - cp's OEIS frontend

A268451 Number of length-n 0..2 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

3, 9, 25, 67, 176, 456, 1169, 2971, 7496, 18796, 46880, 116386, 287775, 709005, 1741247, 4264131, 10415559, 25381773, 61721980, 149801330, 362926749, 877831299, 2120050410, 5112962906, 12314976142, 29625637490, 71188510125

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

			Some solutions for n=9:
..1....0....0....2....2....1....0....2....1....1....2....0....1....0....1....0
..2....0....1....2....2....1....0....2....0....1....0....0....1....2....0....2
..0....2....2....1....2....2....2....1....2....2....2....1....1....2....1....0
..2....1....2....2....2....0....0....0....2....2....2....1....1....2....1....1
..2....1....2....0....2....0....2....1....1....0....2....1....1....2....2....2
..0....1....0....0....0....2....0....1....0....1....2....2....1....2....2....0
..1....0....0....0....0....0....0....1....0....2....0....2....0....1....0....2
..2....1....1....2....0....0....2....2....0....0....0....2....0....0....0....0
..2....2....1....2....0....2....1....0....2....1....1....2....2....0....1....0

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

Column 2 of A268457.

Empirical: a(n) = 6*a(n-1) - 13*a(n-2) + 14*a(n-3) - 10*a(n-4) + 4*a(n-5) - a(n-6).

Empirical g.f.: x*(3 - 9*x + 10*x^2 - 8*x^3 + 3*x^4 - x^5) / (1 - 3*x + 2*x^2 - x^3)^2. - Colin Barker, Jan 13 2019

A268452 Number of length-n 0..3 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

4, 16, 61, 229, 852, 3146, 11536, 42032, 152254, 548568, 1966757, 7019311, 24946486, 88313632, 311507619, 1095064315, 3837407016, 13407618832, 46715533678, 162345449332, 562801287366, 1946559269050, 6717910370343, 23136984979001

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

Column 3 of A268457.

			Some solutions for n=9
..0....1....0....1....1....0....3....3....3....2....1....1....2....1....0....2
..1....2....2....0....1....1....0....2....0....2....0....0....0....2....0....2
..2....3....2....2....0....3....3....3....3....3....0....3....0....2....0....2
..3....0....0....2....2....0....0....3....3....0....2....3....1....0....2....2
..1....2....1....3....0....0....1....1....1....0....0....3....2....0....0....3
..3....3....3....3....0....2....3....3....1....1....1....1....2....3....0....1
..1....3....0....0....2....1....1....1....0....3....1....3....2....0....1....2
..2....0....0....2....0....1....1....1....0....3....3....1....3....0....2....0
..0....3....2....2....0....2....2....3....0....0....0....0....0....3....3....1

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

Cf. A268457.

Empirical: a(n) = 19*a(n-1) -156*a(n-2) +730*a(n-3) -2178*a(n-4) +4446*a(n-5) -6613*a(n-6) +7587*a(n-7) -6948*a(n-8) +5152*a(n-9) -3101*a(n-10) +1503*a(n-11) -573*a(n-12) +163*a(n-13) -31*a(n-14) +3*a(n-15)

A268453 Number of length-n 0..4 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

5, 25, 121, 581, 2776, 13204, 62535, 294967, 1385969, 6488635, 30273074, 140779986, 652648100, 3016745162, 13905372533, 63924885355, 293126854872, 1340883359460, 6119617278729, 27867658231717, 126637380509476, 574312506857594

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

Column 4 of A268457.

			Some solutions for n=7
..4....4....4....0....2....2....1....0....1....3....3....2....2....4....1....2
..0....0....0....3....3....0....4....4....1....3....1....1....4....3....0....1
..1....3....1....2....3....0....4....1....4....1....2....0....4....3....0....1
..4....0....1....2....3....0....2....2....4....4....2....4....2....3....0....3
..2....1....1....2....4....4....1....2....0....2....4....3....4....3....2....1
..2....3....2....1....4....4....4....2....0....1....2....2....0....0....2....3
..2....2....4....2....0....1....3....0....0....0....4....3....0....4....0....0

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

Cf. A268457.

Empirical: a(n) = 39*a(n-1) -693*a(n-2) +7441*a(n-3) -54053*a(n-4) +282341*a(n-5) -1103223*a(n-6) +3324255*a(n-7) -7938129*a(n-8) +15426103*a(n-9) -25019179*a(n-10) +34622055*a(n-11) -41606078*a(n-12) +43991906*a(n-13) -41304450*a(n-14) +34649044*a(n-15) -26064185*a(n-16) +17607461*a(n-17) -10675286*a(n-18) +5792770*a(n-19) -2798790*a(n-20) +1194342*a(n-21) -444872*a(n-22) +142204*a(n-23) -38049*a(n-24) +8201*a(n-25) -1336*a(n-26) +146*a(n-27) -8*a(n-28)

A268454 Number of length-n 0..5 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

6, 36, 211, 1231, 7160, 41526, 240170, 1385338, 7970326, 45742764, 261899866, 1496074920, 8527336280, 48500972230, 275294704851, 1559507239653, 8817578095886, 49763675574292, 280352403988287, 1576707102660739

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

Column 5 of A268457.

			Some solutions for n=6
..4....1....1....0....5....2....5....4....1....4....5....3....0....4....5....0
..0....5....0....0....5....1....5....2....4....5....2....3....2....2....4....4
..3....2....0....0....5....5....4....2....0....3....4....5....3....3....0....5
..5....3....5....2....0....5....2....4....0....2....3....0....5....0....5....1
..0....4....4....3....1....1....5....4....4....1....2....4....2....2....1....1
..4....0....5....4....1....1....0....1....4....1....4....2....1....4....0....3

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 51

Cf. A268457.

Empirical recurrence of order 51 (see link above)

A268455 Number of length-n 0..6 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

7, 49, 337, 2311, 15816, 108032, 736525, 5012171, 34047931, 230889543, 1563104184, 10564826628, 71293105297, 480355314513, 3231663192859, 21709801573961, 145636117720769, 975623832619371, 6526972402359763, 43608650962250889

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

Column 6 of A268457.

			Some solutions for n=6
..4....4....2....2....4....1....0....4....5....1....2....4....2....0....3....1
..4....3....6....4....1....0....0....2....4....0....4....3....3....4....6....1
..4....0....6....5....3....6....3....1....0....6....2....1....3....3....2....2
..0....4....4....6....0....5....5....0....4....6....3....3....4....2....4....0
..6....4....4....4....2....2....3....4....1....6....0....5....5....1....3....5
..3....5....1....4....2....0....5....3....0....5....6....1....1....2....1....2

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 89

Cf. A268457.

Empirical recurrence of order 89 (see link above)

A268456 Number of length-n 0..7 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

8, 64, 505, 3977, 31276, 245626, 1926444, 15089356, 118040270, 922247248, 7196716310, 56092274904, 436681564144, 3395712755366, 26376195676360, 204653424502050, 1586213921445534, 12281469122143732, 94993604706706795

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

Column 7 of A268457.

			Some solutions for n=5
..7....2....3....3....6....4....5....4....1....0....6....7....6....4....2....4
..6....5....2....6....5....4....1....1....6....2....1....4....5....1....4....1
..4....4....1....4....0....5....3....0....4....4....7....5....0....3....7....7
..5....4....1....5....1....0....4....7....3....2....2....3....7....4....3....0
..2....0....1....1....6....6....0....6....6....1....6....6....7....4....2....4

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

Cf. A268457.

A268458 Number of length-4 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

11, 67, 229, 581, 1231, 2311, 3977, 6409, 9811, 14411, 20461, 28237, 38039, 50191, 65041, 82961, 104347, 129619, 159221, 193621, 233311, 278807, 330649, 389401, 455651, 530011, 613117, 705629, 808231, 921631, 1046561, 1183777, 1334059

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

			Some solutions for n=9:
  2  7  0  3  8  5  3  3  4  5  8  9  9  8  2  4
  7  1  3  8  4  1  1  0  8  6  2  5  1  9  2  5
  6  0  7  3  1  1  0  5  8  2  0  8  1  4  0  2
  2  3  1  4  5  0  9  4  9  2  9  4  8  6  2  9

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

Row 4 of A268457.

seq(n^4 + 4*n^3 + 4*n^2 + n + 1, n=1..100); # Robert Israel, Nov 28 2019

Empirical: a(n) = n^4 + 4*n^3 + 4*n^2 + n + 1.
Empirical g.f.: x*(11 + 12*x + 4*x^2 - 4*x^3 + x^4) / (1 - x)^5. - Colin Barker, Jan 13 2019
Proof of empirical formula: There are (n+1)^4 arrays without the constraint. n of them are of the form (x,x+1,x+1,x) with 0 <= x <= n-1, n*(n+1) are of the form (x,x+1,x,y) with 0 <= x<= n-1 and 0<=y<=n, and n*(n+1) are of the form (y,x,x+1,x). That leaves n^4 + 4*n^3 + 4*n^2 + n + 1. - Robert Israel, Nov 28 2019

A268459 Number of length-5 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

16, 176, 852, 2776, 7160, 15816, 31276, 56912, 97056, 157120, 243716, 364776, 529672, 749336, 1036380, 1405216, 1872176, 2455632, 3176116, 4056440, 5121816, 6399976, 7921292, 9718896, 11828800, 14290016, 17144676, 20438152, 24219176

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

			Some solutions for n=9:
  7  5  8  9  3  4  1  2  0  4  3  1  5  1  5  3
  1  4  0  0  3  4  6  4  1  4  6  5  9  8  7  0
  3  1  2  5  2  7  1  2  3  7  4  6  7  6  6  3
  6  1  7  6  1  6  7  1  6  5  4  8  6  6  5  9
  1  1  9  6  5  3  5  8  3  6  4  9  4  6  5  7

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

Row 5 of A268457.

Empirical: a(n) = n^5 + 5*n^4 + 7*n^3 + n^2 + 2*n.
Conjectures from Colin Barker, Jan 13 2019: (Start)
G.f.: 4*x*(4 + 20*x + 9*x^2 - 4*x^3 + x^4) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A268460 Number of length-6 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

22, 456, 3146, 13204, 41526, 108032, 245626, 504876, 959414, 1712056, 2901642, 4710596, 7373206, 11184624, 16510586, 23797852, 33585366, 46516136, 63349834, 84976116, 112428662, 146899936, 189756666, 242556044, 307062646, 385266072

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

			Some solutions for n=6:
  2  0  2  3  6  4  4  4  3  2  3  6  0  3  1  2
  5  2  6  2  3  2  2  2  2  4  1  0  5  2  5  6
  2  4  4  6  4  4  5  1  2  4  1  4  4  2  0  0
  0  0  0  4  6  1  4  2  4  2  3  0  5  0  3  0
  0  4  3  4  1  0  2  4  4  3  6  5  5  0  5  1
  6  2  0  5  0  1  2  5  6  4  5  6  1  2  2  2

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

Row 6 of A268457.

Empirical: a(n) = n^6 + 6*n^5 + 11*n^4 + 2*n^3 + 6*n - 4.
Conjectures from Colin Barker, Jan 13 2019: (Start)
G.f.: 2*x*(11 + 151*x + 208*x^2 - 6*x^3 - 13*x^4 + 11*x^5 - 2*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A268461 Number of length-7 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

Original entry on oeis.org

29, 1169, 11536, 62535, 240170, 736525, 1926444, 4474451, 9476950, 18644745, 34530920, 60809119, 102607266, 166901765, 262977220, 402956715, 602407694, 881028481, 1263420480, 1779951095, 2467712410, 3371580669, 4545381596

Offset: 1

R. H. Hardin, Feb 04 2016

nonn

			Some solutions for n=4:
..3....2....2....3....3....3....4....2....0....0....3....4....4....4....4....4
..4....3....2....1....3....0....4....2....3....4....0....0....0....3....2....1
..2....0....0....1....0....4....4....4....1....3....2....2....2....4....2....3
..0....3....2....3....3....1....4....2....3....0....3....2....4....2....1....3
..1....4....2....2....2....0....0....1....3....2....1....4....1....2....3....0
..3....0....4....1....1....2....2....4....1....0....0....0....3....3....0....2
..4....0....3....2....2....4....3....0....1....3....0....1....4....3....0....0

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

Row 7 of A268457.

Empirical: a(n) = n^7 + 7*n^6 + 16*n^5 + 5*n^4 - 7*n^3 + 18*n^2 - 5*n - 5 for n>1.
Conjectures from Colin Barker, Jan 14 2019: (Start)
G.f.: x*(29 + 937*x + 2996*x^2 + 1355*x^3 - 536*x^4 + 335*x^5 - 88*x^6 + 13*x^7 - x^8) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>9.
(End)

cp's OEIS Frontend

A268451 Number of length-n 0..2 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268452 Number of length-n 0..3 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268453 Number of length-n 0..4 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268454 Number of length-n 0..5 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268455 Number of length-n 0..6 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268456 Number of length-n 0..7 arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268458 Number of length-4 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268459 Number of length-5 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268460 Number of length-6 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.

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A268461 Number of length-7 0..n arrays with no adjacent pair x,x+1 followed at any distance by x+1,x.