A250351 -id:A250351 - cp's OEIS frontend

A250361 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 3 times.

2, 3, 4, 4, 9, 8, 5, 16, 27, 16, 6, 25, 64, 81, 32, 7, 36, 125, 255, 243, 64, 8, 49, 216, 623, 1016, 729, 128, 9, 64, 343, 1293, 3094, 4048, 2187, 256, 10, 81, 512, 2397, 7712, 15365, 16128, 6561, 512, 11, 100, 729, 4091, 16700, 45866, 76300, 64257, 19683, 1024, 12

Offset: 1

R. H. Hardin, Nov 19 2014

Table starts
....2.....3.......4.......5........6.........7..........8..........9.........10
....4.....9......16......25.......36........49.........64.........81........100
....8....27......64.....125......216.......343........512........729.......1000
...16....81.....255.....623.....1293......2397.......4091.......6555.......9993
...32...243....1016....3094.....7712.....16700......32608......58826......99704
...64...729....4048...15365....45866....115963.....259106.....526505.....992530
..128..2187...16128...76300...272760....803382....2052904....4698744....9854280
..256..6561...64257..378880..1621963...5565230...16234706...41828450...97581710
..512.19683..256012.1881364..9644496..38548644..128373416..371780050..964209084
.1024.59049.1020000.9342081.57346376.266998350.1015004124.3304106808.9514922752

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

R. H. Hardin, Table of n, a(n) for n = 1..447

Column 1 is A000079
Column 2 is A000244
Column 3 is A206450
Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Row 3 is A000578(n+1)
Cf. A248944, A250351

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1)
k=3: a(n) = 4*a(n-1) -a(n-4)
k=4: a(n) = 5*a(n-1) -2*a(n-4) -11*a(n-5) +a(n-8)
k=5: [order 13]
k=6: [order 25]
k=7: [order 56]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 3*n + 1
n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 3 for n>1
n=5: a(n) = n^5 + 5*n^4 + 10*n^3 + 5*n^2 + 17*n + 2 for n>2
n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 5*n^3 + 57*n^2 + 13*n + 1 for n>3
n=7: a(n) = n^7 + 7*n^6 + 21*n^5 + 147*n^3 + 49*n^2 + 7*n for n>4

A250346 Number of length n arrays x(i), i=1..n with x(i) in i..i+3 and no value appearing more than 2 times.

Original entry on oeis.org

4, 16, 62, 235, 888, 3349, 12620, 47545, 179104, 674666, 2541362, 9572864, 36059224, 135828387, 511640114, 1927252354, 7259597884, 27345542237, 103005522894, 388002462425, 1461532415920, 5505318159061, 20737499694808, 78114267177504

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

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

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

Column 3 of A250351.

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

Empirical g.f.: x*(4 - 2*x^2 - 5*x^3 + x^5) / (1 - 4*x + 2*x^3 + 5*x^4 - x^6). - Colin Barker, Nov 12 2018

A250347 Number of length n arrays x(i), i=1..n with x(i) in i..i+4 and no value appearing more than 2 times.

Original entry on oeis.org

5, 25, 122, 581, 2724, 12734, 59406, 276816, 1289208, 6002949, 27948445, 130114896, 605739129, 2819935996, 13127753699, 61113960653, 284504996836, 1324460800690, 6165782447251, 28703656674418, 133624541803551, 622064214270768

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

Column 4 of A250351

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

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

Empirical: a(n) = 5*a(n-1) -4*a(n-3) -9*a(n-4) -36*a(n-5) +7*a(n-6) +23*a(n-7) +13*a(n-8) +2*a(n-9) -5*a(n-10) +a(n-12)

A250348 Number of length n arrays x(i), i=1..n with x(i) in i..i+5 and no value appearing more than 2 times.

Original entry on oeis.org

6, 36, 212, 1221, 6900, 38543, 214716, 1193739, 6628042, 36773706, 203951790, 1130937489, 6270501936, 34764775155, 192735759120, 1068507612315, 5923642360728, 32839591629254, 182056176609954, 1009281733768068

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

Column 5 of A250351

			Some solutions for n=6
..5....3....5....3....1....2....0....0....5....4....4....0....0....5....0....0
..6....1....2....5....5....4....5....6....1....2....6....2....4....1....4....6
..2....7....3....2....3....2....4....4....3....4....5....7....3....3....6....6
..7....7....4....6....8....7....4....5....3....5....8....8....4....7....3....7
..9....9....7....7....6....4....9....5....7....7....8....7....7....5....4....9
..9...10....8....5....5....7....5....7...10...10...10....9....9....7....5....7

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

Empirical: a(n) = 6*a(n-1) -8*a(n-3) -12*a(n-4) -66*a(n-5) -332*a(n-6) +132*a(n-7) +447*a(n-8) +294*a(n-9) +630*a(n-10) +396*a(n-11) -484*a(n-12) -222*a(n-13) +192*a(n-14) -188*a(n-15) -144*a(n-16) +114*a(n-17) +50*a(n-18) -12*a(n-19) +21*a(n-20) +16*a(n-21) -a(n-24)

A250349 Number of length n arrays x(i), i=1..n with x(i) in i..i+6 and no value appearing more than 2 times.

Original entry on oeis.org

7, 49, 338, 2287, 15186, 99344, 644040, 4164930, 26882466, 173276640, 1115910270, 7182799173, 46220503936, 297376327082, 1913081363032, 12306454081360, 79161863609011, 509201240096045, 3275346169991511, 21067913854706817

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

Column 6 of A250351

			Some solutions for n=6
..1....2....2....1....0....1....4....2....3....4....3....1....2....2....0....4
..4....3....4....1....3....1....5....7....2....7....5....3....2....2....6....7
..8....3....7....3....5....5....2....4....5....3....6....4....7....4....2....3
..7....9....7....8....5....7....9....6....9....4....7....5....5....4....6....9
.10....7....6....4...10....9...10....4....6....8...10....9....8....6....7....6
..8....8...10....5....6....9....5...11....6....9....8....8....5....8....8....9

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

Empirical: a(n) = 7*a(n-1) -12*a(n-3) -32*a(n-4) -101*a(n-5) -623*a(n-6) -3703*a(n-7) +2630*a(n-8) +10358*a(n-9) +15805*a(n-10) +19448*a(n-11) +41930*a(n-12) +17560*a(n-13) -133023*a(n-14) -179079*a(n-15) -42337*a(n-16) +101951*a(n-17) -34974*a(n-18) -21497*a(n-19) +198775*a(n-20) +430980*a(n-21) +205427*a(n-22) -195761*a(n-23) -191518*a(n-24) -100666*a(n-25) -82593*a(n-26) -182721*a(n-27) -26175*a(n-28) +61736*a(n-29) +73291*a(n-30) +56839*a(n-31) -14702*a(n-32) -18860*a(n-33) -26635*a(n-34) -3079*a(n-35) +4310*a(n-36) +5890*a(n-37) -1330*a(n-38) -882*a(n-39) +37*a(n-40) -321*a(n-41) -191*a(n-42) +119*a(n-43) +25*a(n-44) -9*a(n-45) -a(n-46) -a(n-48)

A250350 Number of length n arrays x(i), i=1..n with x(i) in i..i+7 and no value appearing more than 2 times.

Original entry on oeis.org

8, 64, 506, 3935, 30072, 226247, 1681860, 12411486, 91384716, 671639928, 4929577344, 36146749703, 264885341958, 1940349607935, 14210290469006, 104055436365169, 761877093233402, 5577987879867728, 40836880075623272

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

Column 7 of A250351

			Some solutions for n=5
..2....3....1....0....2....5....3....5....1....1....5....7....4....6....1....0
..8....8....4....2....8....5....6....7....3....2....5....5....7....4....5....2
..5....4....2....5....9....2....8....8....2....2....6....5....8....5....9....2
..9....8....8....3....7....8...10....4...10....4....8...10....6....3....6...10
.10....4....8....8....5....4....4....8....4....8....4...10....7....9....6...10

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 96

Empirical recurrence of order 96 (see link above)

A250352 Number of length 3 arrays x(i), i=1..3 with x(i) in i..i+n and no value appearing more than 2 times.

Original entry on oeis.org

8, 26, 62, 122, 212, 338, 506, 722, 992, 1322, 1718, 2186, 2732, 3362, 4082, 4898, 5816, 6842, 7982, 9242, 10628, 12146, 13802, 15602, 17552, 19658, 21926, 24362, 26972, 29762, 32738, 35906, 39272, 42842, 46622, 50618, 54836, 59282, 63962, 68882, 74048

Offset: 1

R. H. Hardin, Nov 19 2014

a(n) = (n+1)^3 - (n-1), where (n+1)^3 is the number of ways of selecting a triple from n+1 numbers in these subintervals, and there are n-1 of these triples, (3,3,3) up to (n-2,n-2,n-2), where all values are the same, which are discarded. - R. J. Mathar, Oct 09 2020

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

R. H. Hardin, Table of n, a(n) for n = 1..210
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Row 3 of A250351.

a(n) = n^3 + 3*n^2 + 2*n + 2 = 2*A158842(n+1).
From Colin Barker, Nov 12 2018: (Start)
G.f.: 2*x*(4 - 3*x + 3*x^2 - x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A250353 Number of length 4 arrays x(i), i=1..4 with x(i) in i..i+n and no value appearing more than 2 times.

Original entry on oeis.org

16, 75, 235, 581, 1221, 2287, 3935, 6345, 9721, 14291, 20307, 28045, 37805, 49911, 64711, 82577, 103905, 129115, 158651, 192981, 232597, 278015, 329775, 388441, 454601, 528867, 611875, 704285, 806781, 920071, 1044887, 1181985, 1332145

Offset: 1

R. H. Hardin, Nov 19 2014

There are n+1 candidates for any of the 4 values in the 4-tuple. If there were no constraints, there were (n+1)^4 arrays. The constraint of not counting the quadruplets (4,4,4,4), (5,5,5,5), ..... (n+1,n+1,n+1,n+1) discards n-2 of the 4-tuples. [The case n=1 is special because there are not quadruplets]. Adding the constraint of not having triplets discards (3,3,3,*) and (*,n+2,n+2,n+2) where the star represents one of n+1 values; this is a total of 2*(n+1). The constraint of not having triplets also discards the (*,4,4,4), (4,*,4,4), (4,4,*,4), (4,4,4,*), (*,5,5,5),... (*,1+n,1+n,1+n),....(1+n,1+n,1+n,*) where the star represents one of n values (not n+1 here not to account for the quadruplets twice). There are binomial(4,1)*n*(n-2) of these triplets. The result is a(n) = (n+1)^4 -(n-2) -2*(n+1) -4*n*(n-2) = n^4+4*n^3+2*n^2+9*n+1. - R. J. Mathar, Oct 11 2020

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

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

Row 4 of A250351.

a(n) = n^4 + 4*n^3 + 2*n^2 + 9*n + 1 for n>1.
From Colin Barker, Nov 13 2018: (Start)
G.f.: x*(16 - 5*x + 20*x^2 - 4*x^3 - 4*x^4 + x^5) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
(End)

A250354 Number of length 5 arrays x(i), i=1..5 with x(i) in i..i+n and no value appearing more than 2 times.

Original entry on oeis.org

32, 216, 888, 2724, 6900, 15186, 30072, 54888, 93924, 152550, 237336, 356172, 518388, 734874, 1018200, 1382736, 1844772, 2422638, 3136824, 4010100, 5067636, 6337122, 7848888, 9636024, 11734500, 14183286, 17024472, 20303388, 24068724

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

			Some solutions for n=6:
..3....1....2....0....3....1....5....1....2....5....6....4....2....6....3....1
..4....5....1....4....6....1....1....2....2....2....1....1....7....1....1....7
..2....8....3....7....4....4....2....7....5....6....3....2....5....8....5....2
..3....8....8....8....8....4....4....7....5....3....8....9....3....5....6....7
.10....6....6....8....9....7....6....6....9....7....5....5....9....5....7....9

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

Row 5 of A250351.

Empirical: a(n) = n^5 + 5*n^4 + 25*n^2 + 5*n for n>2.
Conjectures from Colin Barker, Nov 13 2018: (Start)
G.f.: 2*x*(16 + 12*x + 36*x^2 - 2*x^3 + 18*x^4 - 33*x^5 + 16*x^6 - 3*x^7) / (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>8.
(End)

A250355 Number of length 6 arrays x(i), i=1..6 with x(i) in i..i+n and no value appearing more than 2 times.

Original entry on oeis.org

64, 622, 3349, 12734, 38543, 99344, 226247, 467642, 894599, 1606988, 2740319, 4473302, 7036127, 10719464, 15884183, 22971794, 32515607, 45152612, 61636079, 82848878, 109817519, 143726912, 185935847, 237993194, 301654823, 378901244

Offset: 1

R. H. Hardin, Nov 19 2014

nonn

			Some solutions for n=6:
..1....4....6....5....1....2....1....6....0....3....0....6....2....1....3....5
..1....3....6....2....7....2....5....6....5....5....4....5....3....1....3....1
..7....8....2....4....5....4....3....5....7....6....8....8....7....8....8....4
..3....8....5....7....5....7....8....7....4....8....9....3....8....5....5....3
..4....9....4....8....9...10....4...10....6....7...10....9....8....6....5....4
..9...11....7....5...10....9....8....9....6....7....8...10...10....9....9....6

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

Row 6 of A250351.

Empirical: a(n) = n^6 + 6*n^5 - 5*n^4 + 55*n^3 + 25*n^2 - 61*n + 98 for n>3.
Conjectures from Colin Barker, Nov 13 2018: (Start)
G.f.: x*(64 + 174*x + 339*x^2 + 113*x^3 + 204*x^4 + 168*x^5 - 847*x^6 + 783*x^7 - 336*x^8 + 58*x^9) / (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>10.
(End)

cp's OEIS Frontend

A250361 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 3 times.

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A250346 Number of length n arrays x(i), i=1..n with x(i) in i..i+3 and no value appearing more than 2 times.

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A250347 Number of length n arrays x(i), i=1..n with x(i) in i..i+4 and no value appearing more than 2 times.

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A250348 Number of length n arrays x(i), i=1..n with x(i) in i..i+5 and no value appearing more than 2 times.

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A250349 Number of length n arrays x(i), i=1..n with x(i) in i..i+6 and no value appearing more than 2 times.

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A250350 Number of length n arrays x(i), i=1..n with x(i) in i..i+7 and no value appearing more than 2 times.

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A250352 Number of length 3 arrays x(i), i=1..3 with x(i) in i..i+n and no value appearing more than 2 times.

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A250353 Number of length 4 arrays x(i), i=1..4 with x(i) in i..i+n and no value appearing more than 2 times.

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A250354 Number of length 5 arrays x(i), i=1..5 with x(i) in i..i+n and no value appearing more than 2 times.

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A250355 Number of length 6 arrays x(i), i=1..6 with x(i) in i..i+n and no value appearing more than 2 times.

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