A228461 -id:A228461 - cp's OEIS frontend

A228740 T(n,k) = number of arrays of the median of three adjacent elements of some length n+2 0..k array.

2, 3, 4, 4, 9, 8, 5, 16, 27, 16, 6, 25, 64, 77, 28, 7, 36, 125, 232, 185, 50, 8, 49, 216, 545, 696, 447, 88, 9, 64, 343, 1096, 1943, 2072, 1071, 156, 10, 81, 512, 1981, 4504, 6797, 6130, 2593, 278, 11, 100, 729, 3312, 9191, 17986, 23627, 18378, 6333, 496, 12, 121

Offset: 1

R. H. Hardin Sep 01 2013

See A228461 for more information about the definition. - N. J. A. Sloane, Sep 02 2013
Table starts
...2....3.....4......5.......6.......7........8........9.......10........11
...4....9....16.....25......36......49.......64.......81......100.......121
...8...27....64....125.....216.....343......512......729.....1000......1331
..16...77...232....545....1096....1981.....3312.....5217.....7840.....11341
..28..185...696...1943....4504....9191....17088....29589....48436.....75757
..50..447..2072...6797...17986...41083....84288...159321...282274....474551
..88.1071..6130..23627...71278..181885...410828...845517..1617004...2913955
.156.2593.18378..83391..287154..819099..2037214..4564455..9418762..18182967
.278.6333.55716.298239.1174282.3749921.10282648.25107493.55950398.115793733

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

R. H. Hardin, Table of n, a(n) for n = 1..811

Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Row 3 is A000578(n+1)
For other rows, columns and diagonals see A228739-A228744.

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-3) +a(n-5)
k=2: [order 14]
k=3: [order 26]
k=4: [order 43]
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) = (2/3)*n^4 + 4*n^3 + (19/3)*n^2 + 4*n + 1
n=5: [polynomial of degree 5]
n=6: [polynomial of degree 6]
n=7: [polynomial of degree 7]

A217883 T(n,k) = number of n-element 0..2 arrays with each element the minimum of k adjacent elements of a random 0..2 array of n+k-1 elements.

Original entry on oeis.org

3, 3, 9, 3, 9, 27, 3, 9, 22, 81, 3, 9, 22, 51, 243, 3, 9, 22, 46, 121, 729, 3, 9, 22, 46, 91, 292, 2187, 3, 9, 22, 46, 86, 183, 704, 6561, 3, 9, 22, 46, 86, 153, 383, 1691, 19683, 3, 9, 22, 46, 86, 148, 274, 819, 4059, 59049, 3, 9, 22, 46, 86, 148, 244, 511, 1749, 9749, 177147

Offset: 1

R. H. Hardin, observation that the diagonal is a polynomial from L. Edson Jeffery in the Sequence Fans Mailing List, Oct 14 2012

See A228461 and A217954 for more information about the definition. - N. J. A. Sloane, Sep 02 2013
Table starts
........3......3......3.....3.....3.....3....3....3....3....3....3....3....3
........9......9......9.....9.....9.....9....9....9....9....9....9....9....9
.......27.....22.....22....22....22....22...22...22...22...22...22...22...22
.......81.....51.....46....46....46....46...46...46...46...46...46...46...46
......243....121.....91....86....86....86...86...86...86...86...86...86...86
......729....292....183...153...148...148..148..148..148..148..148..148..148
.....2187....704....383...274...244...239..239..239..239..239..239..239..239
.....6561...1691....819...511...402...372..367..367..367..367..367..367..367
....19683...4059...1749...993...685...576..546..541..541..541..541..541..541
....59049...9749...3699..1966..1223...915..806..776..771..771..771..771..771
...177147..23422...7772..3880..2263..1520.1212.1103.1073.1068.1068.1068.1068
...531441..56268..16316..7558..4243..2639.1896.1588.1479.1449.1444.1444.1444
..1594323.135166..34325.14544..7910..4711.3107.2364.2056.1947.1917.1912.1912
..4782969.324692..72349.27819.14528..8471.5285.3681.2938.2630.2521.2491.2486
.14348907.779977.152573.53226.26274.15107.9166.5980.4376.3633.3325.3216.3186

			Some solutions for n=4 k=4
..0....0....2....1....0....0....1....2....0....2....2....1....0....2....1....1
..2....2....2....1....0....0....1....1....1....2....1....2....2....2....1....2
..1....2....2....1....0....2....2....0....2....2....1....2....2....2....2....2
..0....0....0....1....1....0....1....0....0....2....1....1....2....1....0....2

R. H. Hardin, Table of n, a(n) for n = 1..902

Column 2 is A202882(n+1). Cf. A228461, A217954, A217878.

Empirical for column k:
k=2: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -a(n-4) +a(n-5)
k=3: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-4) -a(n-5) +a(n-6) +a(n-7)
k=4: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-5) -a(n-6) +a(n-7) +a(n-8) +a(n-9)
k=5: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) +a(n-11)
k=6: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-7) -a(n-8) +a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13)
k=7: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-8) -a(n-9) +a(n-10) +a(n-11) +a(n-12) +a(n-13) +a(n-14) +a(n-15)
Diagonal: a(n) = (1/24)*n^4 + (1/4)*n^3 + (23/24)*n^2 + (3/4)*n + 1

A217954 T(n,k) = number of n-element 0..3 arrays with each element the minimum of k adjacent elements of a random 0..3 array of n+k-1 elements.

Original entry on oeis.org

4, 4, 16, 4, 16, 64, 4, 16, 50, 256, 4, 16, 50, 144, 1024, 4, 16, 50, 130, 422, 4096, 4, 16, 50, 130, 310, 1268, 16384, 4, 16, 50, 130, 296, 736, 3823, 65536, 4, 16, 50, 130, 296, 624, 1821, 11472, 262144, 4, 16, 50, 130, 296, 610, 1289, 4673, 34350, 1048576, 4, 16

Offset: 1

R. H. Hardin, suggestion that the diagonal might be a polynomial from L. Edson Jeffery in the Sequence Fans Mailing List, Oct 15 2012

See A228461 for comments on the definition. - N. J. A. Sloane, Sep 02 2013
Table starts
........4......4......4.....4.....4.....4.....4.....4.....4.....4.....4.....4
.......16.....16.....16....16....16....16....16....16....16....16....16....16
.......64.....50.....50....50....50....50....50....50....50....50....50....50
......256....144....130...130...130...130...130...130...130...130...130...130
.....1024....422....310...296...296...296...296...296...296...296...296...296
.....4096...1268....736...624...610...610...610...610...610...610...610...610
....16384...3823...1821..1289..1177..1163..1163..1163..1163..1163..1163..1163
....65536..11472...4673..2741..2209..2097..2083..2083..2083..2083..2083..2083
...262144..34350..12107..6134..4202..3670..3558..3544..3544..3544..3544..3544
..1048576.102896..31103.14269..8366..6434..5902..5790..5776..5776..5776..5776
..4194304.308419..79039.33577.17569.11666..9734..9202..9090..9076..9076..9076
.16777216.924532.199819.78304.38251.22313.16410.14478.13946.13834.13820.13820

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

R. H. Hardin, Table of n, a(n) for n = 1..1275

Column 2 is A203094(n+1). A217949 is also a column. Cf. A228461, A217883.

Empirical for column k:
k=2: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)
k=3: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) +5*a(n-4) -4*a(n-5) +6*a(n-6) +4*a(n-7) +2*a(n-9) +a(n-10)
k=4: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-5) -4*a(n-6) +6*a(n-7) +4*a(n-8) +5*a(n-9) +a(n-10) +3*a(n-11) +2*a(n-12) +a(n-13)
k=5: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-6) -4*a(n-7) +6*a(n-8) +4*a(n-9) +5*a(n-10) +6*a(n-11) +2*a(n-12) +4*a(n-13) +3*a(n-14) +2*a(n-15) +a(n-16)
k=6: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-7) -4*a(n-8) +6*a(n-9) +4*a(n-10) +5*a(n-11) +6*a(n-12) +7*a(n-13) +3*a(n-14) +5*a(n-15) +4*a(n-16) +3*a(n-17) +2*a(n-18) +a(n-19)
k=7: a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) +6*a(n-8) -4*a(n-9) +6*a(n-10) +4*a(n-11) +5*a(n-12) +6*a(n-13) +7*a(n-14) +8*a(n-15) +4*a(n-16) +6*a(n-17) +5*a(n-18) +4*a(n-19) +3*a(n-20) +2*a(n-21) +a(n-22)
Diagonal: a(n) = (1/720)*n^6 + (1/48)*n^5 + (23/144)*n^4 + (9/16)*n^3 + (241/180)*n^2 + (11/12)*n + 1

A203094 Number of nX1 0..3 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 16, 50, 144, 422, 1268, 3823, 11472, 34350, 102896, 308419, 924532, 2771101, 8305373, 24892609, 74608516, 223618304, 670231838, 2008825312, 6020872062, 18045827096, 54087163859, 162110668160, 485879938474, 1456284886944

Offset: 1

R. H. Hardin, Dec 29 2011

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Lemma 2.1, k=4, no peak.

Column 1 of A203101. Also a column of A228461. Cf. A217883, A217954.

Empirical: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7).

Empirical: G.f.: -x*(1+6*x^2+5*x^4+x^6) / (-1+4*x-6*x^2+10*x^3-5*x^4+6*x^5-x^6+x^7). - R. J. Mathar, May 17 2014

A217949 Number of n-element 0..3 arrays with each element the minimum of 3 adjacent elements of a random 0..3 array of n+2 elements.

Original entry on oeis.org

4, 16, 50, 130, 310, 736, 1821, 4673, 12107, 31103, 79039, 199819, 505477, 1282309, 3259549, 8288613, 21064316, 53497376, 135833020, 344914900, 875983319, 2224986219, 5651490601, 14354263713, 36457137516, 92593166734, 235168023403

Offset: 1

R. H. Hardin, Oct 15 2012

nonn

See A217954 and A228461 for more information. - N. J. A. Sloane, Sep 02 2013

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

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

Column 3 of A217954. Cf. A217883, A217954.

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

Empirical: G.f.: -x*(x^3+x^2+2) *(x^6+2*x^5+x^4+4*x^3+4*x^2+2) / ( (x^5+3*x^2-2*x+1) *(x^5+2*x^4+x^2+2*x-1) ). - R. J. Mathar, May 17 2014

A217878 Number of n-element 0..2 arrays with each element the minimum of 3 adjacent elements of a random 0..2 array of n+2 elements.

Original entry on oeis.org

3, 9, 22, 46, 91, 183, 383, 819, 1749, 3699, 7772, 16316, 34325, 72349, 152573, 321621, 677623, 1427389, 3006930, 6335210, 13348399, 28125235, 59258363, 124851495, 263048937, 554220135, 1167698552, 2460253944, 5183565225, 10921353721

Offset: 1

R. H. Hardin, Oct 14 2012

nonn

See A228461 and A217883 for more information about the definition.

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

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

Column 3 of A217954. Also a column of A217883. Cf. A217954.

Empirical: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +3*a(n-4) -a(n-5) +a(n-6) +a(n-7).

Empirical g.f.: x*(1 + x^2 + x^3)*(3 + x^2 + x^3) / ((1 - x + x^2 + x^3)*(1 - 2*x - x^4)). - Colin Barker, Feb 23 2018

A228464 Number of arrays of maxima of three adjacent elements of some 0..n array of length 9.

Original entry on oeis.org

44, 383, 1821, 6254, 17487, 42386, 92430, 185727, 349558, 623513, 1063283, 1745172, 2771393, 4276212, 6433004, 9462285, 13640784, 19311619, 26895641, 36904010, 49952067, 66774566, 88242330, 115380395, 149387706, 191658429, 243804943

Offset: 1

R. H. Hardin, Aug 22 2013

nonn

See A228461 for explanation of definition.

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

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

Row 7 of A228461. Cf. A217949.

Empirical: a(n) = (4/315)*n^7 + (1/5)*n^6 + (91/45)*n^5 + (63/8)*n^4 + (2557/180)*n^3 + (517/40)*n^2 + (2419/420)*n + 1 = (n+1) *(n+2) *(32*n^5 + 408*n^4 + 3808*n^3 + 7605*n^2 + 5367*n + 1260)/2520.
Conjectures from Colin Barker, Mar 16 2018: (Start)
G.f.: x*(44 + 31*x - 11*x^2 - 54*x^3 + 75*x^4 - 28*x^5 + 8*x^6 - x^7) / (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>8.
(End)

A228457 Number of arrays of maxima of three adjacent elements of some length n+2 0..4 array.

Original entry on oeis.org

5, 25, 95, 295, 821, 2227, 6254, 18394, 55285, 165563, 489488, 1433536, 4188613, 12268735, 36052642, 106118146, 312319987, 918413167, 2698818913, 7929048609, 23298823959, 68477103061, 201284983126, 591668573686, 1739085580405

Offset: 1

R. H. Hardin Aug 22 2013

nonn

Column 4 of A228461

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

R. H. Hardin, Table of n, a(n) for n = 1..191

Empirical: a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) +5*a(n-4) -9*a(n-5) +20*a(n-6) +8*a(n-7) +2*a(n-8) +14*a(n-9) +6*a(n-10) +2*a(n-11) +3*a(n-12) +a(n-13)

A228458 Number of arrays of maxima of three adjacent elements of some length n+2 0..5 array.

Original entry on oeis.org

6, 36, 161, 581, 1847, 5615, 17487, 57303, 194064, 659418, 2212695, 7329603, 24138450, 79571792, 263302273, 873867641, 2902730184, 9635417630, 31950864559, 105883279595, 350872561227, 1163005850963, 3855906404239, 12785544028311

Offset: 1

R. H. Hardin, Aug 22 2013

nonn

Column 5 of A228461.

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Cf. A228461.

Empirical: a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -14*a(n-5) +49*a(n-6) +8*a(n-7) +15*a(n-8) +54*a(n-9) +21*a(n-10) +20*a(n-11) +26*a(n-12) +10*a(n-13) +5*a(n-14) +4*a(n-15) +a(n-16).

A228459 Number of arrays of maxima of three adjacent elements of some length n+2 0..6 array.

Original entry on oeis.org

7, 49, 252, 1036, 3703, 12453, 42386, 151882, 567835, 2147149, 8034340, 29610140, 108114007, 394245755, 1442791435, 5301606775, 19520651849, 71865418075, 264265863322, 970764224658, 3564641145009, 13091320557593, 48095042423647

Offset: 1

R. H. Hardin Aug 22 2013

nonn

Column 6 of A228461

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

R. H. Hardin, Table of n, a(n) for n = 1..95

Empirical: a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -14*a(n-4) -14*a(n-5) +98*a(n-6) -6*a(n-7) +63*a(n-8) +151*a(n-9) +59*a(n-10) +107*a(n-11) +124*a(n-12) +60*a(n-13) +56*a(n-14) +44*a(n-15) +17*a(n-16) +9*a(n-17) +5*a(n-18) +a(n-19)

cp's OEIS Frontend

A228740 T(n,k) = number of arrays of the median of three adjacent elements of some length n+2 0..k array.

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A217883 T(n,k) = number of n-element 0..2 arrays with each element the minimum of k adjacent elements of a random 0..2 array of n+k-1 elements.

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A217954 T(n,k) = number of n-element 0..3 arrays with each element the minimum of k adjacent elements of a random 0..3 array of n+k-1 elements.

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A203094 Number of nX1 0..3 arrays with every nonzero element less than or equal to some horizontal or vertical neighbor.

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A217949 Number of n-element 0..3 arrays with each element the minimum of 3 adjacent elements of a random 0..3 array of n+2 elements.

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A217878 Number of n-element 0..2 arrays with each element the minimum of 3 adjacent elements of a random 0..2 array of n+2 elements.

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A228464 Number of arrays of maxima of three adjacent elements of some 0..n array of length 9.

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A228457 Number of arrays of maxima of three adjacent elements of some length n+2 0..4 array.

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A228458 Number of arrays of maxima of three adjacent elements of some length n+2 0..5 array.

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