cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A228461 Two-dimensional array read by antidiagonals: T(n,k) = number of arrays of maxima of three adjacent elements of some length n+2 0..k array.

Original entry on oeis.org

2, 3, 4, 4, 9, 7, 5, 16, 22, 11, 6, 25, 50, 46, 17, 7, 36, 95, 130, 91, 27, 8, 49, 161, 295, 310, 183, 44, 9, 64, 252, 581, 821, 736, 383, 72, 10, 81, 372, 1036, 1847, 2227, 1821, 819, 117, 11, 100, 525, 1716, 3703, 5615, 6254, 4673, 1749, 189, 12, 121, 715, 2685, 6812
Offset: 1

Views

Author

R. H. Hardin Aug 22 2013

Keywords

Comments

There are two arrays (or lists, or vectors) involved, a length n+2 array with free elements from 0..k (thus (k+1)^(n+2) of them) and an array that is being enumerated of length n, each element of the latter being the maximum of three adjacent elements of the first array.
Many different first arrays can give the same second array.

Examples

			Table starts
...2....3.....4.....5......6......7.......8.......9......10.......11.......12
...4....9....16....25.....36.....49......64......81.....100......121......144
...7...22....50....95....161....252.....372.....525.....715......946.....1222
..11...46...130...295....581...1036....1716....2685....4015.....5786.....8086
..17...91...310...821...1847...3703....6812...11721...19117....29843....44914
..27..183...736..2227...5615..12453...25096...46941...82699...138699...223224
..44..383..1821..6254..17487..42386...92430..185727..349558...623513..1063283
..72..819..4673.18394..57303.151882..357510..768231.1535578..2893605..5191407
.117.1749.12107.55285.194064.567835.1453506.3357985.7152815.14263777.26930773
Some solutions for n=4 k=4
..3....4....4....3....3....4....3....4....3....0....3....3....4....2....0....2
..0....4....1....3....2....0....2....4....1....0....3....3....1....2....0....0
..4....1....1....0....1....0....4....0....0....0....2....1....4....2....2....3
..4....0....3....3....2....0....4....2....3....1....3....1....4....0....4....3

Links

Crossrefs

Column 1 is A005252(n+3)
Column 2 is A217878
Column 3 is A217949.
A228464 is another column.
Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Row 3 is A002412(n+1)
Row 4 is A006324(n+1)
See A217883, A217954 for similar arrays.

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-4)
k=2: 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=3: [order 10]
k=4: [order 13]
k=5: [order 16]
k=6: [order 19]
k=7: [order 22]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = (2/3)*n^3 + (5/2)*n^2 + (17/6)*n + 1
n=4: a(n) = (1/3)*n^4 + 2*n^3 + (25/6)*n^2 + (7/2)*n + 1
n=5: a(n) = (2/15)*n^5 + (7/6)*n^4 + (25/6)*n^3 + (19/3)*n^2 + (21/5)*n + 1
n=6: [polynomial of degree 6]
n=7: [polynomial of degree 7]

Extensions

Edited by N. J. A. Sloane, Sep 02 2013

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

Views

Author

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

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Formula

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

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

Views

Author

R. H. Hardin, Aug 22 2013

Keywords

Comments

See A228461 for explanation of definition.

Examples

			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

Links

Crossrefs

Row 7 of A228461. Cf. A217949.

Formula

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)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.