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 1-9 of 9 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

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

Views

Author

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

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Formula

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

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

Views

Author

R. H. Hardin, Dec 29 2011

Keywords

Examples

			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

Links

Crossrefs

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

Formula

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

Views

Author

R. H. Hardin, Oct 15 2012

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Column 3 of A217954. Cf. A217883, A217954.

Formula

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

Views

Author

R. H. Hardin, Oct 14 2012

Keywords

Comments

See A228461 and A217883 for more information about the definition.

Examples

			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

Links

Crossrefs

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

Formula

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

A217950 Number of n element 0..3 arrays with each element the minimum of 4 adjacent elements of a random 0..3 array of n+3 elements.

Original entry on oeis.org

4, 16, 50, 130, 296, 624, 1289, 2741, 6134, 14269, 33577, 78304, 179671, 406765, 915275, 2061311, 4664237, 10605587, 24181837, 55152622, 125626530, 285683099, 648937238, 1473615534, 3347303199, 7607369029, 17296530794, 39333312344
Offset: 1

Views

Author

R. H. Hardin, Oct 15 2012

Keywords

Comments

Column 4 of A217954.

Examples

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

Links

Crossrefs

Cf. A217954.

Formula

Empirical: 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).
Empirical g.f.: x*(1 - x + x^2)*(2 + 2*x + x^2)*(2 + 4*x^2 + 4*x^3 + 5*x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8) / (1 - 4*x + 6*x^2 - 4*x^3 + x^4 - 6*x^5 + 4*x^6 - 6*x^7 - 4*x^8 - 5*x^9 - x^10 - 3*x^11 - 2*x^12 - x^13). - Colin Barker, Jul 23 2018

A217951 Number of n element 0..3 arrays with each element the minimum of 5 adjacent elements of a random 0..3 array of n+4 elements.

Original entry on oeis.org

4, 16, 50, 130, 296, 610, 1177, 2209, 4202, 8366, 17569, 38251, 84134, 183334, 392284, 824110, 1708779, 3523509, 7276452, 15117730, 31635969, 66571659, 140445543, 296202061, 623355780, 1308333724, 2740013491, 5731815041, 11989696656
Offset: 1

Views

Author

R. H. Hardin Oct 15 2012

Keywords

Comments

Column 5 of A217954

Examples

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

Links

Formula

Empirical: 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)

A217952 Number of n element 0..3 arrays with each element the minimum of 6 adjacent elements of a random 0..3 array of n+5 elements.

Original entry on oeis.org

4, 16, 50, 130, 296, 610, 1163, 2097, 3670, 6434, 11666, 22313, 44907, 93206, 194831, 403141, 818641, 1628770, 3184993, 6157786, 11854873, 22877202, 44459405, 87168326, 172249773, 342083954, 680463827, 1351826291, 2677343192, 5283369867
Offset: 1

Views

Author

R. H. Hardin Oct 15 2012

Keywords

Comments

Column 6 of A217954

Examples

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

Links

Formula

Empirical: 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)

A217953 Number of n element 0..3 arrays with each element the minimum of 7 adjacent elements of a random 0..3 array of n+6 elements.

Original entry on oeis.org

4, 16, 50, 130, 296, 610, 1163, 2083, 3558, 5902, 9734, 16410, 28969, 54049, 105529, 211097, 423650, 840654, 1636964, 3121556, 5839194, 10760322, 19647966, 35780070, 65376302, 120393728, 223926347, 420463887, 795087960, 1508967508, 2864523898
Offset: 1

Views

Author

R. H. Hardin Oct 15 2012

Keywords

Comments

Column 7 of A217954

Examples

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

Links

Formula

Empirical: 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)
Showing 1-9 of 9 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.