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 1-8 of 8 results.

A223718 Number of unimodal functions [1..n]->[0..2].

Original entry on oeis.org

1, 3, 9, 22, 46, 86, 148, 239, 367, 541, 771, 1068, 1444, 1912, 2486, 3181, 4013, 4999, 6157, 7506, 9066, 10858, 12904, 15227, 17851, 20801, 24103, 27784, 31872, 36396, 41386, 46873, 52889, 59467, 66641, 74446, 82918, 92094, 102012, 112711, 124231
Offset: 0

Views

Author

R. H. Hardin, Mar 26 2013

Keywords

Comments

Column 1 of A223725.

Examples

			Some solutions for n=3
..1....2....0....1....0....2....1....2....0....2....0....1....0....1....0....1
..2....1....1....1....0....0....0....1....0....2....2....1....1....2....2....2
..0....1....0....0....1....0....0....0....2....2....1....1....1....2....0....1
From _Joerg Arndt_, Dec 27 2023: (Start)
The a(3) = 22 such functions are (dots for zeros)
   1:  [ . . . ]
   2:  [ . . 1 ]
   3:  [ . . 2 ]
   4:  [ . 1 . ]
   5:  [ . 1 1 ]
   6:  [ . 1 2 ]
   7:  [ . 2 . ]
   8:  [ . 2 1 ]
   9:  [ . 2 2 ]
  10:  [ 1 . . ]
  11:  [ 1 1 . ]
  12:  [ 1 1 1 ]
  13:  [ 1 1 2 ]
  14:  [ 1 2 . ]
  15:  [ 1 2 1 ]
  16:  [ 1 2 2 ]
  17:  [ 2 . . ]
  18:  [ 2 1 . ]
  19:  [ 2 1 1 ]
  20:  [ 2 2 . ]
  21:  [ 2 2 1 ]
  22:  [ 2 2 2 ]
(End)

Links

Crossrefs

Column m=3 of A071920.
Cf. A000124 (unimodal functions [1..n]->[0..1]), A088536 ([1..n] -> [1..n]).

Formula

a(n) = A071920(n,3) = 1+n*(n+1)*(n^2+5*n+18)/24.
G.f.: 1-x*(x^2-3*x+3)*(x^2-x+1) / (x-1)^5 . a(n) = 1+A051744(n). - R. J. Mathar, May 17 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 27 2023

A223719 Number of n X 2 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

9, 81, 484, 2116, 7396, 21904, 57121, 134689, 292681, 594441, 1140624, 2085136, 3655744, 6180196, 10118761, 16104169, 24990001, 37908649, 56340036, 82192356, 117896164, 166513216, 231861529, 318658201, 432681601, 580954609, 771950656
Offset: 1

Views

Author

R. H. Hardin, Mar 26 2013

Keywords

Comments

Column 2 of A223725.

Examples

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

Links

Formula

Empirical: a(n) = (1/576)*n^8 + (1/48)*n^7 + (41/288)*n^6 + (13/24)*n^5 + (793/576)*n^4 + (31/16)*n^3 + (119/48)*n^2 + (3/2)*n + 1.
Conjectures from Colin Barker, Feb 19 2018: (Start)
G.f.: x*(9 + 79*x^2 - 80*x^3 + 106*x^4 - 68*x^5 + 31*x^6 - 8*x^7 + x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A223720 Number of n X 3 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

22, 484, 5883, 46613, 273562, 1285547, 5087912, 17557701, 54161878, 152154419, 394849354, 957192985, 2187192034, 4745347460, 9834757897, 19568773673, 37541913638, 69694254022, 125590862829, 220277446621, 376922731034
Offset: 1

Views

Author

R. H. Hardin, Mar 26 2013

Keywords

Comments

Column 3 of A223725.

Examples

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

Links

Crossrefs

Cf. A223725.

Formula

Empirical: a(n) = (5051/239500800)*n^12 + (61/177408)*n^11 + (91271/21772800)*n^10 + (27/896)*n^9 + (1198121/7257600)*n^8 + (1087/1792)*n^7 + (36617153/21772800)*n^6 + (63719/20160)*n^5 + (26821099/5443200)*n^4 + (305/56)*n^3 + (149993/207900)*n^2 + (3277/770)*n + 1.

A223721 Number of nX4 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

46, 2116, 46613, 608855, 5537147, 38566854, 218619076, 1051051942, 4413826871, 16548850432, 56334327215, 176427795883, 513754867486, 1403147317562, 3620163947216, 8876634938318, 20791393840502, 46723393566999
Offset: 1

Views

Author

R. H. Hardin Mar 26 2013

Keywords

Comments

Column 4 of A223725

Examples

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

Links

Formula

Empirical: a(n) = (456419/5230697472000)*n^16 + (299363/186810624000)*n^15 + (8437981/261534873600)*n^14 + (678959/1868106240)*n^13 + (91942579/28740096000)*n^12 + (161673031/7185024000)*n^11 + (203867119/1828915200)*n^10 + (37055/81648)*n^9 + (59612358509/36578304000)*n^8 + (2850286193/1306368000)*n^7 + (7237625987/718502400)*n^6 + (573765127/143700480)*n^5 - (109277252779/4036032000)*n^4 + (59181202013/216216000)*n^3 - (12927784033/16816800)*n^2 + (36815651/36036)*n - 470 for n>2

A223722 Number of nX5 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

86, 7396, 273562, 5537147, 74140718, 733129227, 5733691150, 37151436091, 206162721823, 1004293423046, 4376877826065, 17322507754495, 63011209241684, 212742488020655, 672145108969951, 2000883664124046, 5644951281387772
Offset: 1

Views

Author

R. H. Hardin Mar 26 2013

Keywords

Comments

Column 5 of A223725

Examples

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

Links

Formula

Empirical: a(n) = (63370093/405483668029440000)*n^20 + (332923627/121645100408832000)*n^19 + (35260507/365849926041600)*n^18 + (11581079/8208171417600)*n^17 + (1549596019/89669099520000)*n^16 + (13415418209/62768369664000)*n^15 + (11281922929/8369115955200)*n^14 + (42712449023/3766102179840)*n^13 + (170811516793/3762339840000)*n^12 + (464597326061/877879296000)*n^11 - (328969296617/78829977600)*n^10 + (13832782422227/241416806400)*n^9 - (125724812428752281/313841848320000)*n^8 + (25263722436660997/11769069312000)*n^7 - (2160123866746163/269007298560)*n^6 + (37001904714133/1676505600)*n^5 - (815520542725887227/15878903040000)*n^4 + (13936490795992651/102918816000)*n^3 - (17701274883426919/53330659200)*n^2 + (9337885696189/21162960)*n - 143729 for n>5

A223723 Number of nX6 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

148, 21904, 1285547, 38566854, 733129227, 9991366555, 105179942478, 899048082946, 6467662339564, 40225236084575, 220829853370074, 1087845990005427, 4872987051273250, 20066702492608999, 76656800237546730
Offset: 1

Views

Author

R. H. Hardin Mar 26 2013

Keywords

Comments

Column 6 of A223725

Examples

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

Links

Formula

Empirical: a(n) = (114400861/793412278431252480000)*n^24 + (66282761/35608838483312640000)*n^23 + (140404438987/963429195237949440000)*n^22 + (1936103317/873349438832640000)*n^21 + (5962068639811/175168944588718080000)*n^20 + (4704445101737/4865804016353280000)*n^19 - (90703908587/1008373858652160000)*n^18 + (138127154730611/896332318801920000)*n^17 - (163329289577203/542318713896960000)*n^16 + (16908411747809/1464595292160000)*n^15 - (217368557681197651/2982752926433280000)*n^14 + (11820863509373563/15064408719360000)*n^13 + (257155619740709796619/41758540970065920000)*n^12 - (25420855801734316747/105450861035520000)*n^11 + (456374229168972649487/135579678474240000)*n^10 - (17588919370618913759/675967057920000)*n^9 + (3304860053076283198397/33894919618560000)*n^8 + (2999124386516753119073/16005934264320000)*n^7 - (9630627493426210141958447/2128789257154560000)*n^6 + (513812787405741819167581/19711011640320000)*n^5 - (356890305042110406199/4740986764800)*n^4 + (6260443467595336664171/82129215168000)*n^3 + (477210241388775897973/2151003254400)*n^2 - (16607355118655295/16997552)*n + 1358946210 for n>10

A223724 Number of nX7 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

239, 57121, 5087912, 218619076, 5733691150, 105179942478, 1461411174370, 16206659678557, 148957213473086, 1167498177877341, 7978569040814189, 48389995885548620, 264233150465396312, 1314507634444102055
Offset: 1

Views

Author

R. H. Hardin Mar 26 2013

Keywords

Comments

Column 7 of A223725

Examples

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

Links

A223717 Number of n X n 0..2 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

3, 81, 5883, 608855, 74140718, 9991366555, 1461411174370, 229731305080429, 38440119395606004
Offset: 1

Views

Author

R. H. Hardin Mar 26 2013

Keywords

Comments

Diagonal of A223725

Examples

			Some solutions for n=3
..2..1..1....2..2..0....2..2..2....1..1..2....2..0..0....2..1..1....1..2..1
..0..0..1....2..1..1....2..2..0....2..1..1....0..1..0....1..2..2....1..2..1
..0..0..0....1..1..1....2..2..0....0..0..1....0..2..0....0..1..0....0..1..0
Showing 1-8 of 8 results.

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