A223718 -id:A223718 - cp's OEIS frontend

A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, 46938, 52956, 59536, 66712, 74519, 82993, 92171, 102091, 112792, 124314, 136698

Offset: 1

N. J. A. Sloane

a(n) is the sum of the first five terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 18 2009
{a(k): 1 <= k <= 5} = divisors of 16. - Reinhard Zumkeller, Jun 17 2009
Equals binomial transform of [1, 1, 1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 02 2010
From Bernard Schott, Apr 05 2021: (Start)
As a(n) = 2^(n-1) for n = 1..5, it is misleading to believe that a(n) = 2^(n-1) for n > 5 (see Patrick Popescu-Pampu link); other curiosities: a(6) = 2^5 - 1 and a(10) = 2^8.
The sequence of the first differences is A000125, the sequence of the second differences is A000124, the sequence of the third differences is A000027 and the sequence of the fourth differences is the all 1's sequence A000012 (see J. H. Conway and R. K. Guy reference, p. 80). (End)
a(n) is the number of binary words of length n matching the regular expression 0*1*0*1*0*. A000124 and A000125 count binary words of the form 0*1*0* and 1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023

			a(7)=99 because the first five terms in the 7th row of Pascal's triangle are 1 + 7 + 21 + 35 + 35 = 99. - _Geoffrey Critzer_, Jan 18 2009
G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 31*x^6 + 57*x^7 + 99*x^8 + 163*x^9 + ...

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 28.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, Chap. 3.
J. H. Conway and R. K. Guy, Le Livre des Nombres, Eyrolles, 1998, p. 80.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 33 pp. 18; 128 Ellipses Paris 2004.
A. Deledicq and D. Missenard, A La Recherche des Régions Perdues, Math. & Malices, No. 22 Summer 1995 issue pp. 22-3 ACL-Editions Paris.
M. Gardner, Mathematical Circus, pp. 177; 180-1 Alfred A. Knopf NY 1979.
M. Gardner, The Colossal Book of Mathematics, 2001, p. 561.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
M. de Guzman, Aventures Mathématiques, Prob. B pp. 115-120 PPUR Lausanne 1990.
Ross Honsberger; Mathematical Gems I, Chap. 9.
Ross Honsberger; Mathematical Morsels, Chap. 3.
Jeux Mathématiques et Logiques, Vol. 3 pp. 12; 51 Prob. 14 FFJM-SERMAP Paris 1988.
J. N. Kapur, Reflections of a Mathematician, Chap.36, pp. 337-343, Arya Book Depot, New Delhi 1996.
C. D. Miller, V. E. Heeren, J. Hornsby, M. L. Morrow and J. Van Newenhizen, Mathematical Ideas, Tenth Edition, Pearson, Addison-Wesley, Boston, 2003, Cptr 1, 'The Art of Problem Solving, page 6.
I. Niven, Mathematics of Choice, pp. 158; 195 Prob. 40 NML 15 MAA 1965.
C. S. Ogilvy, Tomorrow's Math, pp. 144-6 OUP 1972.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 252-255.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 81-87.
A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Alan Calvitti, Illustration of initial terms
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
Colin Defant, Meeting Covered Elements in nu-Tamari Lattices, arXiv:2104.03890 [math.CO], 2021.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy, Letter to N. J. A. Sloane
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Math Forum, Regions of a circle Cut by Chords to n points.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or column, (2014) Table 4 column 1.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Leo Moser and W. Bruce Ross, Mathematical Miscellany, On the Danger of Induction, Mathematics Magazine, Vol. 23, No. 2 (Nov. - Dec., 1949), pp. 109-114.
M. Noy, A Short Solution of a Problem in Combinatorial Geometry, Mathematics Magazine, pp. 52-3 69(1) 1996 MAA.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Patrick Popescu-Pampu, Démarrage trompeur, Images des Mathématiques, CNRS, 2017, rediffusion 2021 (in French).
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
H. P. Robinson, Letter to N. J. A. Sloane, Mar 21 1985
Grant Sanderson, Circle division solution, video, 2015.
Grant Sanderson, Circle division solution, updated video (2023).
K. Uhland, A Blase of Glory
K. Uhland, Moser's Problem
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circle Division by Chords
Eric Weisstein's World of Mathematics, Strong Law of Small Numbers
Reinhart Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000012, A000027, A000124, A000125, A002522, A005408, A016813, A086514, A058331, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261, A007318, A008859-A008863, A219531, A223718.

a000127 = sum . take 5 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

[(n^4-6*n^3+23*n^2-18*n+24)/24: n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

A000127 := n->(n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24;
with (combstruct):ZL:=[S, {S=Sequence(U, card=1)}, unlabeled]: seq(count(subs(r=6, ZL), size=m), m=0..41); # Zerinvary Lajos, Mar 08 2008

f[n_] := Sum[Binomial[n, i], {i, 0, 4}]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Jun 29 2007 *)
Total/@Table[Binomial[n-1,k],{n,50},{k,0,4}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,2,4,8,16},50] (* Harvey P. Dale, Aug 24 2011 *)
Table[(n^4 - 6 n^3 + 23 n^2 - 18 n + 24) / 24, {n, 100}] (* Vincenzo Librandi, Feb 16 2015 *)
a[ n_] := Binomial[n, 4] + Binomial[n, 2] + 1; (* Michael Somos, Dec 23 2017 *)

a(n)=(n^4-6*n^3+23*n^2-18*n+24)/24 \\ Charles R Greathouse IV, Mar 22 2016

{a(n) = binomial(n, 4) + binomial(n, 2) + 1}; /* Michael Somos, Dec 23 2017 */

def A000127(n): return n*(n*(n*(n - 6) + 23) - 18)//24 + 1 # Chai Wah Wu, Sep 18 2021

a(n) = C(n-1, 4) + C(n-1, 3) + ... + C(n-1, 0) = A055795(n) + 1 = C(n, 4) + C(n-1, 2) + n.
a(n) = Sum_{k=0..2} C(n, 2k). - Joel Sanderi (sanderi(AT)itstud.chalmers.se), Sep 08 2004
a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24.
G.f.: (1 - 3*x + 4*x^2 - 2*x^3 + x^4)/(1-x)^5. (for offset 0) - Simon Plouffe in his 1992 dissertation
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24)*exp(x) (for offset 0). [Typos corrected by Juan M. Marquez, Jan 24 2011]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Aug 24 2011
a(n) = A000124(A000217(n-1)) - n*A000217(n-2) - A034827(n), n > 1. - Melvin Peralta, Feb 15 2016
a(n) = A223718(-n). - Michael Somos, Dec 23 2017
For n > 2, a(n) = n + 1 + sum_{i=2..(n-2)}sum_{j=1..(n-i)}(1+(i-1)(j-1)). - Alec Jones, Nov 17 2019

Formula corrected and additional references from torsten.sillke(AT)lhsystems.com

Additional correction from Jonas Paulson (jonasso(AT)sdf.lonestar.org), Oct 30 2003

A223659 Number of unimodal maps [1..n]->[0..3].

Original entry on oeis.org

1, 4, 16, 50, 130, 296, 610, 1163, 2083, 3544, 5776, 9076, 13820, 20476, 29618, 41941, 58277, 79612, 107104, 142102, 186166, 241088, 308914, 391967, 492871, 614576, 760384, 933976, 1139440, 1381300, 1664546, 1994665, 2377673, 2820148, 3329264

Offset: 0

R. H. Hardin, Mar 25 2013

nonn

Column 1 of A223663.

Apparently also column 4 of A071920. - R. J. Mathar, May 17 2014

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..210 from R. H. Hardin)
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Cf. A071920, A223663, A223718.

Empirical: 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 = 1 + n*(n+1)*(n^4 + 14*n^3 + 101*n^2 + 304*n + 660)/720.

Empirical g.f.: 1-x*(x^2-2*x+2)*(x^4-4*x^3+6*x^2-4*x+2) / (x-1)^7. - R. J. Mathar, May 14 2014

a(0)=1 prepended by Alois P. Heinz, Feb 11 2024

A223838 T(n,k) = number of n X k 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 48, 25, 6, 22, 86, 118, 89, 36, 7, 29, 148, 255, 249, 149, 49, 8, 37, 239, 503, 596, 471, 232, 64, 9, 46, 367, 926, 1286, 1240, 824, 342, 81, 10, 56, 541, 1614, 2578, 2884, 2388, 1356, 483, 100, 11, 67, 771, 2690, 4886, 6159, 5992

Offset: 1

R. H. Hardin, Mar 27 2013

			Table starts:
   2   4   7   11    16    22     29     37      46      56      67       79
   3   9  22   46    86   148    239    367     541     771    1068     1444
   4  16  48  118   255   503    926   1614    2690    4318    6712    10146
   5  25  89  249   596  1286   2578   4886    8851   15439   26072    42800
   6  36 149  471  1240  2884   6159  12371   23716   43790   78342   136368
   7  49 232  824  2388  5992  13582  28642   57306  110164  205131   371923
   8  64 342 1356  4325 11749  28369  62869  130891  260219  499470   932402
   9  81 483 2123  7436 21912  56607 132427  287719  591705 1167798  2233350
  10 100 659 3189 12222 39064 108282 268624  611901 1306407 2655964  5202806
  11 121 874 4626 19316 66854 199047 525323 1260152 2805670 5897718 11863338
  ...
Some solutions for n=3 k=4:
..1..0..0..0....0..1..0..0....0..0..0..0....0..0..0..0....0..1..1..0
..1..1..0..0....1..1..1..0....0..0..0..1....0..0..0..0....0..1..1..0
..1..1..0..0....1..1..1..1....0..0..1..1....1..1..1..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..477

Main diagonal is A223832.
Columns 1..7 are A000027(n+1), A000290(n+1), A223833, A223834, A223835, A223836, A223837.
Rows 1..7 are A000124, A223718, A223839, A223840, A223841, A223842, A223843.
Cf. A224262, A224173.

Empirical columns k=1..7 are polynomials of degree k for n>0,0,0,1,2,3,4.

Empirical rows n=1..7 are polynomials of degree 2*n for k>0,0,0,2,4,6,8.

Name corrected by Andrew Howroyd, Mar 18 2025

A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 50, 25, 6, 22, 86, 130, 95, 36, 7, 29, 148, 296, 295, 161, 49, 8, 37, 239, 610, 791, 581, 252, 64, 9, 46, 367, 1163, 1897, 1792, 1036, 372, 81, 10, 56, 541, 2083, 4166, 4900, 3612, 1716, 525, 100, 11, 67, 771, 3544, 8518, 12174, 11088, 6672, 2685, 715, 121, 12

Offset: 1

R. H. Hardin, Apr 23 2013

Table starts
..2...4...7...11....16.....22.....29......37......46.......56.......67
..3...9..22...46....86....148....239.....367.....541......771.....1068
..4..16..50..130...296....610...1163....2083....3544.....5776.....9076
..5..25..95..295...791...1897...4166....8518...16414....30086....52834
..6..36.161..581..1792...4900..12174...27966...60172...122464...237590
..7..49.252.1036..3612..11088..30738...78354..186142...416394...884236
..8..64.372.1716..6672..22716..69498..194634..505912..1233584..2845492
..9..81.525.2685.11517..43065.144111..439791.1241383..3276559..8157227
.10.100.715.4015.18832..76714.278707..920491.2803658..7963384.21280337
.11.121.946.5786.29458.129844.508937.1808521.5911763.17978389.51325352
From Charles A. Lane, Aug 22 2013: (Start)
The first column is also the coefficients of a in y''[x] - a*x^n*y[x] + b*en*y[x] = 0 where n = 0. The recursion yields coefficients of a, a*b*en, a*b^2*en^2 etc.
The second column is obtained when n=1, the third column when n=2. The final column is for n=10.
Example: Write a normal recursion for n=4. For convenience set x to 1. Running the recursion yields
1-(b en)/2+(b^2 en^2)/24+1/30 (a-(b^3 en^3)/24)+(-384 a b en+b^4 en^4)/40320+(2064 a b^2 en^2-b^5 en^5)/3628800+(120960 a^2-7104 a b^3 en^3+b^6 en^6)/479001600+(-4682880 a^2 b en+18984 a b^4 en^4-b^7 en^7)/87178291200+(54268416 a^2 b^2 en^2-43008 a b^5 en^5+b^8 en^8)/20922789888000.
The coefficient of a is 24, the coefficient of a b en is 384 and the coefficient of a b^2 en^2 is 2064. Dividing by 4! yields a sequence of 1,16,86... , the same as column 5 without the leading 1. There is a hint of unity among the oscillators. (End)

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

R. H. Hardin, Table of n, a(n) for n = 1..5304

Column 2 is A000290(n+1).
Column 3 is A002412(n+1).
Column 4 is A006324(n+1).
Row 1 is A000124.
Row 2 is A223718.
Row 3 is A223659.
Cf. A071920, A071921 (larger and reflected versions of table). - Alois P. Heinz, Sep 22 2013

T:= (n, k)-> add(binomial(k+2*j-1, 2*j), j=0..n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 22 2013

T[n_, k_] := Sum[Binomial[k + 2*j - 1, 2*j], {j, 0, n}]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

Empirical: columns k=1..7 are polynomials of degree k.
Empirical: rows n=1..7 are polynomials of degree 2n.
T(n,k) = Sum_{j=0..n} C(k+2*j-1,2*j). - Alois P. Heinz, Sep 22 2013

A223918 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

3, 9, 6, 22, 36, 10, 46, 158, 100, 15, 86, 548, 666, 225, 21, 148, 1600, 3311, 2111, 441, 28, 239, 4102, 13347, 14123, 5548, 784, 36, 367, 9503, 45988, 74040, 48182, 12752, 1296, 45, 541, 20299, 140236, 323394, 319156, 139925, 26494, 2025, 55, 771, 40570

Offset: 1

R. H. Hardin Mar 29 2013

Table starts
..3....9.....22......46.......86.......148........239.........367..........541
..6...36....158.....548.....1600......4102.......9503.......20299........40570
.10..100....666....3311....13347.....45988.....140236......387671.......988447
.15..225...2111...14123....74040....323394....1226491.....4157102.....12856218
.21..441...5548...48182...319156...1721356....7906972....31947798....116289938
.28..784..12752..139925..1147712...7526024...41334135...196691651....831996762
.36.1296..26494..359344..3588729..28247478..183720802..1022160764...4994941017
.45.2025..50863..837243.10031780..93683295..715114500..4620831232..25961964085
.55.3025..91634.1802306.25574524.280338123.2489822632.18558764134.119279565646
.66.4356.156682.3633256.60357024.769012348.7881608430.67311575302.492221386729

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

R. H. Hardin, Table of n, a(n) for n = 1..480

Column 1 is A000217(n+1)
Column 2 is A000537(n+1)
Row 1 is A223718

Empirical: columns k=1..7 are polynomials of degree 2*k for n>0,0,0,1,2,3,4

Empirical: rows n=1..7 are polynomials of degree 4*n

A224262 T(n,k) = number of n X k 0..2 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

3, 9, 6, 22, 36, 10, 46, 158, 100, 15, 86, 548, 648, 225, 21, 148, 1600, 3096, 2017, 441, 28, 239, 4102, 12032, 12467, 5246, 784, 36, 367, 9503, 40182, 59855, 41012, 11990, 1296, 45, 541, 20299, 119367, 240829, 238366, 116692, 24842, 2025, 55, 771, 40570

Offset: 1

R. H. Hardin, Apr 02 2013

			Table starts:
   3    9     22      46       86       148        239         367          541
   6   36    158     548     1600      4102       9503       20299        40570
  10  100    648    3096    12032     40182     119367      322885       808618
  15  225   2017   12467    59855    240829     850875     2717731      8000608
  21  441   5246   41012   238366   1122522    4542734    16423026     54399996
  28  784  11990  116692   816361   4480391   20568693    82733667    301228048
  36 1296  24842  296646  2485967  15921905   83124099   371699763   1478187738
  45 2025  47643  688533  6868203  51343083  306179180  1530419762   6671184875
  55 3025  85838 1482310 17467782 152072846 1038489172  5835731860  28072690614
  66 4356 146878 2995516 41364960 417672794 3266157979 20709405119 110622071553
  ...
Some solutions for n=3 k=4:
..1..1..0..0....1..1..1..1....1..1..2..1....0..2..1..0....0..0..0..0
..2..1..1..0....1..2..2..1....2..2..2..1....0..2..1..1....0..0..2..0
..2..2..1..1....2..2..2..1....2..2..2..2....0..2..2..1....0..0..2..2

R. H. Hardin, Table of n, a(n) for n = 1..241

Main diagonal is A224256.
Columns 1..7 are A000217(n+1), A000537(n+1), A224257, A224258, A224259, A224260, A224261.
Rows 1..7 are A223718, A223919, A224263, A224264, A224265, A224266, A224267.
Cf. A223838.

Empirical: columns k=1..7 are polynomials of order 2*k for n>0,0,0,2,4,6,8.

Empirical: rows n=1..7 are polynomials of degree 4*n for k>0,0,0,2,4,6,8.

Name corrected by Andrew Howroyd, Mar 18 2025

A223789 T(n,k)=Number of nXk 0..2 arrays with rows, diagonals and antidiagonals unimodal.

Original entry on oeis.org

3, 9, 9, 22, 81, 27, 46, 484, 729, 81, 86, 2116, 8635, 6561, 243, 148, 7396, 62365, 151580, 59049, 729, 239, 21904, 334230, 1560013, 2703137, 531441, 2187, 367, 57121, 1455816, 11012718, 39387861, 48302789, 4782969, 6561, 541, 134689, 5425943

Offset: 1

R. H. Hardin Mar 27 2013

Table starts
.....3..........9............22..............46................86
.....9.........81...........484............2116..............7396
....27........729..........8635...........62365............334230
....81.......6561........151580.........1560013..........11012718
...243......59049.......2703137........39387861.........343454446
...729.....531441......48302789......1026135371.......11150023974
..2187....4782969.....862007289.....27088106846......377163884938
..6561...43046721...15379566078....715394830136....12972494260444
.19683..387420489..274427327200..18858304684055...446829906314726
.59049.3486784401.4896915028511.496722962933967.15355124632228358

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

R. H. Hardin, Table of n, a(n) for n = 1..126

Column 1 is A000244
Column 2 is A001019
Row 1 is A223718
Row 2 is A223719

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: [order 15]
k=4: [order 80]
Empirical: rows n=1..5 are polynomials of order 4*n for k>0,0,1,8,15

A223975 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal.

Original entry on oeis.org

3, 9, 9, 22, 81, 27, 46, 484, 729, 81, 86, 2116, 9515, 6561, 243, 148, 7396, 76092, 186004, 59049, 729, 239, 21904, 440628, 2558848, 3628696, 531441, 2187, 367, 57121, 2026448, 22935921, 84988435, 70779056, 4782969, 6561, 541, 134689, 7829639

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
.....3..........9.............22...............46.................86
.....9.........81............484.............2116...............7396
....27........729...........9515............76092.............440628
....81.......6561.........186004..........2558848...........22935921
...243......59049........3628696.........84988435.........1140963027
...729.....531441.......70779056.......2809740785........55803232969
..2187....4782969.....1380511272......92756321858......2708281019793
..6561...43046721....26926081924....3060966419662....131014406127439
.19683..387420489...525177301935..100999995564503...6329626912147424
.59049.3486784401.10243271456697.3332485315028073.305632588672082728

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

R. H. Hardin, Table of n, a(n) for n = 1..286

Column 1 is A000244
Column 2 is A001019
Row 1 is A223718
Row 2 is A223719

Empirical: columns k=1..6 have recurrences of order 1,1,7,18,43,91

Empirical: rows n=1..7 are polynomials of degree 4*n for k>0,0,0,0,2,3,4

A224146 T(n,k)=Number of nXk 0..1 arrays with rows and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 49, 25, 6, 22, 86, 124, 92, 36, 7, 29, 148, 275, 272, 155, 49, 8, 37, 239, 554, 691, 526, 242, 64, 9, 46, 367, 1037, 1573, 1509, 930, 357, 81, 10, 56, 541, 1831, 3296, 3827, 2985, 1536, 504, 100, 11, 67, 771, 3082, 6472, 8838

Offset: 1

R. H. Hardin Mar 31 2013

Table starts
..2...4...7...11....16....22.....29......37......46......56.......67.......79
..3...9..22...46....86...148....239.....367.....541.....771.....1068.....1444
..4..16..49..124...275...554...1037....1831....3082....4984.....7789....11818
..5..25..92..272...691..1573...3296....6472...12058...21506....36961....61517
..6..36.155..526..1509..3827...8838...18969...38392...74053...137204...245636
..7..49.242..930..2985..8375..21183...49365..107697..222603...439909...837071
..8..64.357.1536..5471.16885..46586..117510..275557..608423..1277544..2571226
..9..81.504.2404..9431.31841..95455..259927..653980.1540542..3433254..7299542
.10.100.687.3602.15457.56783.184222..539980.1454873.3651978..8632537.19383022
.11.121.910.5206.24285.96579.337727.1062305.3058144.8167578.20458857.48492125

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

R. H. Hardin, Table of n, a(n) for n = 1..925

Column 1 is A000027(n+1)
Column 2 is A000290(n+1)
Row 1 is A000124
Row 2 is A223718

Empirical: columns k=1..7 are polynomials of degree k for n>0,0,0,1,2,3,4

Empirical: rows n=1..7 are polynomials of degree 2*n

A224310 T(n,k)=Number of nXk 0..2 arrays with diagonals and rows unimodal and antidiagonals nondecreasing.

Original entry on oeis.org

3, 9, 9, 22, 54, 27, 46, 218, 324, 81, 86, 698, 1586, 1944, 243, 148, 1915, 5996, 11361, 11664, 729, 239, 4690, 20214, 45453, 82700, 69984, 2187, 367, 10511, 61953, 164514, 345875, 615481, 419904, 6561, 541, 21919, 174378, 562760, 1258372, 2717759

Offset: 1

R. H. Hardin Apr 03 2013

Table starts
.....3........9.........22..........46..........86..........148..........239
.....9.......54........218.........698........1915.........4690........10511
....27......324.......1586........5996.......20214........61953.......174378
....81.....1944......11361.......45453......164514.......562760......1825800
...243....11664......82700......345875.....1258372......4420701.....15312504
...729....69984.....615481.....2717759.....9829605.....33934344....118317987
..2187...419904....4634768....22071219....80083648....268379906....911404794
..6561..2519424...35003328...182843194...677557164...2215451575...7236130163
.19683.15116544..264487714..1528645389..5882182248..19023816444..59751261572
.59049.90699264.1997888432.12825738594.51821072499.168305254414.512310103541

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

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A000244
Column 2 is 9*6^(n-1)
Row 1 is A223718
Row 2 is A223927

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 6*a(n-1)
k=3: [order 17]
k=4: [order 30] for n>35
k=5: [order 61] for n>69
k=6: [order 88] for n>98
Empirical: rows n=1..7 are polynomials of degree 4*n for k>0,0,3,6,9,12,15

cp's OEIS Frontend

A000127 Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

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A223659 Number of unimodal maps [1..n]->[0..3].

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A223838 T(n,k) = number of n X k 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

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A223918 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal and columns nondecreasing.

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A224262 T(n,k) = number of n X k 0..2 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223789 T(n,k)=Number of nXk 0..2 arrays with rows, diagonals and antidiagonals unimodal.

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A223975 T(n,k)=Number of nXk 0..2 arrays with rows and antidiagonals unimodal.

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