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 1-10 of 12 results. Next

A200865 Number of 0..2 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

17, 37, 77, 163, 343, 723, 1523, 3209, 6761, 14245, 30013, 63235, 133231, 280707, 591427, 1246089, 2625409, 5531525, 11654477, 24555043, 51735495, 109002515, 229659507, 483874057, 1019483609, 2147969733, 4525598973, 9535072003
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Column 2 of A200871.

Examples

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

Links

Programs

  • Mathematica
    a[0,x_,y_] := 1; a[n_,x_,y_] := a[n,x,y] = Sum[If[z <=x<= y || y <=x<= z, a[n-1, z, x], 0], {z, 3}]; a[n_] := Sum[a[n, x, y], {x, 3}, {y, 3}]; Array[a, 25] (* Giovanni Resta, Mar 05 2014 *)

Formula

Empirical: a(n) = 2*a(n-1) +a(n-4).
Empirical g.f.: x*(17 + 3*x + 3*x^2 + 9*x^3) / (1 - 2*x - x^4). - Colin Barker, Oct 15 2017

A200866 Number of 0..3 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

36, 94, 236, 602, 1528, 3882, 9858, 25038, 63592, 161514, 410218, 1041884, 2646208, 6720920, 17069998, 43354902, 110114102, 279671154, 710317326, 1804085608, 4582071648, 11637685314, 29557748082, 75071670020, 190669317026, 484267746348, 1229957991200, 3123884816870
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Examples

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

Links

Crossrefs

Column 3 of A200871.

Programs

  • Mathematica
    a[0,x_,y_] := 1; a[n_,x_,y_] := a[n,x,y] = Sum[If[z <=x<= y || y <=x<= z, a[n-1, z, x], 0], {z, 4}]; a[n_] := Sum[a[n, x, y], {x, 4}, {y, 4}]; Array[a, 25] (* Giovanni Resta, Mar 05 2014 *)

Formula

Empirical: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5).
Empirical g.f.: 2*x*(18 + 11*x + 6*x^2 + 18*x^3 + 8*x^4) / (1 - 2*x - x^2 - 2*x^4 - x^5). - Colin Barker, Oct 15 2017

A200867 Number of 0..4 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

65, 195, 567, 1673, 4917, 14455, 42479, 124851, 366959, 1078565, 3170093, 9317449, 27385589, 80491001, 236577045, 695341043, 2043728099, 6006871845, 17655239697, 51891816107, 152519060911, 448280011791, 1317572818499, 3872575368989
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Column 4 of A200871.

Examples

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

Links

Programs

  • Mathematica
    a[0,x_,y_] := 1; a[n_,x_,y_] := a[n,x,y] = Sum[If[z <=x<= y || y <=x<= z, a[n-1, z, x], 0], {z, 5}]; a[n_] := Sum[a[n, x, y], {x, 5}, {y, 5}]; Array[a, 25] (* Giovanni Resta, Mar 05 2014 *)

Formula

Empirical: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7).
Empirical g.f.: x*(65 + 47*x^2 + 102*x^3 + 10*x^4 + 30*x^5 + 25*x^6) / (1 - 3*x + x^2 - x^3 - 4*x^4 - x^6 - x^7). - Colin Barker, Oct 16 2017

A200868 Number of 0..5 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

106, 356, 1168, 3886, 12890, 42744, 141688, 469726, 1557320, 5163158, 17117854, 56752072, 188154290, 623802050, 2068138180, 6856654898, 22732385492, 75366392740, 249867889178, 828405870894, 2746476505360, 9105600837300
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Column 5 of A200871.

Examples

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

Links

Programs

  • Mathematica
    a[0, x_, y_] := 1; a[n_, x_, y_] := a[n, x, y] = Sum[If[z <= x <= y || y <= x <= z, a[n - 1, z, x], 0], {z, 6}]; a[n_] := Sum[a[n, x, y], {x, 6}, {y, 6}]; Array[a, 25] (* Giovanni Resta, Mar 06 2014 *)

Formula

Empirical: a(n) = 3*a(n-1) + a(n-3) + 7*a(n-4) + 3*a(n-5) + 2*a(n-6) + 3*a(n-7) + a(n-8).
Empirical g.f.: 2*x*(53 + 19*x + 50*x^2 + 138*x^3 + 67*x^4 + 48*x^5 + 57*x^6 + 18*x^7) / (1 - 3*x - x^3 - 7*x^4 - 3*x^5 - 2*x^6 - 3*x^7 - x^8). - Colin Barker, Oct 16 2017

A200869 Number of 0..6 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

161, 595, 2163, 7973, 29325, 107777, 395929, 1454643, 5344795, 19638715, 72159035, 265134245, 974183489, 3579448271, 13151993143, 48324463973, 177558916493, 652405957937, 2397139747601, 8807827288183, 32362661242755, 118910351874471
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Column 6 of A200871.

Examples

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

Links

Programs

  • Mathematica
    a[0,x_,y_] := 1; a[n_,x_,y_] := a[n,x,y] = Sum[If[z <=x<= y || y <=x<= z, a[n-1, z, x], 0], {z, 7}]; a[n_] := Sum[a[n, x, y], {x, 7}, {y, 7}]; Array[a, 25] (* Giovanni Resta, Mar 05 2014 *)

Formula

Empirical: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) +9*a(n-4) +7*a(n-6) +6*a(n-7) +a(n-8) +2*a(n-9) +a(n-10).
Empirical g.f.: x*(161 - 49*x + 266*x^2 + 462*x^3 + 93*x^4 + 389*x^5 + 310*x^6 + 70*x^7 + 105*x^8 + 49*x^9) / (1 - 4*x + 3*x^2 - 4*x^3 - 9*x^4 - 7*x^6 - 6*x^7 - x^8 - 2*x^9 - x^10). - Colin Barker, Oct 16 2017

A200870 Number of 0..7 arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

232, 932, 3704, 14932, 60112, 241718, 971416, 3904290, 15693816, 63085186, 253583656, 1019321774, 4097329180, 16469898636, 66203547550, 266116375576, 1069699742484, 4299838717346, 17283927781406, 69475666835578
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Column 7 of A200871.

Examples

			Some solutions for n=3
..7....7....7....2....2....2....4....1....6....1....6....0....6....5....7....5
..5....1....5....6....4....2....6....5....2....1....6....0....5....3....3....4
..5....1....5....6....7....4....6....5....0....1....4....1....4....3....2....4
..3....4....4....3....7....7....4....5....0....2....0....1....4....3....1....4
..3....6....0....1....1....7....0....6....7....2....0....3....6....0....1....2

Links

Programs

  • Mathematica
    a[0,x_,y_] := 1; a[n_,x_,y_] := a[n,x,y] = Sum[If[z <=x<= y || y <=x<= z, a[n-1, z, x], 0], {z, 8}]; a[n_] := Sum[a[n, x, y], {x, 8}, {y, 8}]; Array[a, 25] (* Giovanni Resta, Mar 05 2014 *)

Formula

Empirical: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) +15*a(n-4) +6*a(n-5) +12*a(n-6) +16*a(n-7) +7*a(n-8) +5*a(n-9) +4*a(n-10) +a(n-11).
Empirical g.f.: 2*x*(116 + 2*x + 220*x^2 + 526*x^3 + 292*x^4 + 473*x^5 + 552*x^6 + 257*x^7 + 180*x^8 + 132*x^9 + 32*x^10) / (1 - 4*x + 2*x^2 - 4*x^3 - 15*x^4 - 6*x^5 - 12*x^6 - 16*x^7 - 7*x^8 - 5*x^9 - 4*x^10 - x^11). - Colin Barker, Oct 16 2017

A200872 Number of 0..n arrays x(0..3) of 4 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

10, 37, 94, 195, 356, 595, 932, 1389, 1990, 2761, 3730, 4927, 6384, 8135, 10216, 12665, 15522, 18829, 22630, 26971, 31900, 37467, 43724, 50725, 58526, 67185, 76762, 87319, 98920, 111631, 125520, 140657, 157114, 174965, 194286, 215155, 237652, 261859
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Row 2 of A200871.

Examples

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

Links

Formula

Empirical: a(n) = (1/12)*n^4 + (3/2)*n^3 + (47/12)*n^2 + (7/2)*n + 1.
Conjectures from Colin Barker, Oct 16 2017: (Start)
G.f.: x*(10 - 13*x + 9*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A200873 Number of 0..n arrays x(0..4) of 5 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

16, 77, 236, 567, 1168, 2163, 3704, 5973, 9184, 13585, 19460, 27131, 36960, 49351, 64752, 83657, 106608, 134197, 167068, 205919, 251504, 304635, 366184, 437085, 518336, 611001, 716212, 835171, 969152, 1119503, 1287648, 1475089, 1683408
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Row 3 of A200871.

Examples

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

Links

Formula

Empirical: a(n) = (1/60)*n^5 + (3/4)*n^4 + (15/4)*n^3 + (25/4)*n^2 + (127/30)*n + 1.
Conjectures from Colin Barker, Oct 16 2017: (Start)
G.f.: x*(16 - 19*x + 14*x^2 - 14*x^3 + 6*x^4 - x^5) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A200874 Number of 0..n arrays x(0..5) of 6 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

26, 163, 602, 1673, 3886, 7973, 14932, 26073, 43066, 67991, 103390, 152321, 218414, 305929, 419816, 565777, 750330, 980875, 1265762, 1614361, 2037134, 2545709, 3152956, 3873065, 4721626, 5715711, 6873958, 8216657, 9765838, 11545361
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Row 4 of A200871.

Examples

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

Links

Formula

Empirical: a(n) = (1/360)*n^6 + (7/24)*n^5 + (197/72)*n^4 + (185/24)*n^3 + (1667/180)*n^2 + 5*n + 1.
Conjectures from Colin Barker, Oct 16 2017: (Start)
G.f.: x*(26 - 19*x + 7*x^2 - 28*x^3 + 22*x^4 - 7*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A200875 Number of 0..n arrays x(0..6) of 7 elements without any interior element greater than both neighbors or less than both neighbors.

Original entry on oeis.org

42, 343, 1528, 4917, 12890, 29325, 60112, 113745, 201994, 340659, 550408, 857701, 1295802, 1905881, 2738208, 3853441, 5324010, 7235599, 9688728, 12800437, 16706074, 21561189, 27543536, 34855185, 43724746, 54409707, 67198888, 82415013
Offset: 1

Views

Author

R. H. Hardin, Nov 23 2011

Keywords

Comments

Row 5 of A200871.

Examples

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

Links

Formula

Empirical: a(n) = (1/2520)*n^7 + (17/180)*n^6 + (281/180)*n^5 + (64/9)*n^4 + (4927/360)*n^3 + (2303/180)*n^2 + (604/105)*n + 1.
Conjectures from Colin Barker, Oct 16 2017: (Start)
G.f.: x*(42 + 7*x - 40*x^2 - 55*x^3 + 70*x^4 - 29*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)
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