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.

A208142 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 12, 81, 108, 64, 10, 16, 144, 324, 240, 100, 12, 20, 256, 720, 900, 450, 144, 14, 25, 400, 1600, 2400, 2025, 756, 196, 16, 30, 625, 3000, 6400, 6300, 3969, 1176, 256, 18, 36, 900, 5625, 14000, 19600, 14112, 7056, 1728, 324, 20, 42, 1296
Offset: 1

Views

Author

R. H. Hardin Feb 23 2012

Keywords

Comments

Table starts
..2...4....6.....9....12.....16.....20......25......30.......36.......42
..4..16...36....81...144....256....400.....625.....900.....1296.....1764
..6..36..108...324...720...1600...3000....5625....9450....15876....24696
..8..64..240...900..2400...6400..14000...30625...58800...112896...197568
.10.100..450..2025..6300..19600..49000..122500..264600...571536..1111320
.12.144..756..3969.14112..50176.141120..396900..952560..2286144..4889808
.14.196.1176..7056.28224.112896.352800.1102500.2910600..7683984.17929296
.16.256.1728.11664.51840.230400.792000.2722500.7840800.22581504.57081024

Examples

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

Links

Crossrefs

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A202195(n-2)
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A202093(n-2)

Formula

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 3*n^3 + 3*n^2
k=4: a(n) = (9/4)*n^4 + (9/2)*n^3 + (9/4)*n^2
k=5: a(n) = n^5 + 4*n^4 + 5*n^3 + 2*n^2
k=6: a(n) = (4/9)*n^6 + (8/3)*n^5 + (52/9)*n^4 + (16/3)*n^3 + (16/9)*n^2
k=7: a(n) = (5/36)*n^7 + (5/4)*n^6 + (155/36)*n^5 + (85/12)*n^4 + (50/9)*n^3 + (5/3)*n^2

A250432 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing sum of every two consecutive values in every row and column.

Original entry on oeis.org

16, 36, 36, 81, 108, 81, 144, 324, 324, 144, 256, 720, 1296, 720, 256, 400, 1600, 3600, 3600, 1600, 400, 625, 3000, 10000, 12000, 10000, 3000, 625, 900, 5625, 22500, 40000, 40000, 22500, 5625, 900, 1296, 9450, 50625, 105000, 160000, 105000, 50625, 9450
Offset: 1

Views

Author

R. H. Hardin, Nov 22 2014

Keywords

Comments

Table starts
...16....36.....81.....144......256.......400.......625........900........1296
...36...108....324.....720.....1600......3000......5625.......9450.......15876
...81...324...1296....3600....10000.....22500.....50625......99225......194481
..144...720...3600...12000....40000....105000....275625.....617400.....1382976
..256..1600..10000...40000...160000....490000...1500625....3841600.....9834496
..400..3000..22500..105000...490000...1715000...6002500...17287200....49787136
..625..5625..50625..275625..1500625...6002500..24010000...77792400...252047376
..900..9450..99225..617400..3841600..17287200..77792400..280052640..1008189504
.1296.15876.194481.1382976..9834496..49787136.252047376.1008189504..4032758016
.1764.24696.345744.2765952.22127616.124467840.700131600.3080579040.13554547776
Essentially the same as A202100; the mapping between the binary arrays in both sequences is by flipping all entries in one set of arrays. - Joerg Arndt, Dec 01 2014

Examples

			Some solutions for n=5 k=4
..0..0..0..0..1....0..0..0..0..0....0..0..1..0..1....0..0..0..1..1
..0..1..0..1..1....0..0..0..1..0....1..1..1..1..1....0..0..1..1..1
..0..0..1..0..1....0..0..0..1..0....0..0..1..1..1....0..0..1..1..1
..0..1..1..1..1....0..0..0..1..0....1..1..1..1..1....0..0..1..1..1
..0..1..1..1..1....0..0..0..1..0....0..1..1..1..1....1..0..1..1..1
..0..1..1..1..1....0..0..0..1..1....1..1..1..1..1....1..1..1..1..1

Links

Crossrefs

Column 1 is A030179(n+3), A202093 - A202099 (further columns).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8); also a polynomial of degree 4 plus a quasipolynomial of degree 2 with period 2
k=2: [order 12; also a polynomial of degree 6 plus a quasipolynomial of degree 4 with period 2]
k=3: [order 16; also a polynomial of degree 8 plus a quasipolynomial of degree 6 with period 2]
k=4: [order 20; also a polynomial of degree 10 plus a quasipolynomial of degree 8 with period 2]
k=5: [order 24; also a polynomial of degree 12 plus a quasipolynomial of degree 10 with period 2]
k=6: [order 28; also a polynomial of degree 14 plus a quasipolynomial of degree 12 with period 2]
k=7: [order 32; also a polynomial of degree 16 plus a quasipolynomial of degree 14 with period 2]

A250426 Number of (n+1)X(2+1) 0..1 arrays with nondecreasing sum of every two consecutive values in every row and column.

Original entry on oeis.org

36, 108, 324, 720, 1600, 3000, 5625, 9450, 15876, 24696, 38416, 56448, 82944, 116640, 164025, 222750, 302500, 399300, 527076, 679536, 876096, 1107288, 1399489, 1739010, 2160900, 2646000, 3240000, 3916800, 4734976, 5659776, 6765201, 8005878
Offset: 1

Views

Author

R. H. Hardin, Nov 22 2014

Keywords

Examples

			Some solutions for n=6:
..0..0..0....0..0..0....0..0..0....0..0..0....0..0..1....0..0..0....0..0..1
..0..1..0....0..0..0....0..0..1....0..0..1....0..0..0....0..1..0....0..0..1
..0..1..1....0..0..0....0..0..1....0..0..0....0..0..1....0..1..0....0..1..1
..0..1..1....0..0..0....0..0..1....0..1..1....0..0..1....0..1..0....0..0..1
..0..1..1....0..0..0....0..0..1....0..0..1....0..1..1....0..1..0....0..1..1
..1..1..1....0..1..1....0..1..1....1..1..1....1..1..1....0..1..1....0..0..1
..0..1..1....0..1..1....0..1..1....1..0..1....1..1..1....1..1..1....1..1..1

Links

Crossrefs

Column 2 of A250432.

Formula

Empirical: a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12).
Empirical for n mod 2 = 0: a(n) = (1/256)*n^6 + (11/128)*n^5 + (49/64)*n^4 + (113/32)*n^3 + (71/8)*n^2 + (23/2)*n + 6.
Empirical for n mod 2 = 1: a(n) = (1/256)*n^6 + (11/128)*n^5 + (199/256)*n^4 + (237/64)*n^3 + (2511/256)*n^2 + (1755/128)*n + (2025/256).
a(n+1)=A202093(n). - R. J. Mathar, Dec 04 2014
Empirical g.f.: x*(36 + 36*x - 36*x^2 + 124*x^4 - 20*x^5 - 115*x^6 + 40*x^7 + 56*x^8 - 26*x^9 - 11*x^10 + 6*x^11) / ((1 - x)^7*(1 + x)^5). - Colin Barker, Nov 14 2018
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