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 1-5 of 5 results.

A202549 T(n,k) = Number of n X k nonnegative integer arrays with each row and column an ascent sequence (interior element no greater than one plus up-steps preceding it).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 18, 8, 1, 1, 16, 86, 86, 16, 1, 1, 32, 422, 1094, 422, 32, 1, 1, 64, 2094, 15184, 15184, 2094, 64, 1, 1, 128, 10438, 219934, 658492, 219934, 10438, 128, 1, 1, 256, 52126, 3249298, 31670778, 31670778, 3249298, 52126, 256, 1, 1, 512
Offset: 1

Views

Author

R. H. Hardin, Dec 20 2011

Keywords

Comments

Table starts
.1...1.....1........1...........1...............1..................1
.1...2.....4........8..........16..............32.................64
.1...4....18.......86.........422............2094..............10438
.1...8....86.....1094.......15184..........219934............3249298
.1..16...422....15184......658492........31670778.........1605067272
.1..32..2094...219934....31670778......5534917394......1078490778110
.1..64.10438..3249298..1605067272...1078490778110....876480920594230
.1.128.52126.48427802.83391232150.222425177005132.794166642153146254

Examples

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

Links

Crossrefs

Column 2 is A000079(n-1).
Column 3 is A082685(n-1).

A268056 T(n,k)=Number of nXk 0..2 arrays with new values introduced in each row and column in sequential order starting with zero.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 18, 8, 1, 1, 16, 86, 86, 16, 1, 1, 32, 422, 1082, 422, 32, 1, 1, 64, 2094, 14554, 14554, 2094, 64, 1, 1, 128, 10438, 200818, 560778, 200818, 10438, 128, 1, 1, 256, 52126, 2796826, 22501266, 22501266, 2796826, 52126, 256, 1, 1, 512
Offset: 1

Views

Author

R. H. Hardin, Jan 25 2016

Keywords

Comments

Table starts
.1...1......1.........1.............1................1....................1
.1...2......4.........8............16...............32...................64
.1...4.....18........86...........422.............2094................10438
.1...8.....86......1082.........14554...........200818..............2796826
.1..16....422.....14554........560778.........22501266............915745002
.1..32...2094....200818......22501266.......2666449962.........322124540754
.1..64..10438...2796826.....915745002.....322124540754......116124404018058
.1.128..52126..39082562...37450557314...39169337042938....42209510913723266
.1.256.260502.546791114.1534144563898.4773343640706306.15385058360607977882

Examples

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

Links

Crossrefs

Column 2 is A000079(n-1).
Column 3 is A082685(n-1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 7*a(n-1) -10*a(n-2)
k=4: a(n) = 21*a(n-1) -108*a(n-2) +140*a(n-3)
k=5: a(n) = 62*a(n-1) -969*a(n-2) +4568*a(n-3) -5740*a(n-4)
k=6: a(n) = 184*a(n-1) -8533*a(n-2) +122786*a(n-3) -563036*a(n-4) +700280*a(n-5)
k=7: a(n) = 549*a(n-1) -75693*a(n-2) +3237331*a(n-3) -45379926*a(n-4) +206208420*a(n-5) -255602200*a(n-6)

A268079 T(n,k)=Number of nXk nonnegative integer arrays with new values introduced in each row and column in sequential order starting with zero.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 18, 8, 1, 1, 16, 86, 86, 16, 1, 1, 32, 422, 1094, 422, 32, 1, 1, 64, 2094, 15106, 15106, 2094, 64, 1, 1, 128, 10438, 216734, 637358, 216734, 10438, 128, 1, 1, 256, 52126, 3168306, 29309170, 29309170, 3168306, 52126, 256, 1, 1, 512
Offset: 1

Views

Author

R. H. Hardin, Jan 25 2016

Keywords

Comments

Table starts
.1...1......1.........1.............1.................1.....................1
.1...2......4.........8............16................32....................64
.1...4.....18........86...........422..............2094.................10438
.1...8.....86......1094.........15106............216734...............3168306
.1..16....422.....15106........637358..........29309170............1412290158
.1..32...2094....216734......29309170........4617638834..........795460720710
.1..64..10438...3168306....1412290158......795460720710.......517992936833258
.1.128..52126..46777214...69903748498...144635795908942....369867566612849678
.1.256.260502.694585586.3516426536462.27199854725237562.280350778114908738774

Examples

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

Links

Crossrefs

Column 2 is A000079(n-1).
Column 3 is A082685(n-1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 2*a(n-1)
k=3: a(n) = 7*a(n-1) -10*a(n-2)
k=4: a(n) = 36*a(n-1) -463*a(n-2) +2640*a(n-3) -6700*a(n-4) +6000*a(n-5)
k=5: [order 14]
k=6: [order 45]

A212596 Number of cards required to build a Menger sponge of level n in origami.

Original entry on oeis.org

12, 192, 3456, 66048, 1296384, 25731072, 513048576, 10248388608, 204867108864, 4096536870912, 81924294967296, 1638434359738368, 32768274877906944, 655362199023255552, 13107217592186044416, 262144140737488355328, 5242881125899906842624
Offset: 0

Views

Author

Daniel de Rauglaudre, May 22 2012

Keywords

Examples

			12 cards (a(0)) are required for a single origami cube: 6 for the cube skeleton, and 6 for panels or possible links to other cubes.

Links

Crossrefs

Cf. A082685.

Programs

Formula

a(n) = 4*(8^n + 2*20^n) = 2^(2*n+3)*5^n+2^(3*n+2).
a(n) = A082685(n)*3*4^(n+1).
From Colin Barker, Apr 10 2014: (Start)
a(n) = 28*a(n-1)-160*a(n-2).
G.f.: -12*(12*x-1) / ((8*x-1)*(20*x-1)). (End)

Extensions

More terms from Colin Barker, Apr 10 2014

A114195 Riordan array (1/(1-3x),x(1-x)/(1-3x)^2).

Original entry on oeis.org

1, 3, 1, 9, 8, 1, 27, 45, 13, 1, 81, 216, 106, 18, 1, 243, 945, 690, 192, 23, 1, 729, 3888, 3915, 1574, 303, 28, 1, 2187, 15309, 20223, 10941, 2993, 439, 33, 1, 6561, 58320, 97524, 67788, 24598, 5072, 600, 38, 1, 19683, 216513, 446148, 385560, 177498
Offset: 0

Views

Author

Paul Barry, Nov 16 2005

Keywords

Comments

Row sums are A082685. Diagonal sums are A114196.

Examples

			Triangle begins
1;
3, 1;
9, 8, 1;
27, 45, 13, 1;
81, 216, 106, 18, 1;
243, 945, 690, 192, 23, 1;

Formula

T(n, k)=sum{j=0..n, C(n, j)C(j+k, 2k)2^(j-k)}; T(n, k)=sum{j=0..n-k, C(k, j)C(n+k-j, 2k)(-1)^j*3^(n-k-j)}.
T(n,k) = 6*T(n-1,k) + T(n-1,k-1) - 9*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 18 2014
Showing 1-5 of 5 results.

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