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.

A221962 Number of -3..3 arrays of length n with the sum ahead of each element differing from the sum following that element by 3 or less.

Original entry on oeis.org

7, 49, 175, 833, 3647, 16513, 73983, 332801, 1495039, 6719489, 30195711, 135700481, 609828863, 2740551681, 12315918335, 55347249153, 248728256511, 1117774675969, 5023233736703, 22574207598593, 101447567933439, 455901232234497
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Column 3 of A221967.

Examples

			Some solutions for n=6:
..1...-3....1....0...-2....2...-3...-2....1....1....1....0...-3...-2....2....2
.-2...-1....1....1....0...-2....2....1....0...-3...-2....0....0....2...-2....0
..2....2...-3...-2....3...-1....1....0....0....1....3....0....1....2....0....0
.-2...-2....0....3...-2....1...-1....3...-1....0....0....0...-1...-2....1...-3
.-2....1....0...-2....2....2....1...-3....3...-1....1....2....0....2...-2....2
..1...-1....1...-1...-2...-1...-2...-1...-3....1....0...-1...-1...-2....3....0

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = 3*a(n-1) + 8*a(n-2) - 4*a(n-3) - 8*a(n-4).
Empirical g.f.: 7*x*(1 + 4*x - 4*x^2 - 8*x^3) / ((1 + x)*(1 - 4*x - 4*x^2 + 8*x^3)). - Colin Barker, Aug 11 2018

A221963 Number of -4..4 arrays of length n with the sum ahead of each element differing from the sum following that element by 4 or less.

Original entry on oeis.org

9, 81, 369, 2241, 12609, 73089, 419841, 2419713, 13930497, 80230401, 462012417, 2660655105, 15322038273, 88236228609, 508131934209, 2926215692289, 16851403538433, 97043367329793, 558850475294721, 3218291629752321
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Column 4 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = 5*a(n-1) + 8*a(n-2) - 20*a(n-3) - 8*a(n-4) + 16*a(n-5).
Empirical g.f.: 9*x*(1 - 2*x)*(1 + 6*x - 8*x^3) / ((1 - x)*(1 + 2*x)*(1 - 6*x + 8*x^3)). - Colin Barker, Aug 11 2018

A221964 Number of -5..5 arrays of length n with the sum ahead of each element differing from the sum following that element by 5 or less.

Original entry on oeis.org

11, 121, 671, 4961, 34111, 241153, 1690623, 11888129, 83512319, 586864641, 4123582463, 28975366145, 203599740927, 1430630760449, 10052572086271, 70636158713857, 496337248845823, 3487600190423041, 24506230080798719
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Column 5 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = 5*a(n-1) + 18*a(n-2) - 20*a(n-3) - 48*a(n-4) + 16*a(n-5) + 32*a(n-6).
Empirical g.f.: 11*x*(1 + 6*x - 12*x^2 - 32*x^3 + 16*x^4 + 32*x^5) / ((1 + x)*(1 - 6*x - 12*x^2 + 32*x^3 + 16*x^4 - 32*x^5)). - Colin Barker, Aug 11 2018

A221965 Number of -6..6 arrays of length n with the sum ahead of each element differing from the sum following that element by 6 or less.

Original entry on oeis.org

13, 169, 1105, 9633, 78273, 653185, 5407233, 44890625, 372332545, 3089205249, 25628045313, 212618141697, 1763923329025, 14633929146369, 121406394466305, 1007215475032065, 8356090693812225, 69324049716609025, 575128245018558465
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Column 6 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = 7*a(n-1) + 18*a(n-2) - 56*a(n-3) - 48*a(n-4) + 112*a(n-5) + 32*a(n-6) - 64*a(n-7).
Empirical g.f.: 13*x*(1 + 6*x - 24*x^2 - 32*x^3 + 80*x^4 + 32*x^5 - 64*x^6) / ((1 - x)*(1 - 6*x - 24*x^2 + 32*x^3 + 80*x^4 - 32*x^5 - 64*x^6)). - Colin Barker, Aug 14 2018

A221966 Number of -7..7 arrays of length n with the sum ahead of each element differing from the sum following that element by 7 or less.

Original entry on oeis.org

15, 225, 1695, 17025, 159615, 1535745, 14661375, 140355585, 1342437375, 12843782145, 122870296575, 1175482548225, 11245539966975, 107583628673025, 1029227717656575, 9846388295204865, 94198150438322175, 901172262043910145
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Column 7 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = 7*a(n-1) + 32*a(n-2) - 56*a(n-3) - 160*a(n-4) + 112*a(n-5) + 256*a(n-6) - 64*a(n-7) - 128*a(n-8).
Empirical g.f.: 15*x*(1 + 2*x)*(1 - 2*x - 4*x^2)*(1 + 8*x - 16*x^2 - 8*x^3 + 16*x^4) / ((1 + x)*(1 - 2*x)*(1 + 2*x - 4*x^2)*(1 - 8*x - 16*x^2 + 8*x^3 + 16*x^4)). - Colin Barker, Aug 14 2018

A221968 Number of -n..n arrays of length 5 with the sum ahead of each element differing from the sum following that element by n or less.

Original entry on oeis.org

63, 705, 3647, 12609, 34111, 78273, 159615, 297857, 518719, 854721, 1345983, 2041025, 2997567, 4283329, 5976831, 8168193, 10959935, 14467777, 18821439, 24165441, 30659903, 38481345, 47823487, 58898049, 71935551, 87186113
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Row 5 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = (20/3)*n^5 + (50/3)*n^4 + 20*n^3 + (40/3)*n^2 + (16/3)*n + 1.
Conjectures from Colin Barker, Aug 14 2018: (Start)
G.f.: x*(9 + 12*x - x^2)*(7 + 27*x + 5*x^2 + x^3) / (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)

A221969 Number of -n..n arrays of length 6 with the sum ahead of each element differing from the sum following that element by n or less.

Original entry on oeis.org

129, 2305, 16513, 73089, 241153, 653185, 1535745, 3246337, 6316417, 11500545, 19831681, 32682625, 51833601, 79545985, 118642177, 172591617, 245602945, 342722305, 469937793, 634290049, 843988993, 1108536705, 1438856449
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Row 6 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = (128/15)*n^6 + (128/5)*n^5 + (112/3)*n^4 + 32*n^3 + (272/15)*n^2 + (32/5)*n + 1.
Conjectures from Colin Barker, Aug 14 2018: (Start)
G.f.: x*(3 + x)*(43 + 453*x + 878*x^2 + 170*x^3 - 9*x^4 + x^5) / (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)

A221970 Number of -n..n arrays of length 7 with the sum ahead of each element differing from the sum following that element by n or less.

Original entry on oeis.org

255, 7425, 73983, 419841, 1690623, 5407233, 14661375, 35111681, 76335103, 153588225, 290033151, 519482625, 889719039, 1466441985, 2337899007, 3620254209, 5463749375, 8059712257, 11648466687, 16528199169, 23064836607
Offset: 1

Views

Author

R. H. Hardin, Feb 01 2013

Keywords

Comments

Row 7 of A221967.

Examples

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

Links

Crossrefs

Cf. A221967.

Formula

Empirical: a(n) = (488/45)*n^7 + (1708/45)*n^6 + (2912/45)*n^5 + (602/9)*n^4 + (2072/45)*n^3 + (952/45)*n^2 + (32/5)*n + 1.
Conjectures from Colin Barker, Aug 14 2018: (Start)
G.f.: x*(255 + 5385*x + 21723*x^2 + 21597*x^3 + 5469*x^4 + 219*x^5 + 9*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)
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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