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

A228212 Number of second differences of arrays of length n + 2 of numbers in 0..2.

Original entry on oeis.org

9, 49, 199, 665, 2059, 6305, 19171, 58025, 175099, 527345, 1586131, 4766585, 14316139, 42981185, 129009091, 387158345, 1161737179, 3485735825, 10458256051, 31376865305, 94134790219, 282412759265, 847255055011, 2541798719465
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

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

Links

Crossrefs

Column 2 of A228218.

Formula

Empirical: a(n) = 5*a(n-1) -6*a(n-2) = A001047(n+2) for n>5.
Conjectures from Colin Barker, Sep 09 2018: (Start)
G.f.: x*(9 + 4*x + 8*x^2 - 36*x^3 - 72*x^4) / ((1 - 2*x)*(1 - 3*x)).
a(n) = 3^(2+n) - 2^(2+n) for n>3.
(End)

A228213 Number of second differences of arrays of length n + 2 of numbers in 0..3.

Original entry on oeis.org

13, 103, 625, 3151, 14053, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

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

Links

Crossrefs

Column 3 of A228218.

Formula

Empirical: a(n) = 7*a(n-1) - 12*a(n-2) = A005061(n+2) for n>7.
Conjectures from Colin Barker, Sep 09 2018: (Start)
G.f.: x*(13 + 12*x + 60*x^2 + 12*x^3 - 504*x^4 - 1584*x^5 - 1728*x^6) / ((1 - 3*x)*(1 - 4*x)).
a(n) = 4^(2+n) - 3^(2+n) for n>5.
(End)

A228215 Number of second differences of arrays of length n+2 of numbers in 0..5.

Original entry on oeis.org

21, 271, 2731, 23351, 176851, 1225631, 8006491, 50556551, 313882531, 1932641711, 11839990891, 72260648471, 439667406451, 2668522016831, 16163719991611, 97745259402791, 590286253682371, 3560791008422351, 21460113482174731
Offset: 1

Views

Author

R. H. Hardin Aug 16 2013

Keywords

Comments

Column 5 of A228218

Examples

			Some solutions for n=4
..1....2...-1...-7...-4..-10...-6....1....2....8...-3....2...-6...-3....5....6
.-4....2...-3....5....0....8....5....2...-5...-2....1...-2....8....0...-3...-9
..1...-2...-1...-1....7...-3....2....0....4...-3....4....6...-1....3...-1....5
..5....3....6....2...-5...-2...-3....5....3....5...-8...-5...-5...-1....6....3

Links

Formula

Empirical: a(n) = 11*a(n-1) -30*a(n-2) for n>11

A228219 Number of second differences of arrays of length 4 of numbers in 0..n.

Original entry on oeis.org

15, 49, 103, 177, 271, 385, 519, 673, 847, 1041, 1255, 1489, 1743, 2017, 2311, 2625, 2959, 3313, 3687, 4081, 4495, 4929, 5383, 5857, 6351, 6865, 7399, 7953, 8527, 9121, 9735, 10369, 11023, 11697, 12391, 13105, 13839, 14593, 15367, 16161, 16975, 17809
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Comments

Row 2 of A228218.

Examples

			Some solutions for n=4:
..1....1....1....5....6....1....2...-5....8....4...-5....8....1....5...-7....0
..4....2...-3....1...-6....1....1...-1...-7....1....0...-4....5...-7....6....6

Links

Crossrefs

Cf. A228218, A272039.

Formula

Empirical: a(n) = 10*n^2 + 4*n + 1 = A272039(n).
Conjectures from Colin Barker, Mar 16 2018: (Start)
G.f.: x*(15 + 4*x + x^2) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)

A228214 Number of second differences of arrays of length n + 2 of numbers in 0..4.

Original entry on oeis.org

17, 177, 1429, 9705, 58141, 320481, 1688101, 8717049, 44633821, 227363409, 1153594261, 5835080169, 29443836301, 148292923329, 745759583941, 3745977788889, 18798608421181, 94267920012849, 472439111692021
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

			Some solutions for n=4:
..8....5...-1...-2....4...-2....3....0...-2...-6....2....0....4...-5....0...-5
.-8....0....2....2...-5...-4...-4...-2....2....5....2...-2...-3....4....2....5
..5....0....0...-2...-1....7....3....2....4....0...-6....3...-1...-1...-6...-6
.-1...-2...-2....4....2...-3...-3....3...-7...-1....4...-3....6...-1....5....4

Links

Crossrefs

Column 4 of A228218.

Formula

Empirical: a(n) = 9*a(n-1) - 20*a(n-2) = A005060(n+2) for n>9.
Conjectures from Colin Barker, Sep 10 2018: (Start)
G.f.: x*(17 + 24*x + 176*x^2 + 384*x^3 - 624*x^4 - 8688*x^5 - 33408*x^6 - 66240*x^7 - 57600*x^8) / ((1 - 4*x)*(1 - 5*x)).
a(n) = 5^(2+n) - 4^(2+n) for n>7.
(End)

A228216 Number of second differences of arrays of length n+2 of numbers in 0..6.

Original entry on oeis.org

25, 385, 4651, 47953, 439927, 3693505, 29066311, 219071473, 1609259287, 11658284065, 83824687591, 599858908753, 4277376525367, 30411820662145, 215703854542471, 1526853641242033, 10789535445362647, 76136107857549025
Offset: 1

Views

Author

R. H. Hardin Aug 16 2013

Keywords

Comments

Column 6 of A228218

Examples

			Some solutions for n=4
..0...-7....0....3...-4....6....6....5....6....2...10...-7...-7....9..-10...-1
..0....1...-2....7....6....4....0...-8...-4...-6..-11....0...12...-4....8....3
..2...-4....5...-6...-8...-4...-9....7....3...-1....8....9...-7....0....0...-8
.-2....4...-7...-2....5....0....6...-6...-3....8...-5...-8....0....3...-2...12

Links

Formula

Empirical: a(n) = 13*a(n-1) -42*a(n-2) for n>13

A228217 Number of second differences of arrays of length n+2 of numbers in 0..7.

Original entry on oeis.org

29, 519, 7309, 88215, 951049, 9399615, 86929081, 766106895, 6537612649, 54701587935, 452558355481, 3719467815855, 30436607366089, 248242046141055, 2019169299698041, 16385984911571535, 132716292890482729
Offset: 1

Views

Author

R. H. Hardin Aug 16 2013

Keywords

Comments

Column 7 of A228218

Examples

			Some solutions for n=4
.-8....0...-8....1....4..-11....3....7..-10...-1...-4....4...-2....8...-3...-9
.13...-2...-1....5...-3....7....0...-4....3...10....8...-2...-8...-4....5...11
-12...-3....4...-7...-4...-3...-5....8...-1..-13....0...-3....7...-8....3...-1
..8....9....2....4....1....3...-2..-11....6...12..-10...-3....7...11...-3...-4

Links

Formula

Empirical: a(n) = 15*a(n-1) -56*a(n-2) for n>15

A228220 Number of second differences of arrays of length 5 of numbers in 0..n.

Original entry on oeis.org

31, 199, 625, 1429, 2731, 4651, 7309, 10825, 15319, 20911, 27721, 35869, 45475, 56659, 69541, 84241, 100879, 119575, 140449, 163621, 189211, 217339, 248125, 281689, 318151, 357631, 400249, 446125, 495379, 548131, 604501, 664609, 728575
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

			Some solutions for n=4:
.-2....1...-5....4....0....1....1....3...-2....4...-5...-8....4...-4....0....3
.-2...-2....6....0....2...-3....4....0....5....3....7....7...-6...-4....1...-7
.-2....5....0...-2...-5....7...-7...-2...-3...-6...-2...-2....3....4...-3....4

Links

Crossrefs

Row 3 of A228218.

Formula

Empirical: a(n) = 20*n^3 + 9*n^2 + 1*n + 1.
Conjectures from Colin Barker, Sep 10 2018: (Start)
G.f.: x*(31 + 75*x + 15*x^2 - x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A228221 Number of second differences of arrays of length 6 of numbers in 0..n.

Original entry on oeis.org

63, 665, 3151, 9705, 23351, 47953, 88215, 149681, 238735, 362601, 529343, 747865, 1027911, 1380065, 1815751, 2347233, 2987615, 3750841, 4651695, 5705801, 6929623, 8340465, 9956471, 11796625, 13880751, 16229513, 18864415, 21807801, 25082855
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

			Some solutions for n=4:
..4....5....6...-2...-5....5...-6....5....7...-2....0....3...-1....4...-2....3
..0...-2...-3...-3...-1....0....3...-3...-5....3...-4...-5....5....0....1...-2
.-4....5....0....1....2...-4....2....3....5...-1....4....0...-6...-3....1...-2
..1...-8....5....1...-1....2...-5...-4...-7...-2....2....3....7....0....4....4

Links

Crossrefs

Row 4 of A228218.

Formula

Empirical: a(n) = 35*n^4 + 14*n^3 - 17*n^2 + 30*n + 1.
Conjectures from Colin Barker, Sep 10 2018: (Start)
G.f.: x*(63 + 350*x + 456*x^2 - 30*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)

A228222 Number of second differences of arrays of length 7 of numbers in 0..n.

Original entry on oeis.org

127, 2059, 14053, 58141, 176851, 439927, 951049, 1854553, 3342151, 5659651, 9113677, 14078389, 21002203, 30414511, 42932401, 59267377, 80232079, 106747003, 139847221, 180689101, 230557027, 290870119, 363188953, 449222281, 550833751
Offset: 1

Views

Author

R. H. Hardin, Aug 16 2013

Keywords

Examples

			Some solutions for n=4:
.-7....5...-3....1...-2...-1....1....5...-1....3....6...-6....5....0....5...-1
..2...-1....1...-2....2....0...-1...-6...-1...-1...-3....5....2....1....0....2
..2...-1...-2....5...-2...-2....2....4....6...-1....4....1...-6....3...-5....0
.-2...-1...-1...-7....3....0....0...-6...-8....3...-3...-2....4...-4....5...-5
..4....2....3....5...-2....4....0....4....6...-6...-3...-3....1....1...-1....8

Links

Crossrefs

Row 5 of A228218.

Formula

Empirical: a(n) = 56*n^5 + 14*n^4 - 108*n^3 + 289*n^2 - 125*n + 1.
Conjectures from Colin Barker, Sep 10 2018: (Start)
G.f.: x*(127 + 1297*x + 3604*x^2 + 2168*x^3 - 475*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)
Showing 1-10 of 12 results. Next

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