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.

Previous Showing 11-16 of 16 results.

A199836 Number of -n..n arrays x(0..6) of 7 elements with zero sum and no two neighbors summing to zero.

Original entry on oeis.org

22, 1650, 20240, 118280, 462234, 1402934, 3579520, 8046928, 16426926, 31082698, 55316976, 93593720, 151783346, 237431502, 360051392, 531439648, 766015750, 1081184994, 1497725008, 2040195816, 2737373450, 3622707110, 4734799872
Offset: 1

Views

Author

R. H. Hardin, Nov 11 2011

Keywords

Comments

Row 5 of A199832.

Examples

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

Links

Crossrefs

Cf. A199832.

Formula

Empirical: a(n) = (5887/180)*n^6 - (1013/60)*n^5 + (245/36)*n^4 - (35/12)*n^3 + (157/45)*n^2 - (6/5)*n.
Conjectures from Colin Barker, May 16 2018: (Start)
G.f.: 2*x*(11 + 748*x + 4576*x^2 + 5240*x^3 + 1167*x^4 + 32*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)

A199837 Number of -n..n arrays x(0..7) of 8 elements with zero sum and no two neighbors summing to zero.

Original entry on oeis.org

34, 6126, 113884, 888420, 4340094, 15805218, 47040968, 120843752, 277500282, 583380598, 1141982292, 2107735180, 3702875670, 6237700074, 10134506112, 15955531856, 24435201362, 36516986238, 53395192396, 76561981236
Offset: 1

Views

Author

R. H. Hardin, Nov 11 2011

Keywords

Comments

Row 6 of A199832.

Examples

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

Links

Crossrefs

Cf. A199832.

Formula

Empirical: a(n) = (19328/315)*n^7 - (1424/45)*n^6 + (704/45)*n^5 - (112/9)*n^4 - (124/45)*n^3 + (229/45)*n^2 - (131/105)*n.
Conjectures from Colin Barker, May 16 2018: (Start)
G.f.: 2*x*(17 + 2927*x + 32914*x^2 + 73486*x^3 + 40405*x^4 + 4819*x^5 + 56*x^6) / (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)

A199838 Number of -n..n arrays x(0..8) of 9 elements with zero sum and no two neighbors summing to zero.

Original entry on oeis.org

66, 23206, 645780, 6715618, 41008804, 179213048, 622300326, 1827026482, 4719970500, 11025201168, 23740333870, 47800415256, 90973748554, 165038447302, 287293180292, 482460245532, 785043786046, 1242210635346, 1917265955424
Offset: 1

Views

Author

R. H. Hardin, Nov 11 2011

Keywords

Comments

Row 7 of A199832.

Examples

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

Links

Crossrefs

Cf. A199832.

Formula

Empirical: a(n) = (259723/2240)*n^8 - (299869/5040)*n^7 + (39757/1440)*n^6 - (8303/360)*n^5 + (31829/2880)*n^4 - (8083/720)*n^3 + (32213/5040)*n^2 - (509/420)*n.
Conjectures from Colin Barker, Mar 02 2018: (Start)
G.f.: 2*x*(33 + 11306*x + 219651*x^2 + 866735*x^3 + 937667*x^4 + 283090*x^5 + 18897*x^6 + 128*x^7) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A199839 Number of -n..n arrays x(0..9) of 10 elements with zero sum and no two neighbors summing to zero.

Original entry on oeis.org

138, 88636, 3685550, 51077518, 389832124, 2044221894, 8281149188, 27785393300, 80752300406, 209581305608, 496408914210, 1090354976530, 2248071267000, 4391976524034, 8191437222152, 14673108432136, 25367684178562
Offset: 1

Views

Author

R. H. Hardin Nov 11 2011

Keywords

Comments

Row 8 of A199832

Examples

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

Links

Formula

Empirical: a(n) = (124952/567)*n^9 - (35524/315)*n^8 + (50588/945)*n^7 - (2494/45)*n^6 + (13739/270)*n^5 - (1927/180)*n^4 - (41254/2835)*n^3 + (3319/420)*n^2 - (781/630)*n

A266913 Denominator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.

Original entry on oeis.org

1, 1, 3, 12, 5, 180, 315, 2240, 567, 907200, 51975, 13305600, 289575, 80720640, 212837625, 3487131648000, 2297295, 64023737057280, 14849255421, 28963119144960000, 17717861581875, 140500090972200960000, 16436269594119375, 6204484017332394393600, 40639128117328125
Offset: 1

Views

Author

Lorenz H. Menke, Jr., Mar 16 2016

Keywords

Comments

Reference A. Dubickas shows that all the volume integrals are rational with V[d] <= 2^d.

Examples

			For d = 3 the volume is 16/3, for each volume we have V[1] = 2, V[2] = 3, V[3] = 16/3, V[4] = 115/12, V[5] = 88/5, V[6] = 5887/180, V[7] = 19328/315, V[8] = 259723/2240, V[9] = 124952/567, V[10] = 381773117/907200, etc.

Links

Crossrefs

The numerator sequence is given by A269067.
Cf. A199832 The rational coefficient of the leading coefficient of the empirical rows duplicates these volume integrals in sequence. This is not a proof.

Programs

  • Mathematica
    V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],
  Table[x[i], {i, 1, d}] \[Element]
   Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]]] (* Lorenz H. Menke, Jr. *)
v[d_] := With[{a = Array[x,d]},RegionMeasure @ ImplicitRegion[a ∈ Cuboid[-Table[1, d], Table[1, d]] && -1 <= Total[a] <= 1,a]] (* Carl Woll *)
v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)

Extensions

a(11)-a(25) from Lorenz H. Menke, Jr., May 10 2018

A269067 Numerator of the volume of d dimensional symmetric convex cuboid with constraints |x1 + x2 + ... xd| <= 1 and |x1|, |x2|, ..., |xd| <= 1.

Original entry on oeis.org

2, 3, 16, 115, 88, 5887, 19328, 259723, 124952, 381773117, 41931328, 20646903199, 866732192, 467168310097, 2386873693184, 75920439315929441, 97261697912, 5278968781483042969, 2387693641959232, 9093099984535515162569, 10872995484706511008, 168702835448329388944396777, 38650653745373963289088
Offset: 1

Views

Author

Lorenz H. Menke, Jr., Feb 19 2016

Keywords

Comments

Reference A. Dubickas shows that all the volume integrals are rational with V[d] <= 2^d.

Examples

			For d = 3 the volume is 16/3, for each volume we have V[1] = 2, V[2] = 3, V[3] = 16/3, V[4] = 115/12, V[5] = 88/5, V[6] = 5887/180, V[7] = 19328/315, V[8] = 259723/2240, V[9] = 124952/567, V[10] = 381773117/907200, etc.

Links

Crossrefs

The denominator sequence is given by A266913.
Cf. A199832 The rational coefficient of the leading coefficient of the empirical rows duplicates these volume integrals in sequence. This is not a proof.

Programs

  • Mathematica
    V[d_] := Integrate[Boole[Abs[Sum[x[i], {i, 1, d}]] <= 1],
  Table[x[i], {i, 1, d}] \[Element]
   Cuboid[Table[-1, {i, 1, d}], Table[+1, {i, 1, d}]]] (* Lorenz H. Menke, Jr. *)
v[d_] := With[{a = Array[x,d]}, RegionMeasure @ ImplicitRegion[ a ∈ Cuboid[-Table[1, d], Table[1, d]] && -1 <= Total[a] <= 1, a]] (* Carl Woll *)
v[d_] := 2^(d+1)/(Pi) Integrate[Sin[t]^(d+1)/t^(n+1), {t, 0, Infinity}] (* Carl Woll *)

Extensions

a(11)-a(23) from Lorenz H. Menke, Jr., May 10 2018
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.