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 10 results.

A221568 Number of 0..3 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 10, 26, 100, 342, 1210, 4240, 14898, 52306, 183684, 645006, 2264978, 7953568, 27929338, 98075178, 344395620, 1209361446, 4246729738, 14912591664, 52366268642, 183886620962, 645726538244, 2267499179678, 7962430263202, 27960449231680, 98184435580010
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Comments

Column 3 of A221573.

Examples

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

Links

Programs

  • PARI
    concat(0, Vec(2*x^2*(5 - 2*x + x^2) / ((1 + x)*(1 - 4*x + 2*x^2 - x^3)) + O(x^30))) \\ Colin Barker, Jan 31 2017

Formula

a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4).
G.f.: 2*x^2*(5 - 2*x + x^2) / ((1 + x)*(1 - 4*x + 2*x^2 - x^3)). - Colin Barker, Jan 31 2017

A221567 Number of 0..2 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 5, 9, 25, 57, 141, 345, 853, 2097, 5149, 12633, 31013, 76161, 187053, 459369, 1128053, 2770065, 6802301, 16704249, 41020357, 100732833, 247366989, 607452297, 1491704341, 3663139761, 8995478557, 22089965337, 54245756261, 133209897153
Offset: 1

Views

Author

R. H. Hardin Jan 20 2013

Keywords

Comments

Column 2 of A221573

Examples

			Some solutions for n=6
..2....1....2....0....0....1....2....0....0....0....0....2....0....2....0....1
..0....1....2....0....0....1....0....2....2....0....0....0....0....2....0....1
..0....1....1....2....2....0....0....2....2....2....2....1....2....1....0....2
..0....0....1....2....0....0....0....0....0....0....2....1....2....1....2....2
..2....0....2....2....2....1....1....2....0....0....1....2....2....2....0....0
..0....0....2....2....2....1....1....2....2....2....1....0....0....0....0....2

Links

Formula

Empirical: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-4).
Empirical g.f.: -x^2*(5-6*x+8*x^2) / ( -1+3*x-2*x^2+4*x^4 ). - R. J. Mathar, Jun 06 2013

A221569 Number of 0..4 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 17, 59, 289, 1293, 5913, 26911, 122621, 558547, 2544357, 11590169, 52796369, 240501763, 1095550873, 4990531051, 22733220441, 103555975477, 471725515497, 2148837489879, 9788536778149, 44589436230083, 203116958964733
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Examples

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

Links

Crossrefs

Column 4 of A221573.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5),seq(a(i)=[0, 17, 59, 289, 1293, 5913][i],i=1..6)},
a(n),remember):
map(f, [$1..50]); # Robert Israel, Jun 04 2018

Formula

Empirical: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6.
Empirical g.f.: -x^2*(17-26*x+45*x^2+8*x^3+x^4) / ( -1+5*x-3*x^2+x^3+15*x^4+3*x^5 ). - R. J. Mathar, Jun 06 2013
Formula verified by Robert Israel, Jun 04 2018: see link.

A221570 Number of 0..5 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 26, 114, 676, 3734, 20944, 117104, 655198, 3665306, 20505052, 114711980, 641737294, 3590092664, 20084178100, 112357602542, 628565969692, 3516408049766, 19671961528672, 110051525562128, 615665004273436
Offset: 1

Views

Author

R. H. Hardin Jan 20 2013

Keywords

Comments

Column 5 of A221573

Examples

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

Links

Formula

Empirical: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6).
Empirical g.f.: -2*x^2*(13-8*x+14*x^2+6*x^3+6*x^4) / ( -1+5*x+3*x^2+9*x^4+6*x^5+3*x^6 ). - R. J. Mathar, Jun 06 2013

A221571 Number of 0..6 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 37, 197, 1369, 8991, 59705, 395641, 2622817, 17385993, 115249117, 763966685, 5064207645, 33569783613, 222528473325, 1475103973253, 9778217146445, 64818163521317, 429668748357261, 2848202159470085, 18880255015594493
Offset: 1

Views

Author

R. H. Hardin Jan 20 2013

Keywords

Comments

Column 6 of A221573

Examples

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

Links

Formula

Empirical: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8).
Empirical: -x^2*(37-62*x+138*x^2-26*x^3+100*x^4-36*x^5+48*x^6) / ( -1+7*x-4*x^2+6*x^3+26*x^4+10*x^5+16*x^6+12*x^8 ). - R. J. Mathar, Jun 06 2013

A221572 Number of 0..7 arrays of length n with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

0, 50, 314, 2500, 19014, 145800, 1116400, 8550512, 65485386, 501533796, 3841097940, 29417832750, 225302467392, 1725524876860, 13215284016064, 101211946587176, 775152325067630, 5936662096954472, 45467136862793520
Offset: 1

Views

Author

R. H. Hardin Jan 20 2013

Keywords

Comments

Column 7 of A221573

Examples

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

Links

Formula

Empirical: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8).
Empirical g.f.: -2*x^2*(25-18*x+51*x^2+4*x^3+66*x^4-18*x^5+6*x^6) / ( -1+7*x+4*x^2+5*x^3+20*x^4+20*x^5+23*x^6-6*x^7+3*x^8 ). - R. J. Mathar, Jun 06 2013

A221574 Number of 0..n arrays of length 3 with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

2, 9, 26, 59, 114, 197, 314, 471, 674, 929, 1242, 1619, 2066, 2589, 3194, 3887, 4674, 5561, 6554, 7659, 8882, 10229, 11706, 13319, 15074, 16977, 19034, 21251, 23634, 26189, 28922, 31839, 34946, 38249, 41754, 45467, 49394, 53541, 57914, 62519, 67362
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Comments

Row 3 of A221573.

Examples

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

Links

Crossrefs

Cf. A221573.

Formula

Empirical: a(n) = 1*n^3 - 1*n^2 + 3*n - 1.
Conjectures from Colin Barker, Aug 08 2018: (Start)
G.f.: x*(2 + x)*(1 + x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
(End)

A221575 Number of 0..n arrays of length 5 with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

6, 57, 342, 1293, 3734, 8991, 19014, 36497, 64998, 109059, 174326, 267669, 397302, 572903, 805734, 1108761, 1496774, 1986507, 2596758, 3348509, 4265046, 5372079, 6697862, 8273313, 10132134, 12310931, 14849334, 17790117, 21179318, 25066359
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Comments

Row 5 of A221573.

Examples

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

Links

Crossrefs

Cf. A221573.

Formula

Empirical: a(n) = 1*n^5 + 1*n^4 - 2*n^3 + 12*n^2 - 15*n + 9 for n>2.
Conjectures from Colin Barker, Aug 08 2018: (Start)
G.f.: x*(6 + 21*x + 90*x^2 - 24*x^3 + 56*x^4 - 39*x^5 + 12*x^6 - 2*x^7) / (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>8.
(End)

A221576 Number of 0..n arrays of length 6 with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

10, 141, 1210, 5913, 20944, 59705, 145800, 317233, 631328, 1170369, 2047960, 3416105, 5473008, 8471593, 12728744, 18635265, 26666560, 37394033, 51497208, 69776569, 93167120, 122752665, 159780808, 205678673, 262069344, 330789025
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Comments

Row 6 of A221573.

Examples

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

Links

Crossrefs

Cf. A221573.

Formula

Empirical: a(n) = 1*n^6 + 2*n^5 - 5*n^4 + 24*n^3 - 41*n^2 + 50*n - 31 for n>3.
Conjectures from Colin Barker, Aug 08 2018: (Start)
G.f.: x*(10 + 71*x + 433*x^2 + 54*x^3 + 378*x^4 - 355*x^5 + 193*x^6 - 80*x^7 + 18*x^8 - 2*x^9) / (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>10.
(End)

A221577 Number of 0..n arrays of length 7 with each element differing from at least one neighbor by something other than 1.

Original entry on oeis.org

16, 345, 4240, 26911, 117104, 395641, 1116400, 2754635, 6127696, 12553189, 24049616, 43584535, 75375280, 125247281, 201055024, 313170691, 475045520, 703848925, 1021190416, 1453929359, 2035077616, 2804800105, 3811518320
Offset: 1

Views

Author

R. H. Hardin, Jan 20 2013

Keywords

Comments

Row 7 of A221573.

Examples

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

Links

Crossrefs

Cf. A221573.

Formula

Empirical: a(n) = 1*n^7 + 3*n^6 - 7*n^5 + 29*n^4 - 41*n^3 + 45*n^2 - 33*n + 19 for n>2.
Conjectures from Colin Barker, Aug 08 2018: (Start)
G.f.: x*(16 + 217*x + 1928*x^2 + 1755*x^3 + 2336*x^4 - 1869*x^5 + 1096*x^6 - 579*x^7 + 160*x^8 - 20*x^9) / (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>10.
(End)
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