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 1-3 of 3 results.

A206455 T(n,k) = number of 0..k arrays of length n avoiding the consecutive pattern 0..k.

Original entry on oeis.org

2, 3, 3, 4, 9, 4, 5, 16, 26, 5, 6, 25, 64, 75, 6, 7, 36, 125, 255, 216, 7, 8, 49, 216, 625, 1016, 622, 8, 9, 64, 343, 1296, 3124, 4048, 1791, 9, 10, 81, 512, 2401, 7776, 15615, 16128, 5157, 10, 11, 100, 729, 4096, 16807, 46655, 78050, 64257, 14849, 11, 12, 121, 1000
Offset: 1

Views

Author

R. H. Hardin, Feb 07 2012

Keywords

Examples

			Table starts
  2    3     4      5       6       7        8        9        10        11 ...
  3    9    16     25      36      49       64       81       100       121 ...
  4   26    64    125     216     343      512      729      1000      1331 ...
  5   75   255    625    1296    2401     4096     6561     10000     14641 ...
  6  216  1016   3124    7776   16807    32768    59049    100000    161051 ...
  7  622  4048  15615   46655  117649   262144   531441   1000000   1771561 ...
  8 1791 16128  78050  279924  823542  2097152  4782969  10000000  19487171 ...
  9 5157 64257 390125 1679508 5764787 16777215 43046721 100000000 214358881 ...
  ...

Links

Crossrefs

Columns 2, 3... are A076264, A206450, A206451, A206452.
Subdiagonal 1 is A048861(n+1)

Programs

  • Maple
    N:= 20: # for the first N antidiagonals
for k from 1 to N-1 do
  F[k]:= gfun:-rectoproc({a(n)=(k+1)*a(n-1) - a(n-k-1), seq(a(j)=(k+1)^j,j=1..k),a(k+1)=(k+1)^(k+1)-1},a(n),remember)
od:
seq(seq(F[m-j](j),j=1..m-1),m=1..N); # Robert Israel, Dec 17 2017
  • Mathematica
    nmax = 20;
col[k_] := col[k] = Module[{a}, a[n_ /; n>2] := a[n] = (k+1)*a[n-1]-a[n-k-1]; a[0]=1; a[1]=k+1; a[2]=(k+1)^2; a[_?Negative]=0; Array[a, nmax]];
T[n_, k_] := If[k == 1, n+1, col[k][[n]]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 22 2022 *)

Formula

Empirical: T(n,k) = sum{i=0..floor(n/(k+1))} ( (-1)^i * (k+1)^(n-(k+1)*i) * binomial(n-k*i,i) ) (after A076264)
Empirical for column k: a(n) = (k+1)*a(n-1) - a(n-(k+1)).
Formula for column k verified by Robert Israel, Dec 17 2017 (see link).

A255660 T(n,k)=Number of length n+k 0..3 arrays with at most two downsteps in every k consecutive neighbor pairs.

Original entry on oeis.org

16, 64, 64, 255, 256, 256, 968, 1016, 1024, 1024, 3340, 3692, 4048, 4096, 4096, 10320, 11752, 14192, 16128, 16384, 16384, 28722, 33042, 42653, 54560, 64257, 65536, 65536, 72920, 83752, 112196, 155144, 209412, 256012, 262144, 262144, 171106, 195020
Offset: 1

Views

Author

R. H. Hardin, Mar 01 2015

Keywords

Comments

Table starts
......16.......64......255.......968......3340.....10320.....28722......72920
......64......256.....1016......3692.....11752.....33042.....83752.....195020
.....256.....1024.....4048.....14192.....42653....112196....265430.....577464
....1024.....4096....16128.....54560....155144....385738....864924....1788660
....4096....16384....64257....209412....564600...1324872...2816673....5555336
...16384....65536...256012....803246...2036844...4542671...9169016...17232696
...65536...262144..1020000...3083292...7323894..15269184..29577432...53275408
..262144..1048576..4063872..11835664..26452984..50963540..92530816..161617336
.1048576..4194304.16191231..45429680..95690028.171784096.286454024..471810032
.4194304.16777216.64508912.174365744.345980784.583245999.900260308.1359483102

Examples

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

Links

Crossrefs

Column 1 is A000302(n+1)
Column 2 is A000302(n+2)
Column 3 is A206450(n+3)

Formula

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 4*a(n-1) -a(n-4)
k=4: [order 12]
k=5: [order 24]
k=6: [order 35]
k=7: [order 48]
Empirical for row n:
n=1: [polynomial of degree 11]
n=2: [polynomial of degree 11]
n=3: [polynomial of degree 11] for n>1
n=4: [polynomial of degree 11] for n>2
n=5: [polynomial of degree 11] for n>3
n=6: [polynomial of degree 11] for n>4
n=7: [polynomial of degree 11] for n>5

A250361 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 3 times.

Original entry on oeis.org

2, 3, 4, 4, 9, 8, 5, 16, 27, 16, 6, 25, 64, 81, 32, 7, 36, 125, 255, 243, 64, 8, 49, 216, 623, 1016, 729, 128, 9, 64, 343, 1293, 3094, 4048, 2187, 256, 10, 81, 512, 2397, 7712, 15365, 16128, 6561, 512, 11, 100, 729, 4091, 16700, 45866, 76300, 64257, 19683, 1024, 12
Offset: 1

Views

Author

R. H. Hardin, Nov 19 2014

Keywords

Comments

Table starts
....2.....3.......4.......5........6.........7..........8..........9.........10
....4.....9......16......25.......36........49.........64.........81........100
....8....27......64.....125......216.......343........512........729.......1000
...16....81.....255.....623.....1293......2397.......4091.......6555.......9993
...32...243....1016....3094.....7712.....16700......32608......58826......99704
...64...729....4048...15365....45866....115963.....259106.....526505.....992530
..128..2187...16128...76300...272760....803382....2052904....4698744....9854280
..256..6561...64257..378880..1621963...5565230...16234706...41828450...97581710
..512.19683..256012.1881364..9644496..38548644..128373416..371780050..964209084
.1024.59049.1020000.9342081.57346376.266998350.1015004124.3304106808.9514922752

Examples

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

Links

Crossrefs

Column 1 is A000079
Column 2 is A000244
Column 3 is A206450
Row 1 is A000027(n+1)
Row 2 is A000290(n+1)
Row 3 is A000578(n+1)
Cf. A248944, A250351

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1)
k=3: a(n) = 4*a(n-1) -a(n-4)
k=4: a(n) = 5*a(n-1) -2*a(n-4) -11*a(n-5) +a(n-8)
k=5: [order 13]
k=6: [order 25]
k=7: [order 56]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 3*n + 1
n=4: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 3 for n>1
n=5: a(n) = n^5 + 5*n^4 + 10*n^3 + 5*n^2 + 17*n + 2 for n>2
n=6: a(n) = n^6 + 6*n^5 + 15*n^4 + 5*n^3 + 57*n^2 + 13*n + 1 for n>3
n=7: a(n) = n^7 + 7*n^6 + 21*n^5 + 147*n^3 + 49*n^2 + 7*n for n>4
Showing 1-3 of 3 results.

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