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 1-4 of 4 results.

A245163 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 8, 1, 1, 2, 4, 9, 16, 1, 1, 2, 4, 9, 23, 32, 1, 1, 2, 4, 9, 23, 64, 64, 1, 1, 2, 4, 9, 23, 65, 186, 128, 1, 1, 2, 4, 9, 23, 65, 199, 551, 256, 1, 1, 2, 4, 9, 23, 65, 199, 653, 1645, 512, 1, 1, 2, 4, 9, 23, 65, 199, 654, 2275, 4926, 1024, 1, 1, 2, 4, 9, 23
Offset: 1

Views

Author

R. H. Hardin, Jul 12 2014

Keywords

Comments

Table starts
.....1........1.........1.........1.........1.........1.........1.........1
.....1........1.........1.........1.........1.........1.........1.........1
.....2........2.........2.........2.........2.........2.........2.........2
.....4........4.........4.........4.........4.........4.........4.........4
.....8........9.........9.........9.........9.........9.........9.........9
....16.......23........23........23........23........23........23........23
....32.......64........65........65........65........65........65........65
....64......186.......199.......199.......199.......199.......199.......199
...128......551.......653.......654.......654.......654.......654.......654
...256.....1645......2275......2296......2296......2296......2296......2296
...512.....4926......8313......8568......8569......8569......8569......8569
..1024....14768.....31439.....33794.....33825.....33825.....33825.....33825
..2048....44293....121637....140039....140580....140581....140581....140581
..4096...132867....477307....605869....612890....612933....612933....612933
..8192...398588...1888721...2718531...2794159...2795181...2795182...2795182
.16384..1195750...7509799..12564289..13280627..13298407..13298464..13298464
.32768..3587235..29940861..59419764..65597882..65851100..65852872..65852873
.65536.10761689.119550419.285878342.335521900.338654554.338694406.338694479

Examples

			Some solutions for n=10 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....0....1....0....1....0....1....1....0....0....1....1....0
..1....0....2....0....1....2....1....0....0....2....2....0....1....1....2....1
..0....1....3....1....2....0....0....2....0....2....1....1....0....1....2....0
..1....2....0....0....0....1....2....3....0....3....1....2....1....1....2....1
..0....2....0....2....2....1....0....3....1....1....0....1....2....2....2....0
..0....1....2....1....2....1....0....2....1....3....2....1....1....1....1....2
..0....1....1....2....1....1....1....0....1....2....1....1....1....2....1....1
..1....1....1....1....1....1....0....1....0....1....1....0....1....1....1....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

Crossrefs

Column 1 is A000079(n-2)
Column 2 is A164039(n-2)
Diagonal is A007476

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 5*a(n-1) -7*a(n-2) +3*a(n-3) for n>4
k=3: a(n) = 10*a(n-1) -37*a(n-2) +64*a(n-3) -52*a(n-4) +16*a(n-5) for n>6
k=4: [order 7] for n>8
k=5: [order 9] for n>10
k=6: [order 11] for n>12
k=7: [order 13] for n>14

A209245 Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.

Original entry on oeis.org

1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497
Offset: 0

Views

Author

Jon Perry, Jan 13 2013

Keywords

Comments

Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.
Slice x(1,j,k) with j,k >= 0 of the cube begins:
1, 1, 1, 1, 1, 1, 1, 1, ... A000012
1, 3, 5, 7, 9, 11, 13, 15, ... A005408
1, 5, 11, 19, 29, 41, 55, 71, ... A028387
1, 7, 19, 39, 69, 111, 167, 239, ... A108766(k+1)
1, 9, 29, 69, 139, 251, 419, 659, ...
1, 11, 41, 111, 251, 503, 923, 1583, ...
1, 13, 55, 167, 419, 923, 1847, 3431, ...
1, 15, 71, 239, 659, 1583, 3431, 6863, ...
The main diagonal of the slice is A134760.

Links

Crossrefs

Cf. A164039, A134760, A209288.
Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 2*n+1,
      ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
       +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
       ((2*n-1)*(7*n-12)*(n-1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 17 2013
  • Mathematica
    b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 3];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *)

Formula

a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.
a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014

A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 25, 16, 5, 1, 0, 1, 32, 65, 56, 25, 6, 1, 0, 1, 64, 161, 176, 105, 36, 7, 1, 0, 1, 128, 385, 512, 385, 176, 49, 8, 1, 0, 1, 256, 897, 1408, 1281, 736, 273, 64, 9, 1, 0, 1, 512, 2049, 3712, 3969, 2752, 1281, 400, 81, 10, 1
Offset: 0

Views

Author

Stefano Spezia, Jan 20 2024

Keywords

Examples

			The array begins:
  0, 1,  1,   1,   1,    1, ...
  0, 1,  2,   3,   4,    5, ...
  0, 1,  4,   9,  16,   25, ...
  0, 1,  8,  25,  56,  105, ...
  0, 1, 16,  65, 176,  385, ...
  0, 1, 32, 161, 512, 1281, ...
  ...

Links

  • Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.

Crossrefs

Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
Cf. A000035, A369325 (main diagonal), A369326.

Programs

  • Mathematica
    A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

Formula

A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).
Sum_{k=0..n} A(n-k,k) = A164039(n-1).

A244762 a(n) = (5*3^n-2*n-1)/4.

Original entry on oeis.org

1, 3, 10, 32, 99, 301, 908, 2730, 8197, 24599, 73806, 221428, 664295, 1992897, 5978704, 17936126, 53808393, 161425195, 484275602, 1452826824, 4358480491, 13075441493, 39226324500, 117678973522, 353036920589, 1059110761791, 3177332285398, 9531996856220, 28595990568687, 85787971706089, 257363915118296
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 06 2014

Keywords

Links

Crossrefs

Cf. A060816 (first differences).
Cf. A108765, A164039, A047926, A000340.

Programs

  • Mathematica
    CoefficientList[Series[(1-2*x+2*x^2)/((1-3*x)*(1-x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2014 *)

Formula

a(n+1) = 3*a(n) + n.
G.f.: (1-2*x+2*x^2) / ((1-3*x)*(1-x)^2).
E.g.f.: exp(x)*(5*exp(2*x) - 2*x - 1)/4. - Stefano Spezia, Aug 28 2023
Showing 1-4 of 4 results.

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