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.

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A221852 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal, diagonal or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 3, 1, 6, 31, 1, 13, 302, 306, 1, 28, 3437, 10085, 2989, 1, 60, 37155, 465305, 354670, 29135, 1, 129, 406612, 19159028, 67228149, 12277568, 283872, 1, 277, 4434129, 811781250, 10799370973, 9607602488, 426752551, 2765627, 1, 595, 48397883
Offset: 1

Views

Author

R. H. Hardin Jan 27 2013

Keywords

Comments

Table starts
.1.......3.........6.........13..........28............60.........129......277
.1......31.......302.......3437.......37155........406612.....4434129.48397883
.1.....306.....10085.....465305....19159028.....811781250.34109464173
.1....2989....354670...67228149.10799370973.1802820316744
.1...29135..12277568.9607602488
.1..283872.426752551
.1.2765627
.1

Examples

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

Links

Crossrefs

Column 2 is A220997
Row 1 is A002478

A222020 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or vertical neighbor, without move-in move-out straight through or left turns.

Original entry on oeis.org

1, 3, 3, 6, 27, 6, 13, 188, 188, 13, 28, 1363, 3875, 1363, 28, 60, 9838, 86088, 86088, 9838, 60, 129, 70994, 1892859, 5925053, 1892859, 70994, 129, 277, 512456, 41636522, 403317044, 403317044, 41636522, 512456, 277, 595, 3698699, 916164266
Offset: 1

Views

Author

R. H. Hardin Feb 05 2013

Keywords

Comments

Table starts
....1..........3.............6...............13.................28
....3.........27...........188.............1363...............9838
....6........188..........3875............86088............1892859
...13.......1363.........86088..........5925053..........403317044
...28.......9838.......1892859........403317044........84799061141
...60......70994......41636522......27460842224.....17838051289367
..129.....512456.....916164266....1870475750419...3753835631741568
..277....3698699...20156123913..127385479747138.789821887272883642
..595...26696370..443464985963.8675741759311142
.1278..192687348.9756801621038
.2745.1390767557
.5896

Examples

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

Links

Crossrefs

Column 1 is A002478

A078007 Expansion of (1-x)/(1-x-2*x^2-x^3).

Original entry on oeis.org

1, 0, 2, 3, 7, 15, 32, 69, 148, 318, 683, 1467, 3151, 6768, 14537, 31224, 67066, 144051, 309407, 664575, 1427440, 3065997, 6585452, 14144886, 30381787, 65257011, 140165471, 301061280, 646649233, 1388937264, 2983297010, 6407820771, 13763352055, 29562290607
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

Let X = the 3x3 matrix [0,1,0; 0,0,1; 1,2,1]. a(n) = center term of X^n; but A002478(n) = term (3,3) of X^n. - Gary W. Adamson, May 30 2008
First bisection of A058278. - Oboifeng Dira, Aug 04 2016

Links

Crossrefs

First differences of A002478.
Cf. A058278.

Programs

  • GAP
    a:=[1,0,2];; for n in [4..40] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 28 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-2*x^2-x^3) )); // G. C. Greubel, Jun 28 2019
  • Mathematica
    LinearRecurrence[{1,2,1}, {1,0,2}, 40] (* or *) CoefficientList[Series[(1 -x)/(1-x-2*x^2-x^3), {x,0,40}], x] (* G. C. Greubel, Jun 28 2019 *)
  • PARI
    Vec((1-x)/(1-x-2*x^2-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012
  • Sage
    ((1-x)/(1-x-2*x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 28 2019

Formula

a(n) = a(n-1) + 2*a(n-2) + a(n-3). - Ilya Gutkovskiy, Aug 06 2016

A116088 Riordan array (1, x*(1+x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 4, 1, 0, 0, 6, 6, 1, 0, 0, 4, 15, 8, 1, 0, 0, 1, 20, 28, 10, 1, 0, 0, 0, 15, 56, 45, 12, 1, 0, 0, 0, 6, 70, 120, 66, 14, 1, 0, 0, 0, 1, 56, 210, 220, 91, 16, 1, 0, 0, 0, 0, 28, 252, 495, 364, 120, 18, 1
Offset: 0

Views

Author

Paul Barry, Feb 04 2006

Keywords

Examples

			Triangle begins as:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 4,  1;
  0, 0, 6,  6,  1;
  0, 0, 4, 15,  8,  1;
  0, 0, 1, 20, 28, 10,  1;
  0, 0, 0, 15, 56, 45, 12, 1;

Links

Crossrefs

Row sums are A002478. Diagonal sums are A094686. Inverse is (-1)^(n-k) * A109971(n,k). Unsigned version of A109970.

Programs

  • GAP
    Flat(List([0..10], n->List([0..n], k-> Binomial(2*k, n-k) ))); # G. C. Greubel, May 09 2019
  • Magma
    [[Binomial(2*k, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019
  • Mathematica
    Flatten[Table[Binomial[2k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Oct 22 2012 *)
  • PARI
    {T(n,k) = binomial(2*k, n-k)}; \\ G. C. Greubel, May 09 2019
  • Sage
    [[binomial(2*k, n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

Formula

G.f.: 1/(1-x*y*(1+x)^2).
Number triangle T(n,k) = C(2*k, n-k) = C(n,k)*C(3*k,n)/C(3*k,k).

A218439 a(n) = A001609(n)^2, where g.f. of A001609 is x*(1+3*x^2)/(1-x-x^3).

Original entry on oeis.org

1, 1, 16, 25, 36, 100, 225, 441, 961, 2116, 4489, 9604, 20736, 44521, 95481, 205209, 440896, 946729, 2033476, 4368100, 9381969, 20151121, 43283241, 92968164, 199685161, 428904100, 921243904, 1978737289, 4250127249, 9128847025, 19607840784, 42115658841
Offset: 1

Views

Author

Paul D. Hanna, Oct 28 2012

Keywords

Comments

A001609 equals the logarithmic derivative of Narayana's cows sequence A000930.

Examples

			O.g.f.: A(x) = x + x^2 + 16*x^3 + 25*x^4 + 36*x^5 + 100*x^6 + 225*x^7 +...
L.g.f.: L(x) = x + x^2/2 + 16*x^3/3 + 25*x^4/4 + 36*x^5/5 + 100*x^6/6 + 225*x^7/7 +...
where exponentiation yields the g.f. of A218438:
exp(L(x)) = 1 + x + x^2 + 6*x^3 + 12*x^4 + 19*x^5 + 48*x^6 + 110*x^7 +...

Links

Crossrefs

Cf. A000930, A001609, A002478, A112455, A218438.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1 + 14*x^2 + 5*x^3 - 9*x^4 - 9*x^5)/((1 + x^2 - x^3)*(1 - x - 2*x^2 - x^3)), {x, 0, 50}], x]] (* G. C. Greubel, Apr 28 2017 *)
  • PARI
    {a(n)=polcoeff(x*(1+14*x^2+5*x^3-9*x^4-9*x^5)/((1+x^2-x^3)*(1-x-2*x^2-x^3+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))

Formula

O.g.f.: x*(1 + 14*x^2 + 5*x^3 - 9*x^4 - 9*x^5)/((1 + x^2 - x^3)*(1 - x - 2*x^2 - x^3)).
Logarithmic derivative of A218438.
a(n) = -2*(-1)^n*A112455(n) +3*A002478(n) -2*A002478(n-1)-2*A002478(n-2), n>1. - R. J. Mathar, Oct 28 2012

A221824 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or vertical neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 3, 3, 6, 31, 6, 13, 252, 252, 13, 28, 2113, 6711, 2113, 28, 60, 17684, 190756, 190756, 17684, 60, 129, 147920, 5374720, 18636989, 5374720, 147920, 129, 277, 1237439, 151592441, 1804472268, 1804472268, 151592441, 1237439, 277, 595, 10351593
Offset: 1

Views

Author

R. H. Hardin Jan 26 2013

Keywords

Comments

Table starts
....1........3............6.............13..............28..............60
....3.......31..........252...........2113...........17684..........147920
....6......252.........6711.........190756.........5374720.......151592441
...13.....2113.......190756.......18636989......1804472268....174820457922
...28....17684......5374720.....1804472268....597819977898.198296512455330
...60...147920....151592441...174820457922.198296512455330
..129..1237439...4274867564.16936005698353
..277.10351593.120554575328
..595.86594795
.1278

Examples

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

Links

Crossrefs

Column 1 is A002478

A221886 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal, vertical or antidiagonal neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 3, 3, 6, 33, 6, 13, 326, 326, 13, 28, 3223, 11878, 3223, 28, 60, 31832, 449069, 449069, 31832, 60, 129, 314426, 16834526, 65039658, 16834526, 314426, 129, 277, 3105733, 632170065, 9356131846, 9356131846, 632170065, 3105733, 277, 595, 30676867
Offset: 1

Views

Author

R. H. Hardin Jan 30 2013

Keywords

Comments

Table starts
...1........3...........6............13............28............60
...3.......33.........326..........3223.........31832........314426
...6......326.......11878........449069......16834526.....632170065
..13.....3223......449069......65039658....9356131846.1347142288446
..28....31832....16834526....9356131846.5123355027782
..60...314426...632170065.1347142288446
.129..3105733.23731120064
.277.30676867
.595

Examples

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

Links

Crossrefs

Column 1 is A002478

A221925 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some king-move neighbor, without move-in move-out straight through or left turns.

Original entry on oeis.org

1, 3, 3, 6, 35, 6, 13, 350, 350, 13, 28, 3487, 13228, 3487, 28, 60, 34714, 508281, 508281, 34714, 60, 129, 345744, 19513309, 75313013, 19513309, 345744, 129, 277, 3443699, 749481703, 11143187999, 11143187999, 749481703, 3443699, 277, 595, 34300167
Offset: 1

Views

Author

R. H. Hardin Jan 31 2013

Keywords

Comments

Table starts
...1.......3.........6..........13..........28............60.........129
...3......35.......350........3487.......34714........345744.....3443699
...6.....350.....13228......508281....19513309.....749481703.28788205897
..13....3487....508281....75313013.11143187999.1648683705320
..28...34714..19513309.11143187999
..60..345744.749481703
.129.3443699
.277

Examples

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

Links

Crossrefs

Column 1 is A002478

A361841 Expansion of 1/(1 - 9*x*(1+x)^2)^(1/3).

Original entry on oeis.org

1, 3, 24, 201, 1809, 16893, 161676, 1574289, 15527052, 154662930, 1552725504, 15688410264, 159355067283, 1625899880673, 16652520666414, 171119405299005, 1763475423260049, 18219685282559559, 188664151412242368, 1957539823296458841, 20347733657193596127
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Links

Crossrefs

Column k=2 of A361839.
Cf. A002478, A137635, A361844.

Programs

  • Maple
    A361841 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], -3/4):
seq(simplify(A361841(n)), n = 0..20); # Peter Luschny, Mar 27 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^2)^(1/3))

Formula

n*a(n) = 3 * ( (3*n-2)*a(n-1) + 2*(3*n-4)*a(n-2) + (3*n-6)*a(n-3) ) for n > 2.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(2*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], -3/4). - Peter Luschny, Mar 27 2023

A104580 Tribonacci convolution triangle.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 7, 12, 9, 4, 1, 13, 26, 25, 14, 5, 1, 24, 56, 63, 44, 20, 6, 1, 44, 118, 153, 125, 70, 27, 7, 1, 81, 244, 359, 336, 220, 104, 35, 8, 1, 149, 499, 819, 864, 646, 357, 147, 44, 9, 1, 274, 1010, 1830, 2144, 1800, 1134, 546, 200, 54, 10, 1
Offset: 0

Views

Author

Paul Barry, Mar 16 2005

Keywords

Examples

			Rows begin
  {1},
  {1,1},
  {2,2,1},
  {4,5,3,1},
  {7,12,9,4,1},
   ...

Crossrefs

First column is A000073(n+2). Row sums are A077939. Diagonal sums are A002478.

Programs

  • Maple
    # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> A000073(n+1)); # Peter Luschny, Oct 19 2022
  • Maxima
    trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i,n-k-i),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

Formula

Riordan array (1/(1-x-x^2-x^3), x/(1-x-x^2-x^3)).
From Paul Barry, Jun 02 2009: (Start)
T(n,m) = T'(n-1,m-1) + T'(n-1,m) + T'(n-2,m) + T'(n-3,m), where T'(n,m) = T(n,m) for n >= 0 and 0 <= m <= n and T'(n,m) = 0 otherwise. (End)
T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*A027907(i,n-k-i). - Emanuele Munarini, Mar 15 2011
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