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 1-5 of 5 results.

A243645 Number of ways two L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 1, 20, 87, 244, 545, 1056, 1855, 3032, 4689, 6940, 9911, 13740, 18577, 24584, 31935, 40816, 51425, 63972, 78679, 95780, 115521, 138160, 163967, 193224, 226225, 263276, 304695, 350812, 401969, 458520, 520831, 589280, 664257, 746164, 835415, 932436
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2014

Keywords

Comments

This sequence also represents the number of edges added to G so that it is complete, where G is a graph of (n-1)^2 nodes arranged in a rhombus and embedded in the hexagonal lattice. G begins with A045944(n-2) edges and a(n) edges are added to form a complete graph. - John Tyler Rascoe, Sep 24 2022

Examples

			a(3) = 1:
._____.
|_| |_|
| |___|
|___|_| .

Links

Crossrefs

Column k=2 of A243608.

Programs

  • Maple
    a:= n-> `if`(n<2, 0, ((((n-4)*n-1)*n+18)*n-16)/2):
seq(a(n), n=0..50);
  • Mathematica
    CoefficientList[Series[x^3 (x^3+3x^2-15x-1)/(x-1)^5,{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,1,20,87,244},40] (* Harvey P. Dale, Mar 06 2016 *)

Formula

G.f.: x^3*(x^3+3*x^2-15*x-1) / (x-1)^5.
a(n) = (n^4-4*n^3-n^2+18*n-16)/2 for n>=2, a(n) = 0 for n<2.
a(n) = A083374(n-1) - A045944(n-2) for n>=2. - John Tyler Rascoe, Sep 24 2022

A243646 Number of ways three L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 0, 11, 196, 1195, 4544, 13215, 32276, 69671, 137120, 251139, 434180, 715891, 1134496, 1738295, 2587284, 3754895, 5329856, 7418171, 10145220, 13657979, 18127360, 23750671, 30754196, 39395895, 49968224, 62801075, 78264836, 96773571, 118788320, 144820519
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2014

Keywords

Examples

			a(4) = 11:
._______.  ._______.  ._______.  ._______.
| |_|_|_|  | |_|_|_|  | |_| |_|  | |_| |_|
|___|_|_|  |___| |_|  |___|___|  |___|___|
| |_| |_|  | |_|___|  | |_|_|_|  |_| |_|_|
|___|___|  |___|_|_|  |___|_|_|  |_|___|_|
._______.  ._______.  ._______.  ._______.
| |_| |_|  |_|_| |_|  |_|_| |_|  |_| |_|_|
|___|___|  | |_|___|  |_|_|___|  |_|___|_|
|_|_| |_|  |___| |_|  | |_| |_|  | |_| |_|
|_|_|___|  |_|_|___|  |___|___|  |___|___|
._______.  ._______.  ._______.
|_|_| |_|  | |_|_|_|  |_| |_|_|
|_| |___|  |___| |_|  | |___|_|
| |___|_|  |_| |___|  |___| |_|
|___|_|_|  |_|___|_|  |_|_|___| .

Links

Crossrefs

Column k=3 of A243608.

Programs

  • Maple
    a:= n-> `if`(n<4, 0, ((((((n-6)*n-6)*n+88)*n-73)*n-310)*n+426)/6):
seq(a(n), n=0..50);

Formula

G.f.: x^4*(x^5-27*x^4+90*x^3-54*x^2-119*x-11) / (x-1)^7.
a(n) = (n^6-6*n^5-6*n^4+88*n^3-73*n^2-310*n+426)/6 for n>=4, a(n) = 0 for n<4.

A243647 Number of ways four L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 0, 1, 176, 3145, 22969, 106819, 376796, 1101151, 2805825, 6438909, 13602304, 26866541, 50186401, 89436655, 153088924, 253052339, 405702361, 633123801, 964595760, 1438347889, 2103619049, 3023051131, 4275452476, 5958967015, 8194686929, 11130748309
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2014

Keywords

Examples

			a(4) = 1:
._______.
| |_| |_|
|___|___|
| |_| |_|
|___|___| .

Links

Crossrefs

Column k=4 of A243608.

Programs

  • Maple
    a:= n-> `if`(n<4, 0, ((((((((n-8)*n-14)*n+244)*n-201)*n
         -2428)*n+4042)*n+7700)*n-15576)/24):
seq(a(n), n=0..50);

Formula

G.f.: -x^4*(9*x^8 -102*x^7 +253*x^6 +179*x^5 -1340*x^4 +916*x^3 +1597*x^2 +167*x+1) / (x-1)^9.
a(n) = (n^8 -8*n^7 -14*n^6 +244*n^5 -201*n^4 -2428*n^3 +4042*n^2 +7700*n -15576) / 24 for n>=4, a(n) = 0 for n<4.

A243648 Number of ways five L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 0, 0, 46, 4431, 73098, 587149, 3125278, 12712329, 42731866, 124522115, 324628878, 773900299, 1714106922, 3569586561, 7053577342, 13321444117, 24185953530, 42413141575, 72121174766, 119308962279, 192546161866, 303861667221, 469873699038, 713211276481
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2014

Keywords

Links

Crossrefs

Column k=5 of A243608.

Programs

  • Maple
    a:= n-> `if`(n<5, 0, ((((((((((n-10)*n-25)*n+520)*n-435)*n
    -9982)*n+19925)*n+82740)*n-215906)*n-244868)*n+728760)/120):
seq(a(n), n=0..40);

Formula

G.f.: x^5*(33*x^10 -293*x^9 +504*x^8 +1350*x^7 -3422*x^6 -7274*x^5 +28906*x^4 -19186*x^3 -26887*x^2 -3925*x -46) / (x-1)^11.
a(n) = (n^10 -10*n^9 -25*n^8 +520*n^7 -435*n^6 -9982*n^5 +19925*n^4 +82740*n^3 -215906*n^2 -244868*n +728760) / 120 for n>=5, a(n) = 0 for n<5.

A243649 Number of ways six L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3161, 147502, 2251309, 19028431, 111126797, 503008566, 1888247929, 6139119795, 17805426945, 47050056470, 115056780421, 263499318031, 570427305781, 1175960541134, 2322552621393, 4416363482851, 8118552261033, 14478163221342, 25121835774173
Offset: 0

Views

Author

Alois P. Heinz, Jun 08 2014

Keywords

Examples

			a(5) = 2:
._________.   ._________.
| |_|_| |_|   |_| |_| |_|
|___| |___|   | |___|___|
|_| |___|_|   |___|_| |_|
| |___| |_|   | |_| |___|
|___|_|___|   |___|___|_| .

Links

Crossrefs

Column k=6 of A243608.

Programs

  • Maple
    a:= n-> `if`(n<6, [0$5,2][n+1], ((((((((((((n-12)*n-39)*n+950)
        *n-815)*n-29672)*n+69499)*n+452518)*n-1454446)*n
        -3319216)*n+12944320)*n+9142512)*n-41687280)/720):
seq(a(n), n=0..40);

Formula

G.f.: -x^5*(97*x^13 -844*x^12 +2143*x^11 -3665*x^10 +26943*x^9 -113864*x^8 +167176*x^7 +102604*x^6 -568735*x^5 +363954*x^4 +579769*x^3 +106565*x^2 +3135*x +2) / (x-1)^13.
a(n) = (n^12 -12*n^11 -39*n^10 +950*n^9 -815*n^8 -29672*n^7 +69499*n^6 +452518*n^5 -1454446*n^4 -3319216*n^3 +12944320*n^2 +9142512*n -41687280) / 720 for n>=6, a(5) = 2, a(n) = 0 for n<5.
Showing 1-5 of 5 results.

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