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.

A127864 Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 5, 11, 33, 87, 241, 655, 1793, 4895, 13377, 36543, 99841, 272767, 745217, 2035967, 5562369, 15196671, 41518081, 113429503, 309895169, 846649343, 2313089025, 6319476735, 17265131521, 47169216511, 128868696065, 352075825151, 961889042433, 2627929735167
Offset: 0

Views

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

Keywords

Comments

The signed version of this sequence appears as A077917.

Examples

			a(2) = 5 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile.

Links

Crossrefs

Cf. A077917, A127865, A127866, A127867, A127868, A127869, A127870, A127871, A127872.
Column k=2 of A220054. - Alois P. Heinz, Dec 03 2012

Programs

  • Magma
    I:=[1,1,5]; [n le 3 select I[n] else Self(n-1) + 4*Self(n-2) + 2*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2022
  • Mathematica
    CoefficientList[Series[1/(1-x-4*x^2-2*x^3), {x,0,30}], x]
  • SageMath
    A028860 = BinaryRecurrenceSequence(2,2,-1,1)
def A127864(n): return A028860(n+2) + (-1)^n
[A127864(n) for n in range(51)] # G. C. Greubel, Dec 08 2022

Formula

a(n) = a(n-1) + 4*a(n-2) + 2*a(n-3).
a(n) = (-1)^n + (1/sqrt(3)) * ((1+sqrt(3))^n - (1-sqrt(3))^n).
G.f.: 1/(1 - x - 4*x^2 - 2*x^3).
a(n) = A028860(n+2) + (-1)^n. - R. J. Mathar, Oct 29 2010
E.g.f.: exp(-x) + (2/sqrt(3))*exp(x)*sinh(sqrt(3)*x). - G. C. Greubel, Dec 08 2022
From Greg Dresden, Nov 10 2024: (Start)
a(n) = 1 + 4*a(n-2) + 6*Sum_{i=0..n-3} a(i) for n>1.
a(2*n) = a(n)^2 + 4*a(n-1)^2 + 4*a(n-1)*a(n-2) for n>1. (End)

A127867 Number of tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 11, 39, 195, 849, 3895, 17511, 79339, 358397, 1620843, 7326991, 33127155, 149766353, 677103839, 3061202815, 13839823275, 62570318397, 282882722979, 1278922980071, 5782057329219, 26140890761969, 118183916056327, 534313772133687, 2415651952691819
Offset: 0

Views

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

Keywords

Examples

			a(2) = 11 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles.

Links

Crossrefs

Cf. A127864, A127865, A127866, A127868, A127869, A127870.
Column k=3 of A220054. - Alois P. Heinz, Dec 03 2012

Programs

  • Mathematica
    Table[Coefficient[Normal[Series[(1 - x)^2/(1 - 3x - 7x^2 + x^3 - 2x^4), {x, 0, 30}]], x, n], {n, 0, 30}]

Formula

G.f.: (1-x)^2/(1-3x-7x^2+x^3-2x^4).

A127870 Number of tilings of a 4 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 33, 195, 2023, 16839, 151817, 1328849, 11758369, 103628653, 914646205, 8068452381, 71189251649, 628067760289, 5541284098945, 48888866203241, 431331449340441, 3805499681885145, 33574725778806817, 296219181642118401, 2613448287490035073
Offset: 0

Views

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

Keywords

Examples

			a(2) = 33 because the 4x2 board can be tiled in one way with only square tiles, in 12 ways using one L-tile and 5 square tiles and in 20 ways with 2 L-tiles and 2 square tiles.

Links

Crossrefs

Cf. A127864, A127865, A127866, A127867, A127868, A127869.
Column k=4 of A220054. - Alois P. Heinz, Dec 03 2012

Programs

  • Mathematica
    Table[Coefficient[Normal[Series[(1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5)/(1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9), {x, 0, 30}]], x, n], {n, 0, 30}]

Formula

G.f.: (1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5) / (1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9).

A220061 Number of tilings of an n X n square using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 5, 39, 2023, 249651, 128938297, 207866584389, 1208344842789831, 23649239068131551559, 1609120545126107661426575, 375082120094104660413783094451, 301522432794951154854984388046484015, 833441700776362178606942848178200903068675, 7931715551857283775957120938092133944383839378911
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Examples

			a(2) = 5, because there are 5 tilings of a 2 X 2 square using right trominoes and 1 X 1 tiles:
  ._._.   ._._.   .___.   .___.   ._._.
  |_|_|   | |_|   | ._|   |_. |   |_| |
  |_|_|   |___|   |_|_|   |_|_|   |___|

Links

Crossrefs

Main diagonal of A220054.
Cf. A233807.

A220055 Number of tilings of a 5 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 87, 849, 16839, 249651, 4134881, 65564239, 1057354073, 16939662301, 272086395449, 4365892578855, 70082433262847, 1124809701807527, 18054055051423891, 289774657566172859, 4651038841674376909, 74651407535212480809, 1198192596525147061411
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

Crossrefs

Column k=5 of A220054.

Programs

  • Maple
    gf:= (20*x^12 +88*x^11 -162*x^10 +26*x^9 +506*x^8 +453*x^7 -605*x^6 -300*x^5 -12*x^4 +47*x^3 +63*x^2 +6*x -1) /
(76*x^14 -152*x^13 -212*x^12 -64*x^11 +1180*x^10 -1738*x^9 -3069*x^8 +308*x^7 +5229*x^6 -756*x^5 -1701*x^4 +144*x^3 +143*x^2 +7*x -1):
a:= n-> coeff (series (gf, x, n+1), x, n):
seq(a(n), n=0..30);

Formula

G.f.: see Maple program.

A220056 Number of tilings of a 6 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 241, 3895, 151817, 4134881, 128938297, 3814023955, 115136505933, 3448441154503, 103598912114381, 3108676107844557, 93324146271938457, 2801146229279170843, 84082823432914559453, 2523871643346500063787, 75758559732310254661669, 2274020749613202850958405
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

Crossrefs

Column k=6 of A220054.

Programs

  • Maple
    gf:= -(128*x^27 -2784*x^26 +11984*x^25 -8672*x^24 -7128*x^23 -34144*x^22 -125640*x^21 +760596*x^20 -718466*x^19 -174758*x^18 +2760675*x^17 -10918043*x^16 +15110507*x^15 -1068879*x^14 -13774618*x^13 +9742272*x^12 +298116*x^11 -1535703*x^10 -168489*x^9 +78558*x^8 +130467*x^7 +2413*x^6 -18124*x^5 -3982*x^4 +368*x^3 +295*x^2 +12*x -1) /
(3072*x^30 -49664*x^29 -102080*x^28 +441952*x^27 +988592*x^26 -641184*x^25 -6795568*x^24 -3529824*x^23 +39110100*x^22 -11698064*x^21 -93989108*x^20 +89810738*x^19 +53775823*x^18 -80969102*x^17 -68370580*x^16 +104188994*x^15 +46195161*x^14 -118554222*x^13 +26901436*x^12 +45228345*x^11 -25090652*x^10 -4432537*x^9 +5299837*x^8 -37118*x^7 -530937*x^6 +10786*x^5 +29450*x^4 -607*x^3 -523*x^2 -13*x +1):
a:= n-> coeff (series (gf, x, n+1), x, n):
seq(a(n), n=0..30);

Formula

G.f. see Maple program.

A220057 Number of tilings of a 7 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 655, 17511, 1328849, 65564239, 3814023955, 207866584389, 11621270470141, 643234164533111, 35743258143250665, 1983110281248178907, 110094091718725808219, 6110504997318928433203, 339180718810796793005395, 18826477870730711026769043
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

  • Alois P. Heinz, Table of n, a(n) for n = 0..300
  • Index entries for linear recurrences with constant coefficients, signature (31, 1688, -11130, -459918, 3166506, 40112677, -345723568, -1333668560, 18915874110, -20132638419, -340706701381, 1131454106758, 1800169591293, -14547686180172, 10887029502746, 77700424321275, -170596686592062, -132867339076434, 814120637502953, -408118945300204, -1874303491278113, 2371447568852003, 2065885358158550, -4510347925558337, -1768155209701568, 5626569971126797, 3093435933277591, -8190510139279992, -2502197734475072, 9273148424569143, -314261646620241, -5569775487698427, 516945589065495, 1847002437609681, 588598730671956, -502432621307848, -485326816160748, 41192473052820, 191502873476778, 53306213357242, -44398553938820, -28149773265720, 14071419586760, 5403011349240, -6450496694256, -6085229826624, -1539089548288, -55320676736, 84819354752, 9551531264, 6799316992, 3599682560, 134983680, -12386304, 1048576).

Crossrefs

Column k=7 of A220054.

Programs

  • Maple
    gf:= (196608*x^53 +15716352*x^52 +82890752*x^51 -81387520*x^50 +2420729856*x^49 -5502464896*x^48 +11135136384*x^47 -2587529280*x^46 -242120057280*x^45 -391566462048*x^44 -677756970360*x^43 -26891699024*x^42 +3419049690968*x^41 -898216784632*x^40 -8630220265938*x^39 +11892744055744*x^38 +12674156157904*x^37
-23326232274170*x^36 -8460520836030*x^35 -49526668612724*x^34 -52615082821909*x^33 +385796302646490*x^32 -89063359404187*x^31 -689833337938642*x^30 +276301559560831*x^29 +939553216589439*x^28 -751421966953043*x^27 -387235565854614*x^26 +367601964623911*x^25 +391200153596741*x^24 -321438046442330*x^23 -254149045627282*x^22
+327959797230961*x^21 -30793906263310*x^20 -105485377717340*x^19 +46988439121753*x^18 +10650397716161*x^17 -12878278811627*x^16 +1803973124746*x^15 +1212527797540*x^14 -447484692550*x^13 +13047687869*x^12 +14482637535*x^11 -3330884126*x^10 +1108885391*x^9 -182374621*x^8 -34669281*x^7 +8700029*x^6 +605086*x^5 -151416*x^4 -6648*x^3 +1064*x^2 +30*x -1) /
(1048576*x^55 -12386304*x^54 +134983680*x^53 +3599682560*x^52 +6799316992*x^51 +9551531264*x^50 +84819354752*x^49 -55320676736*x^48 -1539089548288*x^47 -6085229826624*x^46 -6450496694256*x^45 +5403011349240*x^44 +14071419586760*x^43 -28149773265720*x^42 -44398553938820*x^41 +53306213357242*x^40 +191502873476778*x^39
+41192473052820*x^38 -485326816160748*x^37 -502432621307848*x^36 +588598730671956*x^35 +1847002437609681*x^34 +516945589065495*x^33 -5569775487698427*x^32 -314261646620241*x^31 +9273148424569143*x^30 -2502197734475072*x^29 -8190510139279992*x^28 +3093435933277591*x^27 +5626569971126797*x^26 -1768155209701568*x^25 -4510347925558337*x^24 +2065885358158550*x^23 +2371447568852003*x^22 -1874303491278113*x^21
-408118945300204*x^20 +814120637502953*x^19 -132867339076434*x^18 -170596686592062*x^17 +77700424321275*x^16 +10887029502746*x^15 -14547686180172*x^14 +1800169591293*x^13 +1131454106758*x^12 -340706701381*x^11 -20132638419*x^10 +18915874110*x^9 -1333668560*x^8 -345723568*x^7 +40112677*x^6 +3166506*x^5 -459918*x^4 -11130*x^3 +1688*x^2 +31*x -1):
a:= n-> coeff (series (gf, x, n+1), x, n):
seq(a(n), n=0..30);

Formula

G.f.: see Maple program.

A220058 Number of tilings of an 8 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 1793, 79339, 11758369, 1057354073, 115136505933, 11621270470141, 1208344842789831, 124179182077944123, 12820466607209726137, 1321211811196491541315, 136254474646105474794407, 14047759147701072483029529, 1448476467705491364792194617
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

Crossrefs

Column k=8 of A220054.

A220059 Number of tilings of a 9 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 4895, 358397, 103628653, 16939662301, 3448441154503, 643234164533111, 124179182077944123, 23649239068131551559, 4528155459015015278497, 865169296748334990311763, 165442366105217743521821785, 31626233120646483498726015897, 6046521630766477909485058575551
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

Crossrefs

Column k=9 of A220054.

A220060 Number of tilings of a 10 X n rectangle using right trominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 1, 13377, 1620843, 914646205, 272086395449, 103598912114381, 35743258143250665, 12820466607209726137, 4528155459015015278497, 1609120545126107661426575, 570433530146121798730683231, 202412926394304755588236748481, 71796961505258542492662333092533
Offset: 0

Views

Author

Alois P. Heinz, Dec 03 2012

Keywords

Links

Crossrefs

Column k=10 of A220054.
Showing 1-10 of 10 results.

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