A233333 -id:A233333 - cp's OEIS frontend

A233329 Expansion of (1+4x+x^2)/((1+x)^2(1-x)^5).

1, 7, 21, 51, 102, 186, 310, 490, 735, 1065, 1491, 2037, 2716, 3556, 4572, 5796, 7245, 8955, 10945, 13255, 15906, 18942, 22386, 26286, 30667, 35581, 41055, 47145, 53880, 61320, 69496, 78472, 88281, 98991, 110637, 123291, 136990, 151810, 167790, 185010, 203511

Offset: 0

L. Edson Jeffery, Jan 06 2014

Sequence is related to enumeration of coronas in A233332.

Conjecture: sequence gives column 1 of A233331 (up to an offset).

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A076454 (bisection, up to an offset), A233330-A233333.

[(2*n^4+20*n^3+68*n^2+90*n+37-2*n*(-1)^n-5*(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2014

A233329:=n->(2*n^4+20*n^3+68*n^2+90*n+37-2*n*(-1)^n-5*(-1)^n)/32: seq(A233329(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2014

CoefficientList[Series[(1 + 4*x + x^2)/((1 + x)^2*(1 - x)^5), {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 17 2014 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,7,21,51,102,186,310},50] (* Harvey P. Dale, Jul 05 2019 *)

a(n) = (2*n^4+20*n^3+68*n^2+(90-2*(-1)^n)*n)\/32+1 \\ Charles R Greathouse IV, Oct 28 2014

G.f.: (1+4*x+x^2)/((1+x)^2*(1-x)^5).
a(n) = (2*(n^4+10*n^3+34*n^2+(45+(-1)^(n+1))*n)+37+5*(-1)^(n+1))/32.
a(n) = sum_{j=1..n+1} ( sum_{i=1..j+1} floor(i*j/2) ). - Wesley Ivan Hurt, Nov 17 2014

A233332 Irregular array read by rows: A(n,k) = number of first coronas of a fixed rhombus r_{n,k} with characteristics of n-fold rotational symmetry in the Euclidean plane, n>=2, 1<=k<=floor(n/2), as explained below.

Original entry on oeis.org

1, 83, 1452, 1770, 15587, 19863, 131980, 169716, 182884, 971013, 1245461, 1389317, 6508358, 8289158, 9408838, 9790598, 40813063, 51522567, 58997063, 62834759, 243405576, 304396296, 349949576, 378076936, 387585288

Offset: 2

L. Edson Jeffery, Dec 12 2013

Row index n begins with 2, column index k begins with 1.
Let R_n be the set of floor(n/2) rhombi in which the k-th rhombus r_{n,k} has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (n-k)*Pi/n), n>=2, 1<=k<=floor(n/2). Let T be any tiling of the plane by tiles of R_n. For any tile t in T, let C_m(t) denote the m-th corona of t, m>=0. Equivalently, starting with any tile r in R_n fixed in the plane, we can compose a corona C_m(r) of r of any order m by tessellation using tiles of R_n. This leads to the following problem in the theory of tiles and its reduction for symmetry which seem to have not been addressed before in the literature. (See [Jeffery] for details and definitions.)
Problem 1: For r_{n,k} in R_n fixed in the plane, in how many ways can r_{n,k} be extended to an m-th corona of r_{n,k} using tiles of R_n?
Problem 2: From Problem 1, in how many ways can r_{n,k} be so extended if isometries different from the identity are not counted?
The problems are very difficult, and here A233332 gives a solution only for the simplest case m=1.

			Array begins:
...........1
..........83
........1452.........1770
.......15587........19863
......131980.......169716.......182884
......971013......1245461......1389317
.....6508358......8289158......9408838......9790598
....40813063.....51522567.....58997063.....62834759
...243405576....304396296....349949576....378076936....387585288
..1395618313...1728983049...1990082057...2169422089...2260674313
..7751398922...9515886602..10947167754..12001065994..12646026762..12863117322
.41932226571..51033062411..58616206347..64480008203..68473230347..70495047691

Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.

Dirk Frettlöh, Glossary of tiling terms, Tilings Encyclopedia.
L. E. Jeffery, Constructing A233332.
Eric W. Weisstein, Corona, from MathWorld.
Eric W. Weisstein, Tiling, from MathWorld.

Cf. A233329-A233333.

maxn := 10; t[n_, m_, i_] := 1 + Mod[Floor[m/(n - 1)^(4 - i - 1)], n - 1]; e[n_, k_, m_, i_] := -t[n, m, i] + (-1)^(i)*k + Mod[i, 2]*n + t[n, m, Mod[i - 1, 4]]; a[n_, k_] := Sum[Product[If[e[n, k, m, i] >= 0, 1, 0], {i, 0, 3}]*Product[SeriesCoefficient[Series[(1 - x)/(1 - 2*x + x^n), {x, 0, 2*n - k - 2}], e[n, k, m, i]], {i, 0, 3}], {m, 0, (n - 1)^4 - 1}]; Grid[Table[a[n, k], {n, 2, maxn}, {k, Floor[n/2]}]] (* L. Edson Jeffery, Jul 22 2014 *)

The Jeffery PDF contains an algorithm for constructing this array.

Conjecture: For all n and for all k, A(n,k) == n-2 (mod 2^(n-3)).

A233330 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), for n>=2, 1<=k<=floor(n/2), explained below.

Original entry on oeis.org

1, 16, 64, 81, 169, 225, 361, 484, 527, 676, 900, 1018, 1156, 1521, 1752, 1831, 1849, 2401, 2789, 2989, 2809, 3600, 4195, 4564, 4689, 4096, 5184, 6042, 6634, 6936, 5776, 7225, 8408, 9283, 9820, 10001, 7921, 9801, 11377, 12601, 13437, 13861, 10609, 12996, 15039

Offset: 2

L. Edson Jeffery, Dec 29 2013

For definitions and more details, see the PDF by L. E. Jeffery.
Let n be an integer, n >= 2, and let k in {1,...,floor(n/2)}. Let R_n be the set of floor(n/2) rhombi in which the k-th rhombus r_{n,k} in R_n has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (n-k)*Pi/n). Let T be any tiling of the plane. For any tile t in T, let C_m(t) denote the m-th corona of t, m>=0. Equivalently, starting with any r in R_n fixed in the plane, we can compose a corona C_m(r) of r of any order m by tessellation using tiles of R_n. For any r in R_n fixed in the plane, a disjoint union of r with four tiles t_1,t_2,t_3,t_4 in R_n is called a "candidate." If no tiles overlap in a candidate, then that candidate is called an "r-core;" otherwise that candidate is rejected (since no corona of r can be constructed from it). For each r_{n,k} in R_n, and for m>0, every r_{n,k}-core can be extended to an m-th corona of r_{n,k} using tiles of R_n in a number of ways. For the case m=1, the array A233332 gives the number of ways that this can be done for each n and k. In the theory of tiles the general problem of counting these coronas and its reduction for symmetry seem to have not been addressed before in the literature.
The present array A233330 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332. The array A233331 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332 when isometries different from the identity are not counted.

			Array begins: {1}; {16}; {64, 81}; {169, 225}; {361, 484, 527}; ...

Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.

Dirk Frettlöh, Glossary of tiling-theoretic terms, Tilings Encyclopedia.
L. E. Jeffery, Algorithm for constructing A233332.
Eric W. Weisstein, Corona, from MathWorld.
Eric W. Weisstein, Tiling, from MathWorld.

Cf. A233329-A233333.

maxn := 13; t[n_, m_, i_] := 1 + Mod[Floor[m/(n - 1)^(4 - i)], n - 1]; e[n_, k_, m_, 1] :=  -t[n, m, 1] + k + t[n, m, 4]; e[n_, k_, m_, 2] := -t[n, m, 2] - k + n + t[n, m, 1]; e[n_, k_, m_, 3] :=  -t[n, m, 3] + k + t[n, m, 2]; e[n_, k_, m_, 4] :=  -t[n, m, 4] - k + n + t[n, m, 3]; cores[n_, k_] := Sum[Product[If[e[n, k, m, i] >= 0, 1, 0], {i, 1, 4}], {m, 0, (n - 1)^4 - 1}]; Flatten[Table[cores[n, k], {n, 2, maxn}, {k, 1, Floor[n/2]}]] (* L. Edson Jeffery, Jul 22 2014 *)

The Jeffery PDF contains an algorithm for constructing this array.

Conjecture: The entries of column k satisfy sum{h=0..5} ((-1)^h*binomial(5,h)*A(n+5-h,k)) = 0, for appropriate initial conditions.

A233331 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), reduced for symmetry, for n>=2, 1<=k<=floor(n/2), explained below.

Original entry on oeis.org

1, 7, 21, 15, 51, 66, 102, 136, 75, 186, 244, 274, 310, 406, 462, 246, 490, 631, 729, 780, 735, 939, 1086, 1183, 610, 1065, 1341, 1557, 1707, 1783, 1491, 1861, 2155, 2380, 2511, 1286, 2037, 2512, 2908, 3217, 3427, 3534, 2716, 3322, 3831, 4250, 4548, 4738, 2404

Offset: 2

L. Edson Jeffery, Jan 06 2014

For definitions and more details, see the PDF by L. E. Jeffery.
Let n be an integer, n >= 2, and let k in {1,...,floor(n/2)}. Let R_n be the set of floor(n/2) rhombi in which the k-th rhombus r_{n,k} in R_n has interior angles about its vertices (or corners) given by the pair (k*Pi/n, (n-k)*Pi/n). Let T be any tiling of the plane. For any tile t in T, let C_m(t) denote the m-th corona of t, m>=0. Equivalently, starting with any r in R_n fixed in the plane, we can compose a corona C_m(r) of r of any order m by tessellation using tiles of R_n. For any r in R_n fixed in the plane, a disjoint union of r with four tiles t_1,t_2,t_3,t_4 in R_n is called a "candidate." If no tiles overlap in a candidate, then that candidate is called an "r-core;" otherwise that candidate is rejected (since no corona of r can be constructed from it). For each r_{n,k} in R_n, and for m>0, every r_{n,k}-core can be extended to an m-th corona of r_{n,k} using tiles of R_n in a number of ways. For the case m=1, the array A233332 gives the number of ways that this can be done for each n and k. In the theory of tiles the general problem of counting these coronas and its reduction for symmetry seem to have not been addressed before in the literature.
The array A233330 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332. The present array A233331 gives the number of r_{n,k}-cores associated with the coronas enumerated in A233332 when isometries different from the identity are not counted.

			Array begins: {1}; {7}; {21, 15}; {51, 66}; {102, 136, 75}; ...

Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995, p. 145.

Dirk Frettlöh, Glossary of tiling-theoretic terms, Tilings Encyclopedia.
L. E. Jeffery, Algorithm for constructing A233332.
Eric W. Weisstein, Corona, from MathWorld.
Eric W. Weisstein, Tiling, from MathWorld.

Cf. A076454, A233329-A233333.

Conjecture: G.f. for column 1 is F_1(x) = x^2*(1+4x+x^2)/((1+x)^2*(1-x)^5).
Conjecture: Column 1 = A233329, up to an offset.
Conjecture: Column k has generating function of the form F_k(x) = G_k(x)/((1+x)^2*(1-x)^5), where G_k(x) is a polynomial in x.
Conjecture: A233331(2*M,1) = A076454(M), M>=1.

cp's OEIS Frontend

A233329 Expansion of (1+4*x+x^2)/((1+x)^2*(1-x)^5).

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A233332 Irregular array read by rows: A(n,k) = number of first coronas of a fixed rhombus r_{n,k} with characteristics of n-fold rotational symmetry in the Euclidean plane, n>=2, 1<=k<=floor(n/2), as explained below.

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A233330 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), for n>=2, 1<=k<=floor(n/2), explained below.

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A233331 Irregular array read by rows: T(n,k) = number of r_{n,k}-cores associated with A233332(n,k), reduced for symmetry, for n>=2, 1<=k<=floor(n/2), explained below.

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A233329 Expansion of (1+4x+x^2)/((1+x)^2(1-x)^5).