A076454 -id:A076454 - cp's OEIS frontend

A076459 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.

1, 57, 390, 1510, 4335, 10311, 21532, 40860, 72045, 119845, 190146, 290082, 428155, 614355, 860280, 1179256, 1586457, 2099025, 2736190, 3519390, 4472391, 5621407, 6995220, 8625300, 10545925, 12794301, 15410682, 18438490, 21924435, 25918635, 30474736, 35650032

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002417, A076454-A076458.

[n*(n+1)*(2*n^3+2*n^2-2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/2*n*(n+1)*(2*n^3+2*n^2-2*n-1),n=1..35);

CoefficientList[Series[(1 + 51 x + 63 x^2 + 5 x^3)/(1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(2*n^3+2*n^2-2*n-1)/2.

G.f.: x*(1+51*x+63*x^2+5*x^3)/(1-x)^6.

More terms from Vincenzo Librandi, Dec 30 2013

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

Original entry on oeis.org

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

A076455 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

Original entry on oeis.org

5, 57, 246, 710, 1635, 3255, 5852, 9756, 15345, 23045, 33330, 46722, 63791, 85155, 111480, 143480, 181917, 227601, 281390, 344190, 416955, 500687, 596436, 705300, 828425, 967005, 1122282, 1295546, 1488135, 1701435, 1936880, 2195952, 2480181

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A076454-A076459.

[n*(n+1)*(4*n^2+2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple
```
seq(1/2*n*(n+1)*(4*n^2+2*n-1),n=1..40);
```

CoefficientList[Series[(5 + 32 x + 11 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

G.f.: x*(5 + 32*x + 11*x^2)/(1 - x)^5.

a(n) = n*(n + 1)*(4*n^2 + 2*n - 1)/2.

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.

A076458 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly five ways.

Original entry on oeis.org

17, 165, 678, 1910, 4335, 8547, 15260, 25308, 39645, 59345, 85602, 119730, 163163, 217455, 284280, 365432, 462825, 578493, 714590, 873390, 1057287, 1268795, 1510548, 1785300, 2095925, 2445417, 2836890, 3273578, 3758835, 4296135, 4889072, 5541360

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A076454-A076459.

[n*(n+1)*(10*n^2+8*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/2*n*(n+1)*(10*n^2+8*n-1),n=1..40);

CoefficientList[Series[(17 + 80 x + 23 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(10*n^2+8*n-1)/2.

G.f.: x*(17+80*x+23*x^2)/(1-x)^5.

More terms from Vincenzo Librandi, Dec 30 2013

A076456 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly three ways.

Original entry on oeis.org

9, 93, 390, 1110, 2535, 5019, 8988, 14940, 23445, 35145, 50754, 71058, 96915, 129255, 169080, 217464, 275553, 344565, 425790, 520590, 630399, 756723, 901140, 1065300, 1250925, 1459809, 1693818, 1954890, 2245035, 2566335, 2920944, 3311088, 3739065, 4207245

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A076454-A076459.

[n*(n+1)*(6*n^2+4*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple
```
seq(1/2*n*(n+1)*(6*n^2+4*n-1),n=1..40);
```

CoefficientList[Series[3 (3 + 16 x + 5 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(6*n^2+4*n-1)/2.

G.f.: 3*x*(3+16*x+5*x^2)/(1-x)^5.

A076457 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly four ways.

Original entry on oeis.org

13, 129, 534, 1510, 3435, 6783, 12124, 20124, 31545, 47245, 68178, 95394, 130039, 173355, 226680, 291448, 369189, 461529, 570190, 696990, 843843, 1012759, 1205844, 1425300, 1673425, 1952613, 2265354, 2614234, 3001935, 3431235, 3905008, 4426224, 4997949, 5623345

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A076454-A076459.

[n*(n+1)*(8*n^2+6*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple
```
seq(1/2*n*(n+1)*(8*n^2+6*n-1),n=1..40);
```

CoefficientList[Series[(13 + 64 x + 19 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(8*n^2+6*n-1)/2.

G.f.: x*(13+64*x+19*x^2)/(1-x)^5.

A144066 T(n, k) is the number of order-preserving partial transformations (of an n-element chain) of height k (height(alpha) = |Im(alpha)|); triangle T read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 15, 1, 1, 60, 102, 28, 1, 1, 155, 490, 310, 45, 1, 1, 378, 1935, 2220, 735, 66, 1, 1, 889, 6741, 12285, 7315, 1491, 91, 1

Offset: 0

Abdullahi Umar, Sep 09 2008

T(n, k) is also the number of elements in the Green's J-classes of the monoid of order-preserving partial transformations (of an n-element chain). Sum of rows of T(n, k) is A123164.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   1;
  1,   6,    1;
  1,  21,   15,     1;
  1,  60,  102,    28,    1;
  1, 155,  490,   310,   45,    1;
  1, 378, 1935,  2220,  735,   66,  1;
  1, 889, 6741, 12285, 7315, 1491, 91, 1;
  ...
T(2,1) = 6 because there are exactly 6 order-preserving partial transformations (on a 2-element chain) of height 1, namely: (1)->(1), (1)->(2), (2)->(1), (2)->(2), (1,2)->(1,1), and (1,2)->(2,2) -- the mappings are coordinate-wise.

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8.

Cf. A000384, A066524, A076454, A112857, A123164.

T(n,k) = C(n,k)*A112857(n,k) for 0 <= k <= n.
C(n-1,k-1)*T(n,k) = 2((n-k+1)/(n-k))*T(n-1,k) + C(n,k)*T(n-1,k-1). [This is wrong.]
From Petros Hadjicostas, Feb 14 2021: (Start)
T(n,k) = 2*n*T(n-1,k)/(n-k) + n*T(n-1,k-1)/k for 1 <= k <= n-1 with T(n,0) = T(n,n) = 1 for n >= 0.
T(n,1) = n*(2^n - 1) = A066524(n) for n >= 1.
T(n,n-1) = n*(2*n - 1) = A000384(n) for n >= 1.
T(n,n-2) = A076454(n-1) for n >= 2. (End)

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A076459 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.

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

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A076455 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly two ways.

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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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A076458 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly five ways.

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Keywords

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Magma

Maple

Mathematica

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A076456 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly three ways.

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Magma

Maple

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A076457 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly four ways.

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A076459 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.

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

A076455 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

A076458 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly five ways.

A076456 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly three ways.

A076457 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly four ways.