Steve Butler - cp's OEIS frontend

A260575 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

4, 114, 1492, 13992, 109538, 769632, 5050616, 31702275, 193204684, 1154354559, 6805263818, 39756392269, 230829718918, 1334626765852, 7694795830792, 44279453377166, 254475676808510, 1461211112505546, 8385454709982584, 48102877501302765, 275868835046218560

Offset: 3

Steve Butler, Jul 29 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 3 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that the throws are between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 3..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=3 of A269742.

Cf. A260582, A260583, A260584, A260585, A260727.

Rest[Rest[Rest[CoefficientList[Series[-(625 x^12 - 2206 x^11 + 4397 x^10 - 6648 x^9 + 7058 x^8 - 4674 x^7 + 1748 x^6 - 300 x^5 + 2 x^4 + 4 x^3)/(2600 x^13 - 21100 x^12 + 77590 x^11 - 171025 x^10 + 251874 x^9 - 261466 x^8 + 196626 x^7 - 108337 x^6 + 43682 x^5 - 12713 x^4 + 2592 x^3 - 350 x^2 + 28 x - 1), {x, 0, 33}], x]]]] (* Vincenzo Librandi, Jul 30 2015 *)

G.f.: -(625*x^12 - 2206*x^11 + 4397*x^10 - 6648*x^9 + 7058*x^8 - 4674*x^7 + 1748*x^6 - 300*x^5 + 2*x^4 + 4*x^3)/(2600*x^13 - 21100*x^12 + 77590*x^11 - 171025*x^10 + 251874*x^9 - 261466*x^8 + 196626*x^7 - 108337*x^6 + 43682*x^5 - 12713*x^4 + 2592*x^3 - 350*x^2 + 28*x - 1).

A255305 The largest number that cannot be written as a sum of squarefree numbers that are the product of n primes.

Original entry on oeis.org

6, 23, 299, 3439, 51637, 894211

Offset: 1

Steve Butler, Feb 20 2015

			a(1)=6 because each number that is 7 or larger can be written as the sum of distinct primes.
a(2)=23 because the squarefree numbers less than or equal to 23 that are the product of two primes are {6, 10, 14, 15, 21, 22} and there is no way to add some subset of these together to get 23; on the other hand every number greater than 23 can be so represented, i.e., 24=10+14, 25=10+15, 26=26, 27=6+21, 28=6+22, 30=6+10+14, and so on.

A254152 Number of independent sets in the generalized Aztec diamond E(L_9,L_{2n-1}).

Original entry on oeis.org

1, 32, 1351, 62501, 2976416, 142999897, 6888568813, 332097693792, 16014193762579, 772279980131297, 37243762479698928, 1796118644459454733, 86619824190256627593, 4177339899819872607008, 201457018240598757372431, 9715496740529686006497709, 468541027322402116068858304

Offset: 0

Steve Butler, Jan 26 2015

nonn

E(L_9,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=9, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Independent Vertex Set
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
Index entries for linear recurrences with constant coefficients, signature (74,-1450,10672,-34214,50814,-34671,9772,-936).

Row n=5 of A331406.

Cf. A254124, A254125, A254126, A254150, A254151.

Vec((1 - 42*x + 433*x^2 - 1745*x^3 + 3002*x^4 - 2275*x^5 + 700*x^6 - 72*x^7)/(1 - 74*x + 1450*x^2 - 10672*x^3 + 34214*x^4 - 50814*x^5 + 34671*x^6 - 9772*x^7 + 936*x^8) + O(x^20)) \\ Andrew Howroyd, Jan 16 2020

G.f.: (1 - 42*x + 433*x^2 - 1745*x^3 + 3002*x^4 - 2275*x^5 + 700*x^6 - 72*x^7)/(1 - 74*x + 1450*x^2 - 10672*x^3 + 34214*x^4 - 50814*x^5 + 34671*x^6 - 9772*x^7 + 936*x^8). - Andrew Howroyd, Jan 16 2020

a(10)-a(11) corrected and a(12) and beyond from Andrew Howroyd, Jan 15 2020

A254151 Number of independent sets in the generalized Aztec diamond E(L_7,L_{2n-1}).

Original entry on oeis.org

1, 16, 314, 6556, 139344, 2976416, 63663808, 1362242592, 29151501760, 623849225024, 13350628082560, 285709494797952, 6114316283697408, 130849237522680064, 2800235203724240384, 59926350645878761984, 1282452098548524184576, 27445078313878468469760

Offset: 0

Steve Butler, Jan 26 2015

nonn

E(L_7,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=7, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Independent Vertex Set
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
Index entries for linear recurrences with constant coefficients, signature (30,-202,396,-248,32).

Row n=4 of A331406.

Cf. A254124, A254125, A254126, A254150, A254152.

LinearRecurrence[{30,-202,396,-248,32},{1,16,314,6556,139344},20] (* Harvey P. Dale, May 31 2024 *)

Vec((1 - 14*x + 36*x^2 - 28*x^3 + 4*x^4)/(1 - 30*x + 202*x^2 - 396*x^3 + 248*x^4 - 32*x^5) + O(x^20)) \\ Andrew Howroyd, Jan 16 2020

Empirical g.f.: -(4*x^4-28*x^3+36*x^2-14*x+1) / (32*x^5-248*x^4+396*x^3-202*x^2+30*x-1). - Colin Barker, Jan 26 2015

The above g.f. is correct. See A331406 for bounds on the order of the recurrence. - Andrew Howroyd, Jan 16 2020

Terms a(12) and beyond from Andrew Howroyd, Jan 15 2020

A254150 Number of independent sets in the generalized Aztec diamond E(L_5,L_{2n-1}).

Original entry on oeis.org

1, 8, 73, 689, 6556, 62501, 596113, 5686112, 54239137, 517383521, 4935293524, 47077513469, 449070034657, 4283656560248, 40861585458553, 389776618229969, 3718059650555596, 35466384896440661, 338312070235103473, 3227141903559443792, 30783545081553045457

Offset: 0

Steve Butler, Jan 26 2015

E(L_5,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=5, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Independent Vertex Set
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
Index entries for linear recurrences with constant coefficients, signature (12,-24,5).

Row n=3 of A331406.

Cf. A254124, A254125, A254126, A254151, A254152.

Vec((1 - 4*x + x^2)/(1 - 12*x + 24*x^2 - 5*x^3) + O(x^25)) \\ Andrew Howroyd, Jan 16 2020

Empirical g.f.: -(x^2-4*x+1) / (5*x^3-24*x^2+12*x-1). - Colin Barker, Jan 26 2015

The above g.f. is correct. See A331406 for bounds on the order of the recurrence. - Andrew Howroyd, Jan 16 2020

Terms a(12) and beyond from Andrew Howroyd, Jan 15 2020

A254127 The number of tilings of an n X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are of size (1 X i) or (i X 1) with 1<=i<=n.

Original entry on oeis.org

1, 1, 7, 257, 50128, 50796983, 264719566561, 7063448084710944, 963204439792722969647, 670733745303300958404439297, 2384351527902618144856749327661056, 43263422878945294225852497665519673400479, 4006622856873663241294794301627790673728956619649

Offset: 0

Steve Butler, Jan 25 2015

nonn

Let R(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y|<=n+1, and let two squares be independent if they do not share a common edge. Then a(n) is the number of ways to pick a set of independent cell(s) in R(n). (Note R(n) is also known as the Aztec diamond.)

			a(2)=7 for the following 7 tilings:
   _ _   _ _   _ _   _ _   _ _   _ _   _ _
  |_|_| |_ _| |_|_| | |_| |_| | |_ _| | | |
  |_|_| |_|_| |_ _| |_|_| |_|_| |_ _| |_|_|

Liang Kai, Table of n, a(n) for n = 0..27 (terms 0..15 from Steve Butler)
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.

Cf. A052961, A254124, A254125, A254126.

Main diagonal of A254414.

def matrix_entry(L1, L2, n):
    tally=0
    for i in range(n-1):
        if (not i in L1) and (not i in L2) and (not i+1 in L1) and (not i+1 in L2):
            tally+=1
    return 2^tally
def a(n):
    index_set={}
    counter=0
    for C in Combinations(n):
        index_set[counter]=C
        counter+=1
    current_v=[0]*counter
    current_v[0]=1
    for t in range(n):
        new_v=[0]*counter
        for i in range(counter):
            for j in range(counter):
                new_v[i]+=current_v[j]*matrix_entry(index_set[I], index_set[j], n)
        current_v=new_v
    return current_v[0]
for n in range(0, 10):
    print(a(n), end=', ')

a(0)=1 prepended by Alois P. Heinz, Jan 30 2015

A254126 The number of tilings of a 5 X n rectangle using integer length rectangles with at least one length of size 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1, 4 X 1, 5 X 1.

Original entry on oeis.org

1, 16, 533, 22873, 1064576, 50796983, 2441987149, 117656540512, 5672528575545, 273541357254277, 13191518965300160, 636171495829068099, 30680036092304563369, 1479579136691648516016, 71354395560692698401005, 3441147782121276015384833, 165953315828852845775456128

Offset: 0

Steve Butler, Jan 25 2015

Let G_n be the graph with vertices {(a,b) : 1<=a<=9, 1<=b<=2n-1, a+b odd} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1. Then a(n) is the number of independent sets in G_n.

Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.

Cf. A052961, A254124, A254125, A254127.

Column k=5 of A254414.

G.f: (1 - 58*x + 799*x^2 - 4041*x^3 + 8286*x^4 - 7357*x^5 + 2660*x^6 - 312*x^7)/(1 - 74*x + 1450*x^2 - 10672*x^3 + 34214*x^4 - 50814*x^5 + 34671*x^6 - 9772*x^7 + 936*x^8).

A254125 The number of tilings of a 4 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1, 4 X 1.

Original entry on oeis.org

1, 8, 124, 2408, 50128, 1064576, 22734496, 486248000, 10404289216, 222647030144, 4764694602112, 101966374503680, 2182126445631232, 46698521255409152, 999370260391863808, 21386993399983588352, 457691719382960757760, 9794818132582234683392

Offset: 0

Steve Butler, Jan 25 2015

Let G_n be the graph with vertices {(a,b) : 1<=a<=7, 1<=b<=2n-1, a+b odd} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1. Then a(n) is the number of independent sets in G_n.

Colin Barker, Table of n, a(n) for n = 0..750
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
Index entries for linear recurrences with constant coefficients, signature (30,-202,396,-248,32).

Cf. A052961, A254124, A254126, A254127.

Column k=4 of A254414.

Vec((1-22*x+86*x^2-92*x^3+16*x^4)/(1-30*x+202*x^2-396*x^3 +248*x^4-32*x^5) + O(x^30)) \\ Michel Marcus, Jan 26 2015

G.f.: (1 - 22x + 86x^2 - 92x^3 + 16x^4)/(1 - 30x + 202x^2 - 396x^3 + 248x^4 - 32x^5).

a(n) = 30*a(n-1) - 202*a(n-2) + 396*a(n-3) - 248*a(n-4) + 32*a(n-5) for n>4. - Colin Barker, Jun 07 2020

A254124 The number of tilings of a 3 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1.

Original entry on oeis.org

1, 4, 29, 257, 2408, 22873, 217969, 2078716, 19827701, 189133073, 1804125632, 17209452337, 164160078241, 1565914710964, 14937181915469, 142485030313697, 1359157571347928, 12964936038223753, 123671875897903249, 1179699833714208556, 11253097663211943461

Offset: 0

Steve Butler, Jan 25 2015

Let G_n be the graph with vertices {(a,b) : 1<=a<=5, 1<=b<=2n-1, a+b odd} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1. Then a(n) is the number of independent sets in G_n.

Colin Barker, Table of n, a(n) for n = 0..1000
Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
Index entries for linear recurrences with constant coefficients, signature (12,-24,5).

Cf. A052961, A254125, A254126, A254127.

Column k=3 of A254414.

Vec((1-8*x+5*x^2)/(1-12*x+24*x^2-5*x^3) + O(x^30)) \\ Michel Marcus, Jan 26 2015

G.f.: (1 - 8*x + 5*x^2)/(1 - 12*x + 24*x^2 - 5*x^3).

a(n) = 12*a(n-1) - 24*a(n-2) + 5*a(n-3) for n > 2. - Colin Barker, Jun 07 2020

A253953 Numbers that require three steps to collapse to a single digit in base 4 (written in base 4).

Original entry on oeis.org

223, 1213, 2023, 2122, 2203, 2221, 3133, 11113, 12103, 13033, 20023, 20203, 20221, 21202, 22003, 22021, 22201, 22333, 30313, 31033, 31132, 103033, 110113, 111103, 113032, 121003, 200023, 200203, 200221, 202003, 202021

Offset: 1

Steve Butler, Jan 20 2015

One step consists of taking the number in base 4 and inserting some plus signs between the digits with no restrictions and adding the resulting numbers together in base 4. The numbers given here cannot be taken to a single digit in one or two steps. It is known that three steps always suffice to get to a single digit, and that there are infinitely many numbers that require three steps.

			As an example a(1)=223 (in base 4).  There are then three ways to insert plus signs in the first step:
2+23   22+3   2+2+3
This gives the numbers (in base 4) as 31, 31, and 13 respectively.  In the second step we have one of the following two:
3+1   1+3
In both cases this gives the number (in base 4) of 10.  Finally in the third step we have the following:
1+0
Which gives 1, a single digit, and we cannot get to a single digit in one or two steps.  (Note, the single digit that we reduce to is independent of the sequence of steps taken.)

Steve Butler, Table of n, a(n) for n = 1..637
S. Butler, R. Graham and R. Stong, Partition and sum is fast, arXiv:1501.04067 [math.HO], 2014.

Cf. A253057, A253058, A253952.

cp's OEIS Frontend

User: Steve Butler

A260575 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

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A255305 The largest number that cannot be written as a sum of squarefree numbers that are the product of n primes.

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A254152 Number of independent sets in the generalized Aztec diamond E(L_9,L_{2n-1}).

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A254151 Number of independent sets in the generalized Aztec diamond E(L_7,L_{2n-1}).

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A254150 Number of independent sets in the generalized Aztec diamond E(L_5,L_{2n-1}).

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A254127 The number of tilings of an n X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are of size (1 X i) or (i X 1) with 1<=i<=n.

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A254126 The number of tilings of a 5 X n rectangle using integer length rectangles with at least one length of size 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1, 4 X 1, 5 X 1.

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A254125 The number of tilings of a 4 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1, 4 X 1.

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