A073028 -id:A073028 - cp's OEIS frontend

A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 6, 16, 1, 1, 4, 12, 37, 140, 1, 1, 6, 24, 105, 454, 1987, 1, 1, 10, 40, 250, 1566, 9856, 62266, 1, 1, 15, 80, 726, 5670, 47394, 406168, 3899340, 1, 1, 21, 160, 1824, 18738, 223696, 2916492, 38322758, 508317004

Offset: 0

Pontus von Brömssen, Apr 10 2023

			Triangle begins:
  n\k| 0  1  2  3   4    5     6      7       8
  ---+-----------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  2  4
  4  | 1  1  3  6  16
  5  | 1  1  4 12  37  140
  6  | 1  1  6 24 105  454  1987
  7  | 1  1 10 40 250 1566  9856  62266
  8  | 1  1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+---+---+   +       +---+---+   +---+       +---+   +---+---+       +
  |   |   |       |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |       |   |       |   |   |   |       |   |   |   |       |   |   |
  +       +---+---+   +       +---+---+   +       +---+---+   +       +---+---+
  |       |   |   |   |       |   |   |   |       |   |   |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |       |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |
  +---+       +---+   +---+---+---+---+   +---+---+---+---+   +       +---+---+
  |   |       |   |   |       |   |   |   |   |   |   |   |   |       |   |   |
  +---+---+---+---+   +       +---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |       |   |   |   |   |   |   |   |   |   |       |
  +---+---+---+---+   +---+       +---+   +---+---+---+---+   +---+---+       +
  |   |   |   |   |   |   |       |   |   |       |       |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.

Pontus von Brömssen, Table of n, a(n) for n = 0..90 (rows 0..12)

Main diagonal: A362143.
Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5).
Cf. A219924, A224697, A361216 (rectangular pieces).

A060065 Smallest mode of the sequence { C(n-k,k), k=0..n/2 }.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20

Offset: 0

N. J. A. Sloane, Apr 10 2001

nonn

S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.

Cf. A073028.

A060065 := n->ceil( (5*n-3-sqrt(5*n^2+10*n+9))/10);

a(n)=(5*n-sqrtint(5*n^2+10*n+9)+6)\10 \\ Charles R Greathouse IV, Sep 22 2016

A277282 Max coefficient in n-th Lucas polynomial.

Original entry on oeis.org

2, 1, 2, 3, 4, 5, 9, 14, 20, 30, 50, 77, 112, 182, 294, 450, 672, 1122, 1782, 2717, 4290, 7007, 11011, 16744, 27456, 44200, 68952, 107406, 176358, 281010, 436050, 700910, 1136960, 1797818, 2778446, 4576264, 7354710, 11560835, 18349630, 29910465, 47720400

Offset: 0

Vladimir Reshetnikov, Oct 08 2016

nonn

			For n = 6, L_6(x) = x^6 + 6*x^4 + 9*x^2 + 2, so a(6) = 9.

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000
Vaclav Kotesovec, Plot of a(n) / (goldenratio^n / sqrt(n)) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Polynomial.

Cf. A114525, A073028.

Table[Max@CoefficientList[LucasL[n, x], x], {n, 0, 40}]

A349862 a(n) is the maximum value of binomial(n-2*k,k) with 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 6, 10, 15, 21, 28, 36, 56, 84, 120, 165, 220, 330, 495, 715, 1001, 1365, 2002, 3003, 4368, 6188, 8568, 12376, 18564, 27132, 38760, 54264, 77520, 116280, 170544, 245157, 346104, 490314, 735471, 1081575, 1562275, 2220075, 3124550, 4686825, 6906900, 10015005, 14307150

Offset: 0

Enrique Navarrete, Dec 02 2021

nonn

			a(7) = 5 since row n=7 of A102547 is 1, 5, 3 and the maximum value is 5.
a(20) = 495 since row n=20 of A102547 is 1, 18, 120, 364, 495, 252, 28. The maximum value of 495 occurs at k = 4.

Maximum row values of A102547.

Cf. A073028.

a[n_]:=Max[Table[Binomial[n-2k,k],{k,0,Floor[n/3]}]]; Array[a,49,0] (* Stefano Spezia, Dec 06 2021 *)

a(n) = vecmax(vector(n\3+1, k, k--; binomial(n-2*k, k))); \\ Michel Marcus, Dec 06 2021

from math import comb
def A349862(n): return max(comb(n-2*k,k) for k in range(n//3+1)) # Chai Wah Wu, Jan 04 2022

A362260 Maximum over 0 <= k <= n/2 of the number of permutations of two symbols occurring k and n-2*k times, respectively, where a permutation and its reversal are counted only once.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 9, 12, 19, 28, 44, 66, 110, 170, 255, 396, 651, 1001, 1519, 2520, 4032, 6216, 9752, 15912, 25236, 38760, 63090, 101850, 160050, 248710, 408760, 653752, 1021735, 1634776, 2656511, 4218786, 6562556, 10737090, 17299646, 27313650, 43249115

Offset: 0

Pontus von Brömssen, Apr 15 2023

nonn

Also, a(n) is the maximum number of ways in which a set of integer-sided squares can tile an n X 2 rectangle, up to rotations and reflections.

			For n = 8, the maximum a(8) = 9 is obtained for k = 2. The corresponding permutations of 2 2's and 4 1's are 221111, 212111, 211211, 211121, 211112, 122111, 121211, 121121, and 112211.

Robert Israel, Table of n, a(n) for n = 0..4771

Row maxima of A102541.
Second column of A362258.
Cf. A001224, A073028, A361224 (rectangular pieces).

f:= proc(n) local k, v, m,w;
  m:= 0:
  for k from 0 to n/2 do
    v:= binomial(n-k,k);
    if n:: even and k::even then w:= binomial((n-k)/2,k/2)
    elif (n-k)::odd then w:=binomial((n-k-1)/2, floor(k/2))
    else w:= 0
    fi;
    m:= max(m,(v+w)/2);
  od;
  m
end proc:
map(f, [$0..50]); # Robert Israel, Oct 25 2023

a(n) >= A073028(n)/2.

A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 4, 1, 3, 2, 0, 1, 5, 1, 6, 3, 1, 1, 1, 6, 1, 10, 4, 4, 3, 0, 1, 7, 1, 15, 5, 10, 6, 1, 1, 1, 8, 1, 21, 6, 20, 10, 5, 4, 0, 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1, 1, 10, 1, 36, 8, 56, 21, 35, 20, 6, 5, 0

Offset: 0

Peter Luschny, Jul 21 2024

Member of the family of Lucas-Fibonacci polynomials.

			Triangle starts:
  [ 0]  1;
  [ 1]  1,  0;
  [ 2]  1,  1,  1;
  [ 3]  1,  2,  1,  0;
  [ 4]  1,  3,  1,  1,  1;
  [ 5]  1,  4,  1,  3,  2,  0;
  [ 6]  1,  5,  1,  6,  3,  1,  1;
  [ 7]  1,  6,  1, 10,  4,  4,  3,  0;
  [ 8]  1,  7,  1, 15,  5, 10,  6,  1,  1;
  [ 9]  1,  8,  1, 21,  6, 20, 10,  5,  4,  0;
  [10]  1,  9,  1, 28,  7, 35, 15, 15, 10,  1, 1;

Cf. A374441.

Cf. A000032 (Lucas), A001611 (even sums, Fibonacci + 1), A000071 (odd sums, Fibonacci - 1), A001911 (alternating sums, Fibonacci(n+3) - 2), A025560 (row lcm), A073028 (row max), A117671 & A025174 (central terms), A057979 (subdiagonal), A000217 (column 3).

T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0
elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9);
T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -
binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):

T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling((k + even(k))/2), floor(k/2)) if k > 0, T(n, 0) = 1, where even(k) = 1 if k is even, otherwise 0.

Columns with odd index agree with the odd indexed columns of A374441.

cp's OEIS Frontend

A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n.

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A060065 Smallest mode of the sequence { C(n-k,k), k=0..n/2 }.

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A277282 Max coefficient in n-th Lucas polynomial.

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A349862 a(n) is the maximum value of binomial(n-2*k,k) with 0 <= k <= floor(n/3).

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A362260 Maximum over 0 <= k <= n/2 of the number of permutations of two symbols occurring k and n-2*k times, respectively, where a permutation and its reversal are counted only once.

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A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

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