A074148 -id:A074148 - cp's OEIS frontend

A231064 Coins left after packing X patterns into an n X n array of coins.

4, 4, 11, 15, 16, 24, 24, 31, 35, 41, 44, 49, 51, 55, 56, 64, 69, 71, 75, 76, 84, 89, 91, 95, 96, 104, 109, 111, 115, 116, 124, 129, 131, 135, 136, 144, 149, 151, 155, 156, 164, 169, 171, 175, 176, 184, 189, 191, 195, 196, 204, 209, 211, 215, 216, 224, 229, 231, 235, 236, 244, 249, 251

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the total number of coins left (the coins out side X patterns) after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and voids left is A231065.
a(n) is also the total number of coins left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: x^2*(5*x^15 -5*x^14 -5*x^12 +5*x^11 -5*x^10 +5*x^9 +4*x^5 +x^4 +4*x^3 +7*x^2 +4) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

A231065 Voids left after packing X patterns into an of n X n array of coins.

Original entry on oeis.org

1, 0, 5, 8, 9, 16, 17, 24, 29, 36, 41, 48, 53, 60, 65, 76, 85, 92, 101, 108, 121, 132, 141, 152, 161, 176, 189, 200, 213, 224, 241, 256, 269, 284, 297, 316, 333, 348, 365, 380, 401, 420, 437, 456, 473, 496, 517, 536, 557, 576, 601, 624, 645, 668, 689, 716, 741, 764, 789, 812, 841, 868

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the total number of voids (spaces among coins) left after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and coins left is A231064.
a(n) is also the total number of voids left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: x^2*(4*x^16 -8*x^15 +4*x^14 -4*x^13 +8*x^12 -8*x^11 +8*x^10 -4*x^9 +4*x^6 -5*x^5 +2*x^4 +2*x^3 -6*x^2 +2*x -1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 2, 4, 5, 6, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 8, 1, 2, 3, 5, 6, 7, 8, 9, 1, 2, 3, 4, 6, 7, 8, 9, 10, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13

Offset: 1

Antti Karttunen, Feb 10 2014

This is transpose of A237447, please see comments there.

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 1 1 1 1 1 1 1
  3 3 2 2 2 2 2 2 2
  4 4 4 3 3 3 3 3 3
  5 5 5 5 4 4 4 4 4
  6 6 6 6 6 5 5 5 5
  7 7 7 7 7 7 6 6 6
  8 8 8 8 8 8 8 7 7
  9 9 9 9 9 9 9 9 8

Antti Karttunen, Table of first 144 antidiagonals of array, flattened

Transpose: A237447.
The leftmost column and the topmost row: A000027. Second row: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
Cf. A237265, A000330, A002024, A002260, A004736, A000124, A010054.

(define (A237448 n) (cond ((= 1 (A010054 (- n 1))) (A002024 n)) ((< (A004736 n) (A002260 n)) (A002260 n)) (else (- (A002260 n) 1))))

As a one-dimensional sequence:
If A010054(n-1) = 1 [that is, if n is in A000124], then a(n) = A002024(n), otherwise, if A004736(n) < A002260(n), a(n) = A002260(n), and if A004736(n) >= A002260(n), a(n) = A002260(n)-1.
Equivalently, as a square array T:
When col < row, T(row,col) = row, for 1 < row <= col, T(row,col) = row-1, and for the first row T(1,col) = col = A000027(col).
Can be computed also as a transposed version of the infinite limit of the finite square arrays in sequence A237265: T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(col-1)) + (row-1)).

A082735 Product of n-th group of terms in A074147.

Original entry on oeis.org

1, 8, 105, 5760, 328185, 42577920, 5568833025, 1300252262400, 304513870485825, 111644006842368000, 40992233865440682825, 21695920874860629196800, 11492457771692770753505625, 8291067715225260172247040000

Offset: 1

Amarnath Murthy, Apr 14 2003

nonn

Cf. A074147, A074148, A074149, A082736.

A074148 := proc(n) (2*n^2+4*n+(-1)^n-1)/4 ; end: A082735 := proc(n) doublefactorial(A074148(n))/doublefactorial(A074148(n-2)) ; end: seq(A082735(n),n=1..20) ; # R. J. Mathar, Jul 17 2007

a(1) = 1, a(2n) = product of next 2n even numbers. a(2n+1) = product of next 2n+1 odd numbers.

a(n)=A006882[A074148(n)]/A006882[A074148(n-2)]. a(2n-1)=A062031(n). a(2n)=A062030(n). # R. J. Mathar, Jul 17 2007

Corrected and extended by R. J. Mathar, Jul 17 2007

A082736 LCM of n-th group of terms in A074147.

Original entry on oeis.org

1, 4, 105, 120, 109395, 55440, 1856277675, 42325920, 966710699955, 7210803600, 303646176781042095, 43790142876480, 2432266195067253069525, 6338767304469600, 12793596869123737224933375, 659267412349963697280

Offset: 1

Amarnath Murthy, Apr 14 2003

nonn

Cf. A074147, A074148, A074149, A082735.

A061925 := proc(n) ceil(n^2/2)+1 ; end: A082736 := proc(n) seq(A061925(n-1)+2*k,k=0..n-1) ; ilcm(%) ; end: seq(A082736(n),n=1..20) ; # R. J. Mathar, Jul 17 2007

a(1) = 1, a(2n) = LCM of next 2n even numbers. a(2n+1) = LCM of next 2n+1 odd numbers.

More terms from R. J. Mathar, Jul 17 2007

A109225 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0 < k < n: T(n,k) = T(n-1,k-1) + 1 - T(n-1,k-1) mod 2 + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 6, 1, 1, 5, 12, 13, 8, 1, 1, 6, 17, 26, 21, 10, 1, 1, 7, 24, 43, 48, 31, 12, 1, 1, 8, 31, 68, 91, 80, 43, 14, 1, 1, 9, 40, 99, 160, 171, 124, 57, 16, 1, 1, 10, 49, 140, 259, 332, 295, 182, 73, 18, 1, 1, 11, 60, 189, 400, 591, 628, 477

Offset: 1

Reinhard Zumkeller, Jun 23 2005

Row sums give A084172, the generalized Jacobsthal numbers;
T(n,1) = n for n>0;
T(n,2) = A074148(n) for n>1.

Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318.

A236773 a(n) = n + floor( n^2/2 + n^3/3 ).

Original entry on oeis.org

0, 1, 6, 16, 33, 59, 96, 145, 210, 292, 393, 515, 660, 829, 1026, 1252, 1509, 1799, 2124, 2485, 2886, 3328, 3813, 4343, 4920, 5545, 6222, 6952, 7737, 8579, 9480, 10441, 11466, 12556, 13713, 14939, 16236, 17605, 19050, 20572, 22173, 23855, 25620, 27469

Offset: 0

Bruno Berselli, Feb 07 2014

This sequence follows A074148 and A042965, A236771.
The prime terms are 59, 829, 14939, 35759, 93719, 132409, 155219, 290399, 414179, 487463, ... .
If a(k) is prime then k == 1, 5, 7 or 11 (mod 12).
Third differences: 1, 2, 2, 2, 1, 4 repeated (unsigned terms of A181982).
Fourth differences: 1, 0, 0, -1, 3, -3 repeated (see A131193).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1,-3,3,-1).

Cf. A074148: n+floor(n^2/2).
Cf. A042965: n+floor(1/2+n/3); A236771: n+floor(n/2+n^2/3).
Cf. A236772: floor(sum(i=1..n, n^i/i)).

Magma
```
[n+Floor(n^2/2+n^3/3): n in [0..50]];
```

I:=[0,1,6,16,33,59,96,145,210]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)+Self(n-6)-3*Self(n-7)+3*Self(n-8)-Self(n-9): n in [1..50]]; // Vincenzo Librandi, Feb 08 2014

seq(n+floor(n^2/2+n^3/3),n=0..43); # Paolo P. Lava, Aug 24 2018

Table[n + Floor[n^2/2 + n^3/3], {n, 0, 50}]
CoefficientList[Series[x (1 + 3 x + x^2 + 2 x^3 + 2 x^4 + 2 x^5 + x^7)/((1 + x) (1 - x + x^2) (1 + x + x^2) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 08 2014 *)

vector(60, n, n--; n+floor(n^2/2 +n^3/3)) \\ G. C. Greubel, Aug 12 2018

G.f.: x*(1+3*x+x^2+2*x^3+2*x^4+2*x^5+x^7) / ((1+x)*(1-x+x^2)*(1+x+x^2)*(1-x)^4).
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-6) -3*a(n-7) +3*a(n-8) -a(n-9).
Also, for h>=0:
a(6h)   = 6*h*( 12*h^2 + 3*h + 1 ),
a(6h+1) = 72*h^3 + 54*h^2 + 18*h + 1,
a(6h+2) = 6*( 4*h + 1 )*( 3*h^2 + 3*h + 1 ),
a(6h+3) = 2*( 36*h^3 + 63*h^2 + 39*h + 8 ),
a(6h+4) = 3*( 24*h^3 + 54*h^2 + 42*h + 11 ),
a(6h+5) = 72*h^3 + 198*h^2 + 186*h + 59.

A333520 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length 2*(n-1+k) connecting opposite corners in the n X n grid graph (0 <= k <= floor((n-1)^2/2), n >= 1).

Original entry on oeis.org

1, 2, 6, 4, 2, 20, 36, 48, 48, 32, 70, 224, 510, 956, 1586, 2224, 2106, 732, 104, 252, 1200, 3904, 10560, 25828, 58712, 121868, 217436, 300380, 280776, 170384, 61336, 10180, 924, 5940, 25186, 88084, 277706, 821480, 2309402, 6140040, 15130410, 33339900, 62692432, 96096244, 116826664, 110195700, 78154858, 39287872, 12396758, 1879252, 111712

Offset: 1

Seiichi Manyama, Mar 29 2020

			T(3,1) = 4;
   S--*      S--*--*   S  *--*   S
      |            |   |  |  |   |
   *--*         *--*   *--*  *   *  *--*
   |            |            |   |  |  |
   *--*--E      *--E         E   *--*  E
Triangle starts:
=======================================================
n\k|   0     1     2      3      4 ...      8 ...   12
---|---------------------------------------------------
1  |   1;
2  |   2;
3  |   6,    4,    2;
4  |  20,   36,   48,    48,    32;
5  |  70,  224,  510,   956,  1586, ... , 104;
6  | 252, 1200, 3904, 10560, ................. , 10180;

Seiichi Manyama, Rows n = 1..9, flattened

Row sums give A007764.
T(n,0) gives A000984(n-1).
T(n,1) gives A257888(n).
T(n,floor((n-1)^2/2)) gives A121788(n-1).
T(2*n-1,2*(n-1)^2) gives A001184(n-1).
Cf. A074148, A302337, A329633.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333520(n):
    if n == 1: return [1]
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return [paths.len(2 * (n - 1 + k)).len() for k in range((n - 1) ** 2 // 2 + 1)]
print([i for n in range(1, 8) for i in A333520(n)])

A109857 Next 2n - 1 odd numbers in decreasing order followed by next 2n even numbers in decreasing order.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 12, 10, 8, 6, 17, 15, 13, 11, 9, 24, 22, 20, 18, 16, 14, 31, 29, 27, 25, 23, 21, 19, 40, 38, 36, 34, 32, 30, 28, 26, 49, 47, 45, 43, 41, 39, 37, 35, 33, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 71, 69, 67, 65, 63, 61, 59, 57, 55, 53, 51, 84, 82, 80, 78, 76, 74

Offset: 1

Amarnath Murthy, Jul 08 2005

This sequence is a permutation of the positive integers. - Werner Schulte, Jul 29 2023

			 1;
 4,  2;
 7,  5,  3;
12, 10,  8,  6;
17, 15, 13, 11,  9;
24, 22, 20, 18, 16, 14;
31, 29, 27, 25, 23, 21, 19;
40, 38, 36, 34, 32, 30, 28, 26;

Cf. A074147 (row reversed), A074149 (row sums), A074148 (column 1), A001844, A061925 (main diagonal).

T(n,k)=n*(n+1)/2+floor(n/2)-2*(k-1) \\ Werner Schulte, Jul 29 2023

From Werner Schulte, Jul 29 2023: (Start)
T(n, k) = n*(n+1)/2 + floor(n/2) - 2*(k-1)  for 1 <= k <= n.
T(n, n) = (n^2-3*n+4)/2 + floor(n/2)  for n > 0.
T(2*n-1, n) = n^2 + (n-1)^2 = A001844(n-1)  for n > 0. (End)

More terms from Joshua Zucker, May 05 2006

A120413 Largest even number strictly less than n^2.

Original entry on oeis.org

0, 2, 8, 14, 24, 34, 48, 62, 80, 98, 120, 142, 168, 194, 224, 254, 288, 322, 360, 398, 440, 482, 528, 574, 624, 674, 728, 782, 840, 898, 960, 1022, 1088, 1154, 1224, 1294, 1368, 1442, 1520, 1598, 1680, 1762, 1848, 1934, 2024, 2114, 2208, 2302, 2400, 2498, 2600

Offset: 1

Henry Bottomley, Jul 06 2006

Longest non-intersecting route from (0, 0) to (n - 1, n - 1) staying in an (n - 1) X (n - 1) box (shortest route is length 2n A005843).

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Maple
```
seq(2*ceil(n^2/2)-2,n=1..50);
```

Flatten[Table[{(2n - 1)^2 - 1, 4n^2 - 2}, {n, 25}]] (* Alonso del Arte, Apr 15 2016 *)

lista(nn) = for(n=0, nn, print1((-1+(-1)^n+4*n+2*n^2)/2, ", ")); \\ Altug Alkan, Apr 15 2016

a(n) = 2*ceiling[n^2/2] - 2 = 2*A074148(n) = A085046(n) - 1.
From Colin Barker, Jul 29 2012: (Start)
a(n) = (-1 + (-1)^n + 4*n + 2*n^2)/2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: 2*x*(1 + 2*x - x^2)/((1-x)^3*(1+x)). (End)
a(n) = n^2 - 2 for even n; a(n) = n^2 - 1 for odd n. -Dennis P. Walsh, Apr 15 2016

Offset corrected by N. J. A. Sloane, Apr 15 2016

cp's OEIS Frontend

A231064 Coins left after packing X patterns into an n X n array of coins.

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A231065 Voids left after packing X patterns into an of n X n array of coins.

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A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

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A082735 Product of n-th group of terms in A074147.

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A082736 LCM of n-th group of terms in A074147.

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A109225 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0 < k < n: T(n,k) = T(n-1,k-1) + 1 - T(n-1,k-1) mod 2 + T(n-1,k).

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A236773 a(n) = n + floor( n^2/2 + n^3/3 ).

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A333520 Triangle read by rows: T(n,k) is the number of self-avoiding paths of length 2*(n-1+k) connecting opposite corners in the n X n grid graph (0 <= k <= floor((n-1)^2/2), n >= 1).

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A109857 Next 2*n - 1 odd numbers in decreasing order followed by next 2*n even numbers in decreasing order.

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A109857 Next 2n - 1 odd numbers in decreasing order followed by next 2n even numbers in decreasing order.