A204557 -id:A204557 - cp's OEIS frontend

A079326 a(n) = the largest number m such that if m monominoes are removed from an n X n square then an L-tromino must remain.

1, 2, 7, 9, 17, 20, 31, 35, 49, 54, 71, 77, 97, 104, 127, 135, 161, 170, 199, 209, 241, 252, 287, 299, 337, 350, 391, 405, 449, 464, 511, 527, 577, 594, 647, 665, 721, 740, 799, 819, 881, 902, 967, 989, 1057, 1080, 1151, 1175, 1249, 1274, 1351, 1377, 1457

Offset: 2

Mambetov Timur (timur_teufel(AT)mail.ru), Feb 13 2003

			a(3)=2 because if a middle row of 3 monominoes are removed from the 3 X 3, no L remains.

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

Frobenius number for k successive numbers: A028387 (k=2), this sequence (k=3), A138984 (k=4), A138985 (k=5), A138986 (k=6), A138987 (k=7), A138988 (k=8).

Cf. A093353, A104519.

Table[FrobeniusNumber[{a, a + 1, a + 2}], {a, 2, 54}] (* Zak Seidov, Jan 08 2015 *)

a(n) = (n^2)/2 - 1 (n even), (n^2-n)/2 - 1 (n odd).
a(n) = A204557(n-1) / (n-1). - Reinhard Zumkeller, Jan 18 2012
From Bruno Berselli, Jan 18 2011: (Start)
G.f.: x^2*(1+x+3*x^2-x^4)/((1+x)^2*(1-x)^3).
a(n) = n*(2*n+(-1)^n-1)/4 - 1.
a(n) = A105638(-n+2). (End)

Edited by Don Reble, May 28 2007

A045975 Take the first odd integer and multiple of 1, the next 2 even integers and multiples of 2, the next 3 odd integers and multiples of 3, the next 4 even integers and multiples of 4, ...

Original entry on oeis.org

1, 2, 4, 9, 15, 21, 24, 28, 32, 36, 45, 55, 65, 75, 85, 90, 96, 102, 108, 114, 120, 133, 147, 161, 175, 189, 203, 217, 224, 232, 240, 248, 256, 264, 272, 280, 297, 315, 333, 351, 369, 387, 405, 423, 441, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 561, 583, 605, 627, 649, 671, 693

Offset: 1

Fang-kuo Huang (gsyps(AT)ms17.hinet.net)

A generalized Connell sequence.

			Triangle begins:
    1;
    2,   4;
    9,  15,  21;
   24,  28,  32,  36;
   45,  55,  65,  75,  85;
   90,  96, 102, 108, 114, 120;
  133, 147, 161, 175, 189, 203, 217;
  ...

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A001614, A033291.

Seen as a triangle read by rows: cf. A204558 (row sums), A005917 (central terms), A204556 (left edge), A204557 (right edge).

a045975 n k = a045975_tabl !! (n-1) !! (k-1)
a045975_row n = a045975_tabl !! (n-1)
a045975_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys | even k    = take k ms
        | otherwise = take k $ filter odd ms
     ms = filter ((== 0) . (`mod` k)) xs
-- Reinhard Zumkeller, Jan 18 2012

first[n_?EvenQ] := (n - 1)*n^2/2; first[n_?OddQ] := n*(n^2 - 2n + 3)/2; row[n_] := (ro = {first[n]}; next = first[n] + n; While[ Length[ro] < n, If[Mod[next , 2] == Mod[n, 2], AppendTo[ro, next]]; next = next + n]; ro); Flatten[ Table[row[n], {n, 1, 11}]](* Jean-François Alcover, Jun 08 2012 *)

More terms from James Sellers

Keyword tabl added by Reinhard Zumkeller, Jan 18 2012

A204556 Left edge of the triangle A045975.

Original entry on oeis.org

1, 2, 9, 24, 45, 90, 133, 224, 297, 450, 561, 792, 949, 1274, 1485, 1920, 2193, 2754, 3097, 3800, 4221, 5082, 5589, 6624, 7225, 8450, 9153, 10584, 11397, 13050, 13981, 15872, 16929, 19074, 20265, 22680, 24013, 26714, 28197, 31200, 32841, 36162, 37969, 41624

Offset: 1

Reinhard Zumkeller, Jan 18 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Haskell
```
a204556 = head . a045975_row
```

[n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4: n in [1..50]]; // G. C. Greubel, Jun 15 2018

Table[n*(2*n^2 - 3*n + (-1)^n*(n - 3) + 3)/4, {n, 1, 50}] (* G. C. Greubel, Jun 15 2018 *)

Vec(x*(1+x+4*x^2+12*x^3+3*x^4+3*x^5)/((1+x)^3*(x-1)^4) + O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

for(n=1, 50, print1(n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4, ", ")) \\ G. C. Greubel, Jun 15 2018

a(n) = A045975(n,1);
a(n) = A031940(n-1) * n for n > 1;
a(n) = A204557(n) - A045895(n).
G.f.: x*(1+x+4*x^2+12*x^3+3*x^4+3*x^5) / ((1+x)^3*(x-1)^4). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jan 28 2016: (Start)
a(n) = n*(2*n^2-3*n+(-1)^n*(n-3)+3)/4.
a(n) = (n^3-n^2)/2 for n even.
a(n) = (n^3-2*n^2+3*n)/2 for n odd.
(End)

A045895 Period length of pairs (a,b) where a has period 2n-2 and b has period n.

Original entry on oeis.org

0, 2, 12, 12, 40, 30, 84, 56, 144, 90, 220, 132, 312, 182, 420, 240, 544, 306, 684, 380, 840, 462, 1012, 552, 1200, 650, 1404, 756, 1624, 870, 1860, 992, 2112, 1122, 2380, 1260, 2664, 1406, 2964, 1560

Offset: 1

Ralf W. Grosse-Kunstleve

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]

Cf. A045896, A204556, A204557.

[Lcm(2*n-2, n): n in [1..50]]; // G. C. Greubel, Jun 15 2018

Mathematica
```
Table[ LCM[ 2*n-2, n ], {n, 40} ]
```

for(n=1, 50, print1(lcm(2*n-2, n), ", ")) \\ G. C. Greubel, Jun 15 2018

a(n) = A204557(n) - A204556(n). - Reinhard Zumkeller, Jan 18 2012
From Amiram Eldar, Sep 14 2022: (Start)
a(n) = n*(n-1) for n even.
a(n) = 2*n*(n-1) for n odd.
a(n) = lcm(2*n-2, n).
a(n) = 2*A045896(n-2).
Sum_{n>=2} 1/a(n) = (log(2)+1)/2. (End)

cp's OEIS Frontend

A079326 a(n) = the largest number m such that if m monominoes are removed from an n X n square then an L-tromino must remain.

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A045975 Take the first odd integer and multiple of 1, the next 2 even integers and multiples of 2, the next 3 odd integers and multiples of 3, the next 4 even integers and multiples of 4, ...

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A204556 Left edge of the triangle A045975.

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Magma

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PARI

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A045895 Period length of pairs (a,b) where a has period 2n-2 and b has period n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula