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A309038 Irregular triangle T read by rows: given a square made of n^2 squares of unit area, T(n, k) is the longest perimeter that can be obtained by removing k of n^2 squares such that the modified figure remains connected and without holes (n >= 0 and 0 <= k <= n^2). 4
0, 4, 0, 8, 8, 8, 4, 0, 12, 14, 16, 18, 20, 16, 12, 8, 4, 0, 16, 18, 20, 22, 24, 26, 28, 28, 28, 26, 24, 20, 16, 12, 8, 4, 0, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 42, 40, 38, 36, 32, 28, 24, 20, 16, 12, 8, 4, 0, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 56, 56, 56, 56, 56, 56, 52, 48, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

All the terms of this sequence are even numbers (A005843).

In the figure, two unit area squares can be connected in a corner or sideways.

Every n-th row of the triangle is made of almost four successive finite arithmetic progressions characterized respectively by the following common differences: 2, 0, -2, -4. If we let h_i(n) be the number of first differences of i-th progression (i = 1,2,3,4), we have that 4*n + 2*h_1(n) - 2*h_3(n) - 4*h_4(n) = 0 and h_1(n) + h_2(n) + h_3(n) + h_4(n) = n^2.

LINKS

Table of n, a(n) for n=0..97.

FORMULA

T(n, 0) = A008586(n).

T(n, k) = 2*(2*n + k) for 0 <= k <= h_1(n), T(n, k) = 2*(2*n + h_1(n)) for h_1(n) <= k <= h_1(n) + h_2(n), T(n, k) = 2*(2*(n + h_1(n)) + h_2(n) - k) for h_1(n) + h_2(n) <= k <= h_1(n) + h_2(n) + h_3(n), T(n, k) = 2*(2*(n + h_2(n) - k) + 3*h_1(n) + h_3(n)), for h_1(n) + h_2(n) + h_3(n) <= k <= n^2, where h_4(n) = n for 0 <= n <= 2 and h_4(n) = (1/8)*(-29 + 12*n + 2*n^2 - 3*cos(n*Pi) - 12*sin(n*Pi/2)) for n > 2, h_3(n) = 2*delta(n, 4) - 4*delta(n, 1) + 1 - cos(n*Pi) + 2*sin(n*Pi/2) and delta(i, j) is the Kronecker delta, h_2(n) = 2*(delta(n, 2) + delta(n, 4)) for 0 <= n <= 4 and h_2(n) = (1/8)*(71 - 20*n + 2*n^2 + 25*cos(n*Pi) + 4*sin(n*Pi/2)) for n > 4, h_1(n) = n^2 - (h_1(n) + h_2(n) + h_3(n)).

EXAMPLE

The triangle T(n, k) begins:

---+-------------------------------------------------------------------

n\k|  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16

---+-------------------------------------------------------------------

0  |  0

1  |  4   0

2  |  8   8   8   4   0

3  | 12  14  16  18  20  16  12   8   4   0

4  | 16  18  20  22  24  26  28  28  28  26  24  20  16  12   8   4   0

...

Here are the values of h_i's for the first seven rows of the triangle T:

n    h_1(n)   h_2(n)   h_3(n)   h_4(n)

--------------------------------------

0         0        0        0        0

1         0        0        0        1

2         0        2        0        2

3         4        0        0        5

4         6        2        2        6

5        12        0        4        9

6        16        6        0       14

...

Illustrations for n = 4, k=0..15 by Andrew Howroyd, Sep 01 2019: (Start)

   __.__.__.__    __.__.__.__    __.__.__.__    __.__.__.__

  |           |  |           |  |           |  |           |

  |           |  |__         |  |__         |  |__       __|

  |           |   __|        |   __|__      |   __|__   |__

  |__.__.__.__|  |__.__.__.__|  |__|  |__.__|  |__|  |__.__|

       (16)           (18)          (20)           (22)

   __.__.__.__    __.  .__.__    __    __.__    __    __.__

  |           |  |  |__|     |  |  |  |     |  |  |  |   __|

  |__    __.__|  |__    __.__|  |__|__|__.__|  |__|__|__|

   __|__|__.__    __|__|__.__    __|__|__.__    __|__|__.__

  |__|  |__.__|  |__|  |__.__|  |__|  |__.__|  |__|  |__.__|

      (24)            (26)          (28)           (28)

   __       __             __             __

  |  |   __|__|   __    __|__|   __    __|__|   __    __

  |__|__|__|     |__|__|__|     |__|__|__|     |__|__|__|

   __|__|__.__    __|__|__.__    __|__|__       __|__|__

  |__|  |__.__|  |__|  |__.__|  |__|  |__|     |__|  |__|

      (28)            (26)         (24)           (20)

   __    __             __

  |__|__|__|         __|__|         __             __

   __|__|         __|__|         __|__|           |__|

  |__|           |__|           |__|

      (16)         (12)            (8)             (4)

(End)

MATHEMATICA

h4[n_]:=If[n>2, (1/8)(-29+12n+2n^2-3*Cos[n*Pi]-12*Sin[n*Pi/2]), n]; h3[n_]:=1-Cos[n*Pi]-4*KroneckerDelta[n, 1]+2*KroneckerDelta[n, 4]+2*Sin[n*Pi/2]; h2[n_]:=If[n>4, (1/8)(71-20n+2n^2+25Cos[n*Pi]+4Sin[n*Pi/2]), 2*(KroneckerDelta[n, 2]+KroneckerDelta[n, 4])]; h1[n_]:=n^2-(h2[n]+h3[n]+h4[n]); T[n_, k_]:=If[0<=k<=h1[n], 2(2n+k), If[h1[n]<k<=(h1[n]+h2[n]), 2(2n+h1[n]), If[(h1[n]+h2[n])<k<=(h1[n]+h2[n]+h3[n]), 2(2(n+h1[n])+h2[n]-k), 2(2(n+h2[n]-k)+3h1[n]+h3[n])]]]; Flatten[Table[T[n, k], {n, 0, 6}, {k, 0, n^2}]]

CROSSREFS

Cf. A000290, A005843, A008586, A048759, A326118 (h_4).

Sequence in context: A154847 A165738 A200604 * A159468 A010769 A145831

Adjacent sequences:  A309035 A309036 A309037 * A309039 A309040 A309041

KEYWORD

nonn,tabf

AUTHOR

Stefano Spezia, Jul 08 2019

STATUS

approved

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Last modified January 24 16:32 EST 2020. Contains 331204 sequences. (Running on oeis4.)