A033585 -id:A033585 - cp's OEIS frontend

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

Offset: 3

Stefano Spezia, Mar 19 2021

			The array begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  21   45   78  120  171 ...
4  |  36   78  136  210  300 ...
5  |  55  120  210  325  465 ...
6  |  78  171  300  465  666 ...
7  | 105  231  406  630  903 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten

O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

A317948 An example of a morphic word: the sorted (by length, then alphabetically) sequence of words of the form ab under the action of a finite automaton defined as follows: start state is 0; a and b map states [0, 1, 2, 3] to states [1, 2, 3, 0] and [0, 3, 1, 2], respectively.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 1, 2, 0, 0, 2, 3, 1, 0, 1, 0, 1, 2, 3, 0, 2, 3, 0, 3, 1, 2, 0, 3, 1, 2, 0, 2, 3, 1, 0, 0, 2, 3, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 3, 1, 2, 0, 2, 3, 0, 3, 1, 2, 0, 2, 3, 1, 0, 3, 1, 2, 0, 2, 3, 1, 0, 1, 2, 3, 0, 0, 2, 3, 1, 0, 1, 2, 3, 0

Offset: 1

N. J. A. Sloane, Aug 12 2018

The Duchêne et al. (2011) reference mentions many other sequences that are of great interest.
From Kevin Ryde, Dec 26 2020: (Start)
This sequence can be taken as a square array S(m,k) read by upwards antidiagonals with rows m >= 0 and columns k >= 0.  S(m,k) is the final state for input word a^m b^k where ^ denotes repetition.  Input a's cycle around states 0,1,2,3 so S(m,0) = m mod 4.  Within an array row, input b's are no change at state 0 (so a row of 0's), or a repeating cycle 3,2,1 starting at m mod 4.
Antidiagonal d of the array is input words of length d = m+k, so terms S(d-k,k).  These are words a^(d-k) b^k and the combination of d-k mod 4 and k mod 3 is 12-periodic within a diagonal.  The sequence can also be taken as a triangle read by rows T(d,k) = S(d-k,k) for d >= 0 and 0 <= k <= d.
Rigo (2000, section 2.1 remark 2) notes that the sequence (flat sequence) is not periodic because pairs of terms 0,0 are at ever-increasing distances apart.  They are a(n)=a(n+1)=0 iff n = 2*t*(4*t+1) = A033585(t) for t >= 1, which is every fourth triangular number.
(End)

			From _Kevin Ryde_, Dec 26 2020: (Start)
Array S(m,k) begins
       k=0  1  2  3  4  5  6  7
      +-------------------------
  m=0 |  0, 0, 0, 0, 0, 0, 0, 0,
  m=1 |  1, 3, 2, 1, 3, 2, 1,
  m=2 |  2, 1, 3, 2, 1, 3,     sequence by upwards
  m=3 |  3, 2, 1, 3, 2,          antidiagonals,
  m=4 |  0, 0, 0, 0,
  m=5 |  1, 3, 2,          12-periodic in diagonals
  m=6 |  2, 1,             (3 or 1-periodic in rows)
  m=7 |  3,                 (4-periodic in columns)
(End)

Eric Duchêne, Aviezri S. Fraenkel, Richard J. Nowakowski, and Michel Rigo, Extensions and restrictions of Wythoff's game preserving Wythoff's sequence as set of P-positions, Slides from a talk, LIAFA, Paris, October 21, 2011. See around the 35th slide, a slide with first line "In fact, this is a special case of the following result...".
Michel Rigo, Generalization of automatic sequences for numeration systems on a regular language, Theoret. Comput. Sci., 244 (2000) 271-281. See section 2.1 sequence u.
Michel Rigo and Arnaud Maes, More on generalized automatic sequences, Journal of Automata, Languages and Combinatorics 7.3 (2002): 351-376. See Fig. 3.

S(m,k) = if(m%=4, (m-k-1)%3+1, 0); \\ Kevin Ryde, Dec 26 2020

aut0, aut1 = [1, 2, 3, 0], [0, 3, 1, 2]
a, row = [0], [0]
for i in range(1, 10):
    row = [aut0[row[0]]] + [aut1[x] for x in row]
    a += row
print(a)
# Andrey Zabolotskiy, Aug 17 2018

From Kevin Ryde, Dec 26 2020: (Start)
S(m,k) = 0 if m==0 (mod 4), otherwise S(m,k) = (((m mod 4) - k - 1) mod 3) + 1.
T(d,k) = S(d-k,k) = p(3*d+k mod 12) where p(0..11) = 0,2,3,1, 0,1,2,3, 0,3,1,2.
(End)

New name and terms a(51) and beyond from Andrey Zabolotskiy, Aug 17 2018

A361258 Irregular triangle read by rows in which row n lists the print order of a 4n-page booklet.

Original entry on oeis.org

2, 3, 4, 1, 2, 7, 8, 1, 4, 5, 6, 3, 2, 11, 12, 1, 4, 9, 10, 3, 6, 7, 8, 5, 2, 15, 16, 1, 4, 13, 14, 3, 6, 11, 12, 5, 8, 9, 10, 7, 2, 19, 20, 1, 4, 17, 18, 3, 6, 15, 16, 5, 8, 13, 14, 7, 10, 11, 12, 9, 2, 23, 24, 1, 4, 21, 22, 3, 6, 19, 20, 5, 8, 17, 18, 7, 10, 15, 16, 9, 12, 13, 14, 11

Offset: 1

Ole Palnatoke Andersen, Mar 06 2023

Row n is a permutation of the first 4*n positive integers.

			Triangle begins:
  2,  3,  4, 1;
  2,  7,  8, 1, 4,  5,  6, 3;
  2, 11, 12, 1, 4,  9, 10, 3, 6,  7,  8, 5;
  2, 15, 16, 1, 4, 13, 14, 3, 6, 11, 12, 5, 8, 9, 10, 7;
  ...

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A008586 (row lengths), A033585 (row sums).

T:= n-> seq([2*m, 4*n-2*m+1, 4*n-2*m+2, 2*m-1][], m=1..n):
seq(T(n), n=1..6);  # Alois P. Heinz, Mar 06 2023

A343155 Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.

Original entry on oeis.org

1, 10, 36, 9, 78, 58, 136, 164, 25, 210, 318, 138, 300, 520, 356, 49, 406, 770, 654, 250, 528, 1068, 1032, 612, 81, 666, 1414, 1490, 1086, 394, 820, 1808, 2028, 1672, 932, 121, 990, 2250, 2646, 2370, 1614, 570, 1176, 2740, 3344, 3180, 2440, 1316, 169, 1378, 3278, 4122, 4102, 3410, 2238, 778

Offset: 1

Stefano Spezia, Apr 07 2021

			The triangle T(n, k) begins:
n\k|   1    2    3    4
---+-------------------
1  |   1
2  |  10
3  |  36    9
4  |  78   58
5  | 136  164   25
6  | 210  318  138
7  | 300  520  356   49
...
For n = 1 the matrix is
      1
and T(1, 1) = 1.
For n = 2 the matrix is
      1, 2
      4, 3
and T(2, 1) = 1 + 2 + 3 + 4 = 4*5/2 = 10.
For n = 3 the matrix is
      1, 2, 3
      8, 9, 4
      7, 6, 5
and T(3, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 8*9/2 = 36; T(3, 2) = 9.
For n = 4 the matrix is
      1,  2,  3,  4
     12, 13, 14,  5
     11, 16, 15,  6
     10,  9,  8,  7
and T(4, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 12*13/2 = 78; T(4, 2) = 13 + 14 + 15 + 16 = (13 + 16)*4/2 = 58.
...

Cf. A000007, A000290, A033585, A037270 (row sums), A110654, A126224.

Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n,2k-1],{n,14},{k,Ceiling[n/2]}]//Flatten

T(n, k) = 2*(2*k - n - 1)*(3 + 8*k*(k - n - 1) + 4*n) + n^2*0^(n+1-2*k) with 0 < k <= ceiling(n/2).

T(n, 1) = A033585(n-1) for n > 1.

cp's OEIS Frontend

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

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A317948 An example of a morphic word: the sorted (by length, then alphabetically) sequence of words of the form a*b* under the action of a finite automaton defined as follows: start state is 0; a and b map states [0, 1, 2, 3] to states [1, 2, 3, 0] and [0, 3, 1, 2], respectively.

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A361258 Irregular triangle read by rows in which row n lists the print order of a 4n-page booklet.

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Maple

A343155 Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.

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A317948 An example of a morphic word: the sorted (by length, then alphabetically) sequence of words of the form ab under the action of a finite automaton defined as follows: start state is 0; a and b map states [0, 1, 2, 3] to states [1, 2, 3, 0] and [0, 3, 1, 2], respectively.