A355320 -id:A355320 - cp's OEIS frontend

A355339 a(n) is the number of central even terms in the n-th row of triangle A355320; a(0) = 0.

0, 3, 1, 3, 3, 7, 1, 3, 5, 3, 5, 11, 3, 11, 1, 3, 5, 7, 7, 3, 5, 7, 9, 19, 7, 3, 5, 19, 3, 11, 1, 3, 5, 7, 9, 11, 13, 11, 7, 3, 5, 7, 9, 11, 13, 15, 17, 35, 15, 11, 7, 3, 5, 7, 9, 35, 7, 3, 5, 19, 3, 11, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 23, 19, 15

Offset: 0

Rémy Sigrist, Jun 29 2022

nonn

			The first terms, alongside the corresponding rows of A355320, are:
   n  a(n)  n-th row of A355320
   -  ----  -------------------------------------------------------------
   0     0                                1
   1     3                          1 (0  0  0) 1
   2     1                    1  0  0  1 (2) 1  0  0  1
   3     3              1  0  0  2  3 (2  0  2) 3  2  0  0  1
   4     3        1  0  0  3  4  3  1 (6  8  6) 1  3  4  3  0  0  1
   5     7  1  0  0  4  5  4  3(12 16 12  6 12 16 12) 3  4  5  4  0  0  1

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A000918, A355320.

{ rr = 0; r = 1; for (n=0, 76, print1 (max(0, 2*valuation(r >> (2*n), 2)-1)", "); [rr, r] = [r, bitxor(bitxor(r, r<<4), bitxor(rr<<3, rr<<5))]) }

Conjecture: a(n) = 1 iff n > 0 and n belongs to A000918.

A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 2, 3, 2, 0, 0, 1, 8, 6, 1, 3, 4, 3, 0, 0, 1, 6, 12, 16, 12, 3, 4, 5, 4, 0, 0, 1, 44, 33, 18, 21, 27, 20, 6, 5, 6, 5, 0, 0, 1, 60, 76, 95, 72, 40, 34, 41, 30, 10, 6, 7, 6, 0, 0, 1, 256, 210, 154, 155, 177, 135, 75, 52, 58, 42, 15, 7, 8, 7, 0, 0, 1, 460, 520, 581, 480

Offset: 0

Clark Kimberling, Jun 29 2004

			Rows:
  1;
  0, 0, 1;
  2, 1, 0, 0, 1;
  0, 2, 3, 2, 0, 0, 1;
T(3,2) counts these paths:
  (0,0)-(1,-2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,4)-(3,2).

Paolo Xausa, Table of n, a(n) for n = 0..9999 (rows 0..99 of triangle, flattened)
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Rémy Sigrist, Representation of the odd terms for n = 0..2^11

Cf. A096587, A096609, A096610, A096611, A096612, A355320.

A096608[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[k<=2n,T[n-1,Abs[k-2]]+T[n-2,Abs[k-1]]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,k],{n,0,rowmax},{k,0,2n}]]; A096608[10] (* Generates 11 rows *) (* Paolo Xausa, May 09 2023 *)

row(n) = { my (rr=0, r=1); for (k=1, n, [rr, r]=[r, r*(1+'X^4)+rr*('X^3+'X^5)]); Vec(r)[1+2*n..1+4*n] } \\ Rémy Sigrist, Jun 29 2022

T(0, 0) = 1, T(0, 1) = 0, T(0, 2) = 0; T(1, 0) = 0, T(1, 1) = 0, T(1, 2) = 1.
For n >= 2, T(n, 0) = 2*T(n-2, 1) + 2*T(n-1, 2); T(n, 1) = T(n-2, 0) + T(n-2, 2) + T(n-1, 3) + T(n-1, 1); for 2 <= k <= 2n, T(n, k) = T(n-2, k-1) + T(n-2, k+1) + T(n-1, k-2) + T(n-1, k+2).
T(n, 0) + 2*Sum_{k = 1..2*n} T(n, k) = A002605(k). - Rémy Sigrist, Jun 29 2022

Offset changed to 0 by Rémy Sigrist, Jun 29 2022

A361292 Square array A(n, k), n, k >= 0, read by antidiagonals; A(0, 0) = 1, and otherwise A(n, k) is the sum of all terms in previous antidiagonals at one knight's move away.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 4, 2, 0, 2, 5, 4, 7, 7, 4, 5, 2, 5, 5, 10, 14, 12, 14, 10, 5, 5, 5, 10, 21, 23, 30, 30, 23, 21, 10, 5, 10, 23, 35, 49, 62, 60, 62, 49, 35, 23, 10, 23, 40, 69, 100, 119, 137, 137, 119, 100, 69, 40, 23

Offset: 0

Rémy Sigrist, Mar 12 2023

			Square array A(n, k) begins:
  n\k |  0   1    2    3     4     5     6      7      8      9      10
  ----+----------------------------------------------------------------
    0 |  1   0    0    0     1     1     0      2      5      5      10
    1 |  0   0    1    1     0     2     5      5     10     23      40
    2 |  0   1    0    2     4     4    10     21     35     69     138
    3 |  0   1    2    2     7    14    23     49    100    190     382
    4 |  1   0    4    7    12    30    62    119    250    512    1031
    5 |  1   2    4   14    30    60   137    290    599   1263    2639
    6 |  0   5   10   23    62   137   298    662   1430   3043    6502
    7 |  2   5   21   49   119   290   662   1472   3281   7181   15569
    8 |  5  10   35  100   250   599  1430   3281   7410  16585   36699
    9 |  5  23   69  190   512  1263  3043   7181  16585  37700   84939
   10 | 10  40  138  382  1031  2639  6502  15569  36699  84939  194154

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Rémy Sigrist, PARI program

See A355320 for a similar sequence.

A361292list[dmax_]:=Module[{A},A[0,0]=1;A[n_,k_]:=A[n,k]=A[k,n]=If[n>=0&&k>=0,A[n-2,k-1]+A[n-2,k+1]+A[n-1,k-2]+A[n+1,k-2],0];Table[A[n-k,k],{n,0,dmax-1},{k,0,n}]];A361292list[15] (* Generates 15 antidiagonals *) (* Paolo Xausa, Oct 17 2023 *)

PARI
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See Links section.
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A(n, k) = A(k, n).

A(n, k) = A'(n-2, k-1) + A'(n-2, k+1) + A'(n-1, k-2) + A'(n+1, k-2) for n + k > 0 (where A' extends A with 0's outside its domain of definition).

cp's OEIS Frontend

A355339 a(n) is the number of central even terms in the n-th row of triangle A355320; a(0) = 0.

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A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

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A361292 Square array A(n, k), n, k >= 0, read by antidiagonals; A(0, 0) = 1, and otherwise A(n, k) is the sum of all terms in previous antidiagonals at one knight's move away.

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Mathematica

PARI

Formula