A291896 -id:A291896 - cp's OEIS frontend

A291905 Row sums of A291904.

1, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 8, 8, 11, 14, 16, 21, 26, 32, 39, 49, 60, 75, 93, 114, 142, 176, 217, 268, 334, 411, 510, 632, 779, 967, 1196, 1477, 1832, 2266, 2801, 3470, 4291, 5310, 6572, 8129, 10061, 12449, 15401, 19058, 23581, 29178, 36102, 44668

Offset: 0

Seiichi Manyama, Sep 05 2017

nonn

Number of compositions of n where the first part is 1 and the absolute difference between consecutive parts is 1.

			The a(6)=2 compositions of 6 are:
:
:  o o|
: oooo|
:
:   o|
:  oo|
: ooo|
:
The a(9)=3 compositions of 9 are:
:
:   o  |
:  ooo |
: ooooo|
:
:  o o o|
: oooooo|
:
:     o|
:  o oo|
: ooooo|

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A291896, A291904.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
     `if`(j=i, 0, b(n-j, j)), j=max(1, i-1)..min(i+1, n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 05 2017

T[0, 0] = 1; T[, 0] = 0; T[n?Positive, k_] /; 0 < k <= Floor[(Sqrt[8n+1] - 1)/2] := T[n, k] = T[n-k, k-1] + T[n-k, k+1]; T[, ] = 0;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8n+1] - 1)/2]}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2019 *)

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==0 else sum(b(n - j, j) for j in range(max(1, i - 1), min(i + 1, n) + 1) if j != i)
def a(n): return b(n, 0)
print([a(n) for n in range(61)]) # Indranil Ghosh, Sep 06 2017, after Maple program

A291895 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) + T(n-k,k+1) with T(0,0) = 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 2, 0, 5, 3, 1, 0, 8, 5, 1, 0, 13, 9, 2, 0, 22, 14, 4, 0, 36, 24, 6, 1, 0, 60, 40, 11, 1, 0, 100, 66, 18, 2, 0, 166, 111, 31, 4, 0, 277, 184, 52, 7, 0, 461, 308, 86, 12, 1, 0, 769, 513, 146, 20, 1, 0, 1282, 855, 243, 35, 2

Offset: 0

Seiichi Manyama, Sep 05 2017

			First few rows are:
  1;
  0,  1;
  0,  1;
  0,  1,  1;
  0,  2,  1;
  0,  3,  2;
  0,  5,  3, 1;
  0,  8,  5, 1;
  0, 13,  9, 2;
  0, 22, 14, 4;
  0, 36, 24, 6, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A291896.

Columns 0-1 give A000007, A186085 (for n>0).

A291930 Row sums of A291929.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 60, 139, 321, 743, 1716, 3965, 9158, 21152, 48848, 112808, 260500, 601553, 1389096, 3207660, 7406989, 17103860, 39495306, 91200333, 210594475, 486292240, 1122916743, 2592971247, 5987531168, 13826041086, 31926247578, 73722134145, 170234630412

Offset: 0

Seiichi Manyama, Sep 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2752

Cf. A291896, A291905.

A291942 Row sums of A291940.

Original entry on oeis.org

1, 1, -2, 5, -9, 14, -24, 47, -87, 151, -272, 505, -918, 1656, -3020, 5512, -10020, 18221, -33172, 60388, -109899, 199992, -364002, 662529, -1205777, 2194488, -3994057, 7269275, -13230124, 24078998, -43824226, 79760817, -145165804, 264203948, -480855437, 875164374

Offset: 0

Seiichi Manyama, Sep 06 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..3841

Cf. A291896, A291905, A291930.

A351106 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

Original entry on oeis.org

1, 1, 3, 1, 6, 23, 2, 17, 118, 1470, 3, 47, 762, 23878, 914525, 5, 133, 5724, 420894, 40285572

Offset: 1

Pontus von Brömssen, Jan 31 2022

			Triangle begins:
  m\n| 1   2    3      4        5  6
  ---+------------------------------
  1  | 1
  2  | 1   3
  3  | 1   6   23
  4  | 2  17  118   1470
  5  | 3  47  762  23878   914525
  6  | 5 133 5724 420894 40285572  ?

Wikipedia, Racetrack

Cf. A291896 (column n=1), A329118, A351041, A351107 (main diagonal), A351108, A351110.

A351108 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 2, 3, 8, 3, 3, 7, 12, 40, 5, 7, 13, 26, 160, 1380, 9, 13, 28, 61, 918, 12940, 211164, 14, 27, 61, 161, 7260, 142453, 4997155, 205331148

Offset: 1

Pontus von Brömssen, Feb 01 2022

			Triangle begins:
  m\n|  1  2  3   4    5      6       7         8
  ---+-------------------------------------------
  1  |  1
  2  |  1  0
  3  |  1  1  2
  4  |  2  2  3   8
  5  |  3  3  7  12   40
  6  |  5  7 13  26  160   1380
  7  |  9 13 28  61  918  12940  211164
  8  | 14 27 61 161 7260 142453 4997155 205331148

Wikipedia, Racetrack

Cf. A064297, A291896 (column n=1), A351042, A351106, A351109 (main diagonal).

cp's OEIS Frontend

A291905 Row sums of A291904.

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A291895 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) + T(n-k,k+1) with T(0,0) = 1.

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A291930 Row sums of A291929.

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A291942 Row sums of A291940.

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A351106 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

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A351108 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

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