A254029 -id:A254029 - cp's OEIS frontend

A362359 Triangle T read by rows, obtained from the array A for the solutions of the Monkey and Coconuts Problem (s sailors and one coconut to the monkey).

1, 2, 7, 3, 15, 79, 4, 23, 160, 1021, 5, 31, 241, 2045, 15621, 6, 39, 322, 3069, 31246, 279931, 7, 47, 403, 4093, 46871, 559867, 5764795, 8, 55, 484, 5117, 62496, 839803, 11529596, 134217721, 9, 63, 565, 6141, 78121, 1119739, 17294397, 268435449, 3486784393, 10, 71, 646, 7165, 93746, 1399675, 23059198, 402653177, 6973568794, 99999999991

Offset: 1

Richard S. Fischer and Wolfdieter Lang, Jun 20 2023

For the five sailors and one monkey problem see A254029.

The rows s of the array A give the positive solutions to the following problem: Recurrence co(k) = ((s-1)/s)*(co(k-1) - 1), for k >= 0, with co(0) = a, and the requirement c0(s) - 1 == 0 (mod s), for s >= 1. Then a = a(s, n) = A(s, n), for n >= 1.

			The array A begins:
s\n     1      2      3       4       5       6       7       8       9 ...
---------------------------------------------------------------------------
1:      1      2      3       4       5       6       7       8       9 ...
2:      7     15     23      31      39      47      55      63      71 ...
3:     79    160    241     322     403     484     565     646     727 ...
4:   1021   2045   3069    4093    5117    6141    7165    8189    9213 ...
5:  15621  31246  46871   62496   78121   93746  109371  124996  140621 ...
6: 279931 559867 839803 1119739 1399675 1679611 1959547 2239483 2519419 ...
...
s = 7: 5764795 11529596 17294397 23059198 28823999 34588800 40353601 46118402 51883203 57648004, ...
...
-----------------------------------------------------------------------------
The triangle begins:
  n\k  1  2   3    4     5       6        7         8          9          10
  ---------------------------------------------------------------------------
  1:   1
  2:   2  7
  3:   3 15  79
  4    4 23 160 1021
  5:   5 31 241 2045 15621
  6:   6 39 322 3069 31246  279931
  7:   7 47 403 4093 46871  559867  5764795
  8:   8 55 484 5117 62496  839803 11529596 134217721
  9:   9 63 565 6141 78121 1119739 17294397 268435449 3486784393
 10:  10 71 646 7165 93746 1399675 23059198 402653177 6973568794 99999999991
 ...
-----------------------------------------------------------------------------

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).

Rows of array A (columns of triangle T starting with index n): A000027, A004771(n-1), A362360, A362361, A254029.

First column of array A (diagonal of triangle T): A014293.

A362359row[n_]:=Array[(n-#+1)#^(#+1)-#+1&,n];Array[A362359row,10] (* Paolo Xausa, Nov 17 2023 *)

T(n, k) = A(k, n - k + 1), with the array A(s, n) = n*s^(s+1) - (s - 1), for s >= 1 and n >= 1. (Array read by antidiagonals downwards.)
T(n, k) = (n - k + 1)*k^(k+1) - (k - 1), for k = 1, 2, ..., n.
O.g.f. for row s of array A: (x/(1 - x)^2)*(s^(s + 1) - (s - 1)*(1 - x)).
E.g.f. for column n of array A: n*(-W(-x)/(1 - (-W(-x)))^3) - (1 - (1 - x)*exp(x)), with the principal branch of Lambert's W-function

A362360 a(n) = 81*n - 2.

Original entry on oeis.org

79, 160, 241, 322, 403, 484, 565, 646, 727, 808, 889, 970, 1051, 1132, 1213, 1294, 1375, 1456, 1537, 1618, 1699, 1780, 1861, 1942, 2023, 2104, 2185, 2266, 2347, 2428, 2509, 2590, 2671, 2752, 2833, 2914, 2995, 3076, 3157

Offset: 1

Richard S. Fischer and Wolfdieter Lang, Jun 20 2023

This gives the solution to the Monkey and Coconut Problem (three sailors, one coconut to the monkey). For the five sailors, one monkey problem see A254029.

This is row s = 3 of the array A given in A362359, hence the third column of the corresponding triangle T with offset 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A254029, A362359.

Range[50]81-2 (* Paolo Xausa, Nov 17 2023 *)

a(n) = 3^4*n - 2, for n >= 1.
O.g.f.: (x/(1-x)^2)*(3^4 - 2*(1-x)).
E.g.f.: 2 + exp(x)*(81*x - 2). - Stefano Spezia, Jun 24 2023

A362361 a(n) = n*2^10 - 3.

Original entry on oeis.org

1021, 2045, 3069, 4093, 5117, 6141, 7165, 8189, 9213, 10237, 11261, 12285, 13309, 14333, 15357, 16381, 17405, 18429, 19453, 20477, 21501, 22525, 23549, 24573, 25597, 26621, 27645, 28669, 29693, 30717, 31741, 32765

Offset: 1

Richard S. Fischer and Wolfdieter Lang, Jun 20 2023

This gives the solution to the Monkey and Coconut Problem (four sailors one coconut to the monkey). For the five sailors one monkey problem see A254029.

This is row s = 4 of the array given in A362359, hence the fourth column of the corresponding triangle T with offset 4.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A254029, A362359, A362360.

Range[50]2^10-3 (* Paolo Xausa, Nov 17 2023 *)

def A362361(n): return (n<<10)-3 # Chai Wah Wu, Jun 25 2023

a(n) = n*4^5 - 3, for n >= 1.
O.g.f: (x/(1-x)^2)*(4^5 - 3*(1-x)).
E.g.f.: 3 + exp(x)*(1024*x - 3). - Stefano Spezia, Jun 24 2023

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A362359 Triangle T read by rows, obtained from the array A for the solutions of the Monkey and Coconuts Problem (s sailors and one coconut to the monkey).

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A362360 a(n) = 81*n - 2.

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A362361 a(n) = n*2^10 - 3.

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