A081493 -id:A081493 - cp's OEIS frontend

A159797 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n-1.

0, 1, 1, 2, 3, 4, 3, 5, 7, 9, 4, 7, 10, 13, 16, 5, 9, 13, 17, 21, 25, 6, 11, 16, 21, 26, 31, 36, 7, 13, 19, 25, 31, 37, 43, 49, 8, 15, 22, 29, 36, 43, 50, 57, 64, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101

Offset: 0

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th square A000290(n).
See also A162611, A162614 and A162622.
The triangle sums, see A180662 for their definitions, link the triangle A159797 with eleven sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
T(n,k) is the number of distinct sums in the direct sum of {1, 2, ... n} with itself k times for 1 <= k <= n+1, e.g., T(5,3) = the number of distinct sums in the direct sum {1,2,3,4,5} + {1,2,3,4,5} + {1,2,3,4,5}. The sums range from 1+1+1=3 to 5+5+5=15. So there are 13 distinct sums. - Derek Orr, Nov 26 2014

			Triangle begins:
0;
1, 1;
2, 3, 4;
3, 5, 7, 9;
4, 7,10,13,16;
5, 9,13,17,21,25;
6,11,16,21,26,31,36;

Harvey P. Dale, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A000290, A001477, A081493, A159798, A162609, A162610, A162611, A162614, A162622.
Cf.: A006002 (row sums). - R. J. Mathar, Jul 17 2009
Cf. A163282, A163283, A163284, A163285. - Omar E. Pol, Nov 18 2009
From Johannes W. Meijer, May 20 2011: (Start)
Triangle sums (see the comments): A006002 (Row1), A050187 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Fi1 and Ze1), A006918 (Related to Kn21, Kn22, Kn23, Fi2 and Ze2), A000330 (Kn3), A016061 (Kn4), A190717 (Related to Ca1 and Ze3), A144677 (Related to Ca2 and Ze4), A000292 (Related to Ca3, Ca4, Gi3 and Gi4) A190718 (Related to Gi1) and A144678 (Related to Gi2). (End)

A159797:=proc(n) local m: m := floor( (sqrt(8*n+1)-1)/2 ): A159797(n):= m + (n - m*(m+1)/2)*(m-1) end: seq(A159797(n),n=0..75); # Johannes W. Meijer, May 20 2011

Flatten[Table[NestList[#+n-1&,n,n],{n,0,12}]] (* Harvey P. Dale, Aug 04 2014 *)

Given m = floor( (sqrt(8*n+1)-1)/2 ), then a(n) = m + (n - m*(m+1)/2)*(m-1). - Carl R. White, Jul 24 2010

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Nov 18 2009
More terms from Carl R. White, Jul 24 2010

A159798 Triangle read by rows in which row n lists n terms, starting with 1, such that the difference between successive terms is equal to n-3.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 4, 1, 3, 5, 7, 9, 1, 4, 7, 10, 13, 16, 1, 5, 9, 13, 17, 21, 25, 1, 6, 11, 16, 21, 26, 31, 36, 1, 7, 13, 19, 25, 31, 37, 43, 49, 1, 8, 15, 22, 29, 36, 43, 50, 57, 64, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 1, 11

Offset: 1

Omar E. Pol, Jul 09 2009

Note that for n>1 the last term of the n-th row is the square A000290(n-2).
Row sums are n*(n^2-4*n+5)/2 = 1, 1, 3, 10, 25, 51, 91, 148, 225, ... - R. J. Mathar, Jul 17 2009, Jul 20 2009
Row sums are the positive terms of A162607. - Omar E. Pol, Jul 24 2009

			Triangle begins:
  1;
  1,  0;
  1,  1,  1;
  1,  2,  3,  4;
  1,  3,  5,  7,  9;
  1,  4,  7, 10, 13, 16;
  1,  5,  9, 13, 17, 21, 25;
  1,  6, 11, 16, 21, 26, 31, 36;
  1,  7, 13, 19, 25, 31, 37, 43, 49;
  1,  8, 15, 22, 29, 36, 43, 50, 57, 64;
  1,  9, 17, 25, 33, 41, 49, 57, 65, 73, 81;
  1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000290, A081493, A159797, A162607, A162609, A162610, A162611.

/* As triangle */ [[1 + k*(n-3): k in [0..n-1]]: n in [1..15]]; // G. C. Greubel, Apr 21 2018

Table[1 + k*(n-3), {n, 1, 20}, {k, 0, n-1}]// Flatten (* G. C. Greubel, Apr 21 2018 *)

for(n=1, 20, for(k=0,n-1, print1(1 + k*(n-3), ", "))) \\ G. C. Greubel, Apr 21 2018

T(n,k) = 1 + k*(n-3), 0<=kR. J. Mathar, Jul 17 2009

More terms from R. J. Mathar, Jul 17 2009
Typo in row sums corrected by R. J. Mathar, Jul 20 2009
Edited by Omar E. Pol, Jul 24 2009

A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 7, 1, 4, 7, 10, 13, 1, 5, 9, 13, 17, 21, 1, 6, 11, 16, 21, 26, 31, 1, 7, 13, 19, 25, 31, 37, 43, 1, 8, 15, 22, 29, 36, 43, 50, 57, 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81

Offset: 1

Omar E. Pol, Jul 09 2009

Equals A081493 when first column is removed. - Georg Fischer, Jul 25 2023

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,  7;
  1,  4,  7, 10, 13;
  1,  5,  9, 13, 17, 21;
  1,  6, 11, 16, 21, 26, 31;

Cf. A002061, A159797, A159798, A162610, A161611, A162612, A162613.

Cf. A060354 (row sums), A081493 (without first column).

Table[NestList[#+(n-2)&,1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Oct 23 2017 *)

T(n,n) = A002061(n-1).

T(n,k) = A076110(n-1,k) = 1+(n-2)*(k-1). - R. J. Mathar, Mar 30 2023

A382532 a(n) = Product_{k=1..n} (n*k-k+1).

Original entry on oeis.org

1, 1, 6, 105, 3640, 208845, 17873856, 2131900225, 337767408000, 68586144251625, 17361688356812800, 5359035747797893161, 1980990543353657472000, 863884504344556052483125, 438824910158909833337856000, 256841080519120696725634418625, 171586094900260237697765926076416

Offset: 0

Wesley Ivan Hurt, Mar 30 2025

nonn

Fill an n X n square array with the numbers 1..n^2 in increasing order by rows. a(n) is the product of the numbers along the main antidiagonal (see example).

			                                                       [1   2  3  4  5]
                                       [1   2  3  4]   [6   7  8  9 10]
                             [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                    [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
            [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
  ------------------------------------------------------------------------
   n         1        2         3            4                 5
  ------------------------------------------------------------------------
   a(n)      1        6         105         3640             208845

Row products of A081493.

Cf. A006003, A370753.

Table[Product[n*k - k + 1, {k, n}], {n, 0, 20}]

a(n) = (n-1)^n * Pochhammer(n/(n-1), n) for n>=2.
a(n) = Product_{k=1..n} A081493(n,k).
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n+1). - Vaclav Kotesovec, Apr 01 2025

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A159797 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n-1.

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A159798 Triangle read by rows in which row n lists n terms, starting with 1, such that the difference between successive terms is equal to n-3.

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A162609 Triangle read by rows in which row n lists n terms, starting with 1, with gaps = n-2 between successive terms.

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Programs

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A382532 a(n) = Product_{k=1..n} (n*k-k+1).

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