A162609 -id:A162609 - cp's OEIS frontend

A351704 Sums of the ascending diagonals of the triangle A162609.

1, 1, 2, 3, 7, 10, 20, 26, 45, 55, 86, 101, 147, 168, 232, 260, 345, 381, 490, 535, 671, 726, 892, 958, 1157, 1235, 1470, 1561, 1835, 1940, 2256, 2376, 2737, 2873, 3282, 3435, 3895, 4066, 4580, 4770, 5341, 5551, 6182, 6413, 7107, 7360, 8120, 8396, 9225, 9525, 10426, 10751, 11727, 12078, 13132, 13510, 14645

Offset: 0

Eddie Gutierrez, May 05 2022

nonn

Each term is the sum of an ascending diagonal of the triangle A162609.

			a(4) = (64 + 8 + 12)/12 = 7
a(5) = (250 - 75 + 50 + 15)/24 = 10.

Eddie Gutierrez, The Wheel Octagon Triangle - Ascending Diagonals (Part VIIIb)

Cf. A162609.

// Calculates and prints out the triangle and terms of ascending diagonals (on first line). To get more terms increment j.
#include 
int main()
{
   int n, j=8, k, C, F1, F2,s;
   F1=1; F2=1;
   printf("%d ", F1);
   printf("%d ", F2);
   for (s=0;s<=j;s++)
   {
      F1=F1 + 2*s*s + 2*s + 1;
      F2=F2 + 2*s*s + 3*s + 2;
      printf("%d ", F1);
      printf("%d ", F2);
   }
   printf("\n");
   return 0;
}

a(n) = (n^3 + 2*n + 12)/12, for even n.

a(n) = (2*n^3 - 3*n^2 + 10*n + 15)/24, for odd n.

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.

Original entry on oeis.org

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

A162614 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^3 - 1.

Original entry on oeis.org

0, 1, 1, 2, 9, 16, 3, 29, 55, 81, 4, 67, 130, 193, 256, 5, 129, 253, 377, 501, 625, 6, 221, 436, 651, 866, 1081, 1296, 7, 349, 691, 1033, 1375, 1717, 2059, 2401, 8, 519, 1030, 1541, 2052, 2563, 3074, 3585, 4096, 9, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833

Offset: 0

Omar E. Pol, Jul 15 2009

Note that the last term of the n-th row is the fourth power of n, A000583(n).

See also the triangles of A162615 and A162616.

			Triangle begins:
  0;
  1,   1;
  2,   9,  16;
  3,  29,  55,  81;
  4,  67, 130, 193, 256;
  5, 129, 253, 377, 501,  625;
  6, 221, 436, 651, 866, 1081, 1296;
  ...

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162615, A162616, A162622, A162623, A162624.

def A162614(n,k):
    return n+k*(n**3-1)
print([A162614(n,k) for n in range(20) for k in range(n+1)])
# R. J. Mathar, Oct 20 2009

Sum_{k=0..n} T(n,k) = n*(n^2-n+1)*(n+1)^2/2 (row sums). - R. J. Mathar, Jul 20 2009

T(n,k) = n + k*(n^3-1). - R. J. Mathar, Oct 20 2009

More terms from R. J. Mathar, Oct 20 2009

A162622 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^4 - 1.

Original entry on oeis.org

0, 1, 1, 2, 17, 32, 3, 83, 163, 243, 4, 259, 514, 769, 1024, 5, 629, 1253, 1877, 2501, 3125, 6, 1301, 2596, 3891, 5186, 6481, 7776, 7, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 9, 6569, 13129

Offset: 0

Omar E. Pol, Jul 15 2009

Note that the last term of the n-th row is the 5th power of n, A000584(n).

See also the triangles of A162623 and A162624.

			Triangle begins:
  0;
  1,    1;
  2,   17,    32;
  3,   83,   163,   243;
  4,  259,   514,   769,  1024;
  5,  629,  1253,  1877,  2501,  3125;
  6, 1301,  2596,  3891,  5186,  6481,  7776;
  7, 2407,  4807,  7207,  9607, 12007, 14407, 16807;
  8, 4103,  8198, 12293, 16388, 20483, 24578, 28673, 32768;
  9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 59049; etc.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162623, A162624.

/* Triangle: */ [[n+k*(n^4-1): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Dec 14 2012

A162622 := proc(n,k) n+k*(n^4-1) ; end proc: seq(seq( A162622(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 11 2010

Flatten[Table[NestList[#+n^4-1&,n,n],{n,0,9}]] (* Harvey P. Dale, Jun 23 2013 *)

Sum_{k=0..n} T(n,k) = n*(n+1)*(1+n^4)/2 (row sums). [R. J. Mathar, Jul 20 2009]

7th and later rows from R. J. Mathar, Feb 11 2010

A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.

Original entry on oeis.org

1, 1, 3, 28, 585, 22176, 1339975, 118514880, 14454403425, 2326680294400, 478015854767451, 122087424094272000, 37947924636264267625, 14105590169042424729600, 6178966019176767549393375, 3150334059785191453342744576, 1849556085478041490537172810625

Offset: 0

Amarnath Murthy, Mar 28 2004

We have the triangle (chopped versions of A076110, A162609)
  1;
  1 3;
  1 4  7;
  1 5  9 13;
  1 6 11 16 21;
  1 7 13 19 25 31;
...
Sequence contains the product of the terms of the rows.
a(n) = b(n-1) where b(n) = n^n*Gamma(n+1/n)/Gamma(1/n) and b(0) is limit n->0+ of b(n). - Gerald McGarvey, Nov 10 2007
Product of the entries in the first column of an n X n square array with elements 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Apr 02 2025

			a(5) = 1*6*11*16*21 = 22176.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A057237, A092987.

Main diagonal of A256268.

List([0..20], n-> Product([0..n-1], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

[1] cat [ (&*[j*n+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Mar 04 2020

a:= n-> mul(n*j+1, j=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 24 2015

Flatten[{1, Table[n^n * Pochhammer[1/n, n], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 05 2018 *)

vector(21, n, my(m=n-1); prod(j=0,m-1, j*m+1)) \\ G. C. Greubel, Mar 04 2020

[product(j*n+1 for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Mar 04 2020

a(n) = Product_{k=1..n} (1+(k-1)*n) = 1*(1+n)*(1+2n)*...*(n^2-n+1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*n^(n-k). - Vladeta Jovovic, Jan 28 2005
a(n) = n! * [x^n] 1/(1 - n*x)^(1/n) for n > 0. - Ilya Gutkovskiy, Oct 05 2018
a(n) ~ sqrt(2*Pi) * n^(2*n - 3/2) / exp(n). - Vaclav Kotesovec, Oct 05 2018

More terms from Erich Friedman, Aug 08 2005

Offset corrected by Alois P. Heinz, Nov 24 2015

A162615 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^3 - 1 = A068601(n).

Original entry on oeis.org

1, 2, 9, 3, 29, 55, 4, 67, 130, 193, 5, 129, 253, 377, 501, 6, 221, 436, 651, 866, 1081, 7, 349, 691, 1033, 1375, 1717, 2059, 8, 519, 1030, 1541, 2052, 2563, 3074, 3585, 9, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833, 10, 1009, 2008, 3007, 4006, 5005, 6004

Offset: 1

Omar E. Pol, Jul 12 2009

See also the triangles of A162614 and A162616.

			Triangle begins:
  1;
  2,   9;
  3,  29,  55;
  4,  67, 130, 193;
  5, 129, 253, 377, 501;
  6, 221, 436, 651, 866, 1081;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162616.

A162615 := proc(n,k) n+(k-1)*(n^3-1) ; end proc: seq(seq(A162615(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2010

Flatten[Table[c=n^3-1;NestList[#+c&,n,n-1],{n,10}]] (* Harvey P. Dale, Nov 13 2011 *)

Row sums: n*(n^4 - n^3 + n + 1)/2. - R. J. Mathar, Jul 20 2009

Terms beyond the 6th row from R. J. Mathar and Max Alekseyev, Feb 05 2010

A162616 Triangle read by rows in which row n lists n terms, starting with n^3 + n - 1, such that the difference between successive terms is equal to n^3 - 1 = A068601(n).

Original entry on oeis.org

1, 9, 16, 29, 55, 81, 67, 130, 193, 256, 129, 253, 377, 501, 625, 221, 436, 651, 866, 1081, 1296, 349, 691, 1033, 1375, 1717, 2059, 2401, 519, 1030, 1541, 2052, 2563, 3074, 3585, 4096, 737, 1465, 2193, 2921, 3649, 4377, 5105, 5833, 6561, 1009, 2008

Offset: 1

Omar E. Pol, Jul 12 2009

Note that the last term of the n-th row is the fourth power of n, A000583(n).

See also the triangles of A162614 and A162615.

			Triangle begins:
    1;
    9,  16;
   29,  55,  81;
   67, 130, 193, 256;
  129, 253, 377, 501,  625;
  221, 436, 651, 866, 1081, 1296;
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000583, A068601, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615.

A162616 := proc(n,k) n^3+n-1+(k-1)*(n^3-1) ; end proc: seq(seq(A162616(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2010

Table[NestList[#+n^3-1&,n^3+n-1,n-1],{n,10}]//Flatten (* Harvey P. Dale, Dec 17 2021 *)

Row sums: n*(n^2 + n - 1)*(n^2+1)/2. - R. J. Mathar, Jul 20 2009

More terms from R. J. Mathar, Feb 05 2010

A162624 Triangle read by rows in which row n lists n terms, starting with n^4 + n - 1, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

Original entry on oeis.org

1, 17, 32, 83, 163, 243, 259, 514, 769, 1024, 629, 1253, 1877, 2501, 3125, 1301, 2596, 3891, 5186, 6481, 7776, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 6569, 13129, 19689, 26249, 32809

Offset: 1

Omar E. Pol, Jul 12 2009

Note that the last term of the n-th row is the 5th power of n, A000584(n).

See also the triangles of A162622 and A162623.

			Triangle begins:
     1;
    17,   32;
    83,  163,  243;
   259,  514,  769, 1024;
   629, 1253, 1877, 2501, 3125;
  1301, 2596, 3891, 5186, 6481, 7776;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162622, A162623.

A162624 := proc(n,k) return n+k*(n^4-1): end: seq(seq(A162624(n,k), k=1..n), n=1..10); # Nathaniel Johnston, Apr 30 2011

Table[NestList[#+n^4-1&,n^4+n-1,n-1],{n,10}]//Flatten (* Harvey P. Dale, Apr 28 2022 *)

Row sums: n*(n^5 + n^4 + n - 1)/2. - R. J. Mathar, Jul 20 2009

A162623 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

Original entry on oeis.org

1, 2, 17, 3, 83, 163, 4, 259, 514, 769, 5, 629, 1253, 1877, 2501, 6, 1301, 2596, 3891, 5186, 6481, 7, 2407, 4807, 7207, 9607, 12007, 14407, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 10

Offset: 1

Omar E. Pol, Jul 12 2009

See also the triangles of A162622 and A162624.

			Triangle begins:
  1;
  2,   17;
  3,   83,  163;
  4,  259,  514,  769;
  5,  629, 1253, 1877, 2501;
  6, 1301, 2596, 3891, 5186, 6481;

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162622, A162624.

A162623 := proc(n,k) n+k*(n^4-1) ; end: seq(seq(A162623(n,k),k=0..n-1),n=1..15) ; # R. J. Mathar, Sep 27 2009

dst[n_]:=Module[{c=n^4-1},Range[n,n*c,c]]; Flatten[Join[{1},Table[dst[n],{n,2,10}]]] (* Harvey P. Dale, Jul 29 2014 *)

Row sums: n*(n^5 - n^4 + n + 1)/2. - R. J. Mathar, Jul 20 2009

More terms from R. J. Mathar, Sep 27 2009

cp's OEIS Frontend

A351704 Sums of the ascending diagonals of the triangle A162609.

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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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A162614 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^3 - 1.

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A162622 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^4 - 1.

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A092985 a(n) is the product of the first n terms of an arithmetic progression with the first term 1 and common difference n.

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