A162611 -id:A162611 - 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

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

A162612 Triangle read by rows in which row n lists n terms, starting with n^2+n-1, with gaps = n^2-1 between successive terms.

Original entry on oeis.org

1, 5, 8, 11, 19, 27, 19, 34, 49, 64, 29, 53, 77, 101, 125, 41, 76, 111, 146, 181, 216, 55, 103, 151, 199, 247, 295, 343, 71, 134, 197, 260, 323, 386, 449, 512, 89, 169, 249, 329, 409, 489, 569, 649, 729, 109, 208, 307, 406, 505, 604, 703, 802, 901, 1000, 131, 251

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th cube A000578(n).

			Triangle begins:
   1;
   5,   8;
  11,  19,  27;
  19,  34,  49,  64;
  29,  53,  77, 101, 125;
  41,  76, 111, 146, 181, 216;

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

Cf. A000578, A028387, A159797, A159798, A162610, A162611, A162613.

Table[NestList[#+n^2-1&,n^2+n-1,n-1],{n,20}]//Flatten (* Harvey P. Dale, Mar 02 2024 *)

T(n,k)=n+k*(n^2-1) \\ Franklin T. Adams-Watters, Aug 06 2009

Sum_{k=1..n} T(n,k)= n*(n^3 + n^2 + n - 1)/2 (row sums). - R. J. Mathar, Jul 20 2009

More terms from Franklin T. Adams-Watters, Aug 06 2009

A162613 Triangle read by rows in which row n lists n terms, starting with n, with gaps = n^2-1 between successive terms.

Original entry on oeis.org

1, 2, 5, 3, 11, 19, 4, 19, 34, 49, 5, 29, 53, 77, 101, 6, 41, 76, 111, 146, 181, 7, 55, 103, 151, 199, 247, 295, 8, 71, 134, 197, 260, 323, 386, 449, 9, 89, 169, 249, 329, 409, 489, 569, 649, 10, 109, 208, 307, 406, 505, 604, 703, 802, 901, 11, 131, 251, 371, 491, 611

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is A100104(n).

			Triangle begins:
  1;
  2,   5;
  3,  11,  19;
  4,  19,  34,  49;
  5,  29,  53,  77, 101;
  6,  41,  76, 111, 146, 181;

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

Cf. A000578, A100104, A159797, A159798, A162610, A162611, A162612.

Cf. A100855 (row sums). - R. J. Mathar, Jul 20 2009

Table[NestList[#+n^2-1&,n,n-1],{n,11}]//Flatten (* Harvey P. Dale, Feb 24 2016 *)

More terms from Vincenzo Librandi, Aug 02 2010

A100104 a(n) = n^3 - n^2 + 1.

Original entry on oeis.org

1, 1, 5, 19, 49, 101, 181, 295, 449, 649, 901, 1211, 1585, 2029, 2549, 3151, 3841, 4625, 5509, 6499, 7601, 8821, 10165, 11639, 13249, 15001, 16901, 18955, 21169, 23549, 26101, 28831, 31745, 34849, 38149, 41651, 45361, 49285, 53429, 57799, 62401, 67241, 72325

Offset: 0

N. J. A. Sloane, Jan 12 2005

Appears to be the number of possible distinct sums of a set of n distinct integers between 1 and n^2. Checked up to n=6. - Dylan Hamilton, Sep 21 2010

a(n) = A100104(n+1) - A100104(n). - Reinhard Zumkeller, Jul 07 2012

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A162611. - Vincenzo Librandi, May 27 2010

Cf. A049451 (first differences).

a049451 n = n * (3 * n + 1)  -- Reinhard Zumkeller, Jul 07 2012

f[n_]:=n^3-n^2+1;Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Array[#^3-#^2+1&,50,0] (* or *) LinearRecurrence[{4,-6,4,-1},{1,1,5,19},50] (* Harvey P. Dale, Sep 11 2011 *)

a(n)=n^3-n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

From Harvey P. Dale, Sep 11 2011: (Start)
a(0)=1, a(1)=1, a(2)=5, a(3)=19, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x^3+7*x^2-3*x+1)/(x-1)^4. (End)

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

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.

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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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A162612 Triangle read by rows in which row n lists n terms, starting with n^2+n-1, with gaps = n^2-1 between successive terms.

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A162613 Triangle read by rows in which row n lists n terms, starting with n, with gaps = n^2-1 between successive terms.

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A100104 a(n) = n^3 - n^2 + 1.

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