A159798 -id:A159798 - 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

A162610 Triangle read by rows in which row n lists n terms, starting with 2n-1, with gaps = n-1 between successive terms.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 7, 10, 13, 16, 9, 13, 17, 21, 25, 11, 16, 21, 26, 31, 36, 13, 19, 25, 31, 37, 43, 49, 15, 22, 29, 36, 43, 50, 57, 64, 17, 25, 33, 41, 49, 57, 65, 73, 81, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121

Offset: 1

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th square A000290(n).

Row sums are n*(n^2+2*n-1)/2, apparently in A127736. - R. J. Mathar, Jul 20 2009

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

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

Cf. A000027, A000290, A159797, A159798.

Cf. A209297; A005408 (left edge), A000290 (right edge), A127736 (row sums), A056220 (central terms), A026741 (number of odd terms per row), A142150 (number of even terms per row), A221491 (number of primes per row).

a162610 n k = k * n - k + n
a162610_row n = map (a162610 n) [1..n]
a162610_tabl = map a162610_row [1..]
-- Reinhard Zumkeller, Jan 19 2013

Flatten[Table[NestList[#+n-1&,2n-1,n-1], {n,15}]] (* Harvey P. Dale, Oct 20 2011 *)

# From R. J. Mathar, Oct 20 2009
def A162610(n, k):
    return 2*n-1+(k-1)*(n-1)
print([A162610(n,k) for n in range(1,20) for k in range(1,n+1)])

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

T(n,k) = (k+1)*(n-1)+1. - Reinhard Zumkeller, Jan 19 2013

More terms from R. J. Mathar, Oct 20 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

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

A159788 a(n) = A159786(n+1)/2.

Original entry on oeis.org

0, 0, 2, 4, 4, 6, 16, 24, 24, 26, 32, 36, 38, 52, 88, 112, 112, 114, 120, 124, 126, 140, 168, 184, 186, 196, 212, 222, 240, 304, 432, 480, 480, 482, 488, 492, 494, 508, 536, 552, 554, 564, 580, 590, 608, 672, 768, 816, 818, 828

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 2009

nonn

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

Toothpick sequence: A139250.

Cf. A139251, A152998, A159786, A159787, A159789, A159798.

a(11)-a(49) from Robert Price, May 10 2019

A162607 a(n) = n(n^2 - 4n + 5)/2.

Original entry on oeis.org

0, 1, 1, 3, 10, 25, 51, 91, 148, 225, 325, 451, 606, 793, 1015, 1275, 1576, 1921, 2313, 2755, 3250, 3801, 4411, 5083, 5820, 6625, 7501, 8451, 9478, 10585, 11775, 13051, 14416, 15873, 17425, 19075, 20826, 22681, 24643, 26715, 28900, 31201, 33621

Offset: 0

R. J. Mathar and Omar E. Pol, Jul 21 2009

Positive values are the row sums of triangle A159798.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A159798.

[n*(n^2 - 4*n + 5)/2: n in [0..50]]; // Vincenzo Librandi, Dec 19 2012

f[n_]:=(n^3-n^2)/2+1; Table[f[n],{n,-1,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
CoefficientList[Series[x*(1 - 3*x + 5*x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 19 2012 *)

for(n=0,50, print1(n*(n^2-4*n+5)/2, ", ")) \\ G. C. Greubel, Apr 21 2018

G.f.: x*(1 - 3*x + 5*x^2)/(1 - x)^4. - Vincenzo Librandi, Dec 19 2012
a(n) = A057145(n-1,n) = A072277(n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: x*(x^2 - x + 2)*exp(x)/2. - G. C. Greubel, Apr 21 2018

A162619 Triangle read by rows in which row n lists n consecutive natural numbers A000027, starting with A014689(n) = A000040(n)-n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 14, 15, 16, 11, 12, 13, 14, 15, 16, 17, 18, 14, 15, 16, 17, 18, 19, 20, 21, 22, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 25, 26, 27, 28, 29, 30, 31, 32, 33

Offset: 1

Omar E. Pol, Jul 10 2009

Note that the last term of the n-th row is A000040(n)-1 = A006093(n).

A162618 Triangle read by rows in which row n lists n consecutive natural numbers A000027, starting with A008578(n-1) - n + 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 3, 4, 5, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 15, 16, 17, 11, 12, 13, 14, 15, 16, 17, 18, 19, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30

Offset: 1

Omar E. Pol, Jul 10 2009

Note that the last term of the n-th row is the noncomposite number A008578(n-1).

			Contribution from _Omar E. Pol_, Jul 15 2009: (Start)
Triangle begins:
   1;
   1,  2;
   1,  2,  3;
   2,  3,  4,  5;
   3,  4,  5,  6,  7;
   6,  7,  8,  9, 10, 11;
   7,  8,  9, 10, 11, 12, 13;
  10, 11, 12, 13, 14, 15, 16, 17;
  11, 12, 13, 14, 15, 16, 17, 18, 19;
  14, 15, 16, 17, 18, 19, 20, 21, 22, 23;
  19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29;
  20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31;
(End)

Cf. A000040, A008578, A159797, A159798, A162611, A162619, A162620.

Cf. A162614, A162622. - Omar E. Pol, Jul 15 2009

A162620 Triangle read by rows in which row n lists n consecutive natural numbers A000027, starting with A000040(n)-n+1.

Original entry on oeis.org

2, 2, 3, 3, 4, 5, 4, 5, 6, 7, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12, 13, 11, 12, 13, 14, 15, 16, 17, 12, 13, 14, 15, 16, 17, 18, 19, 15, 16, 17, 18, 19, 20, 21, 22, 23, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 26, 27, 28, 29, 30, 31, 32, 33

Offset: 1

Omar E. Pol, Jul 10 2009

Note that the last term of the n-th row is the n-th prime A000040(n).

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A162610 Triangle read by rows in which row n lists n terms, starting with 2n-1, with gaps = n-1 between successive terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A159788 a(n) = A159786(n+1)/2.

Views

Author

Keywords

Links

Crossrefs

Extensions

A162607 a(n) = n*(n^2 - 4*n + 5)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A162607 a(n) = n(n^2 - 4n + 5)/2.