A162614 -id:A162614 - 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

A162611 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^2 - 1.

Original entry on oeis.org

0, 1, 1, 2, 5, 8, 3, 11, 19, 27, 4, 19, 34, 49, 64, 5, 29, 53, 77, 101, 125, 6, 41, 76, 111, 146, 181, 216, 7, 55, 103, 151, 199, 247, 295, 343, 8, 71, 134, 197, 260, 323, 386, 449, 512, 9, 89, 169, 249, 329, 409, 489, 569, 649, 729, 10, 109, 208, 307, 406, 505, 604

Offset: 0

Omar E. Pol, Jul 09 2009

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

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

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

A163284 Triangle read by rows in which row n lists n+1 terms, starting with n^4 and ending with n^5, such that the difference between successive terms is equal to n^4 - n^3.

Original entry on oeis.org

0, 1, 1, 16, 24, 32, 81, 135, 189, 243, 256, 448, 640, 832, 1024, 625, 1125, 1625, 2125, 2625, 3125, 1296, 2376, 3456, 4536, 5616, 6696, 7776, 2401, 4459, 6517, 8575, 10633, 12691, 14749, 16807, 4096, 7680, 11264, 14848, 18432, 22016, 25600, 29184, 32768

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000583(n) and the last term of row n is A000584(n).

			Triangle begins:
0;
1,1;
16,24,32;
81,135,189,243;
256,448,640,832,1024;
625,1125,1625,2125,2625,3125;
1296,2376,3456,4536,5616,6696,7776;
2401,4459,6517,8575,10633,12691,14749,16807;
4096,7680,11264,14848,18432,22016,25600,29184,32768;
6561,12393,18225,24057,29889,35721,41553,47385,53217,59049;
10000,19000,28000,37000,46000,55000,64000,73000,82000,91000,100000;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000583, A000584, A085537, A159797, A162611, A162614, A162622, A163282, A163283, A163285.

Table[n^4 + k*(n^4 - n^3), {n,0,15}, {k,0,n}] // Flatten (* G. C. Greubel, Dec 17 2016 *)

A163284(n, k)=n^4 +k*(n^4 -n^3) \\ G. C. Greubel, Dec 17 2016

A163285 Triangle read by rows in which row n lists n+1 terms, starting with n^5 and ending with n^6, such that the difference between successive terms is equal to n^5 - n^4.

Original entry on oeis.org

0, 1, 1, 32, 48, 64, 243, 405, 567, 729, 1024, 1792, 2560, 3328, 4096, 3125, 5625, 8125, 10625, 13125, 15625, 7776, 14256, 20736, 27216, 33696, 40176, 46656, 16807, 31213, 45619, 60025, 74431, 88837, 103243, 117649, 32768, 61440, 90112, 118784, 147456

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000584(n) and the last term of row n is A001014(n).
The main entry for this sequence is A159797. See also A163282, A163283 and A163284.
Row sums give A163275. - Omar E. Pol, Mar 18 2012

			Triangle begins:
0;
1,1;
32,48,64;
243,405,567,729;
1024,1792,2560,3328,4096;
3125,5625,8125,10625,13125,15625;
7776,14256,20736,27216,33696,40176,46656;
16807,31213,45619,60025,74431,88837,103243,117649;
32768,61440,90112,118784,147456,176128,204800,233472,262144;
59049,111537,164025,216513,269001,321489,373977,426465,478953,531441;
100000,190000,280000,370000,460000,550000,640000,730000,820000,910000,1000000;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000584, A001014, A085538, A159797, A162611, A162614, A162622, A163282, A163283, A163284.

rw[n_]:=Range[n^5,n^6,n^5-n^4]; Join[{0,1},Flatten[Array[rw,10]]] (* Harvey P. Dale, Mar 18 2012 *)

A163285(n, k)=n^5 +k*(n^5 -n^4) \\ G. C. Greubel, Dec 17 2016

A162608 Triangle read by rows in which row n lists n+1 terms, starting with n!, such that the difference between successive terms is also equal to n!.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 6, 12, 18, 24, 24, 48, 72, 96, 120, 120, 240, 360, 480, 600, 720, 720, 1440, 2160, 2880, 3600, 4320, 5040, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880

Offset: 0

Omar E. Pol, Jul 22 2009

Note that the last term of the n-th row is the factorial of (n+1) = (n+1)! = A000142(n+1).
Sequence A178883 (with shape A000041) is a "refinement" of Table A162608; as expected, both sequences have row sums A001710(n+2). - Alford Arnold, Sep 28 2010
From Dennis P. Walsh, May 18 2020: (Start)
T(n,k) provides the number of length (n+2) permutations with elements 1 and 2 as cycle-mates in a (k+1)-cycle. We note that 1 and 2 are cycle-mates if they are elements of the same cycle in the permutation.
For example, T(3,2) counts the 12 permutations of length 5 that have 1 and 2 in the same 3 cycle, namely, (1 2 3)(4)(5), (1 3 2)(4)(5), (1 2 3)(4 5), (1 3 2)(4 5), (1 2 4)(3)(5), (1 4 2)(3)(5), (1 2 4)(3 5), (1 4 2)(3 5),(1 2 5)(3)(4), (1 5 2)(3)(4), (1 2 5)(3 4), and (1 5 2)(3 4).
Note that there are binomial(n,k-1) ways to choose the other (k-1) cycle-mates of 1 and 2 in the (k+1)-cycle and then k! different (k+1)-cycles with these elements. Since there are (n+1-k)! ways to permute the remaining elements, we obtain T(n,k) = (n+1-k)!*k!*binomial(n,k-1) = n!*k. (End)

			Triangle begins:
1;
1,     2;
2,     4,     6;
6,     12,    18,     24;
24,    48,    72,     96,     120;
120,   240,   360,    480,    600,    720;
720,   1440,  2160,   2880,   3600,   4320,   5040;
5040,  10080, 15120,  20160,  25200,  30240,  35280,  40320;
40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880;
362880,725760,1088640,1451520,1814400,2177280,2540160,2903040,3265920,3628800;
...
Observation: It appears that rows sums = A001710(n+2).

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A000142, A001710, A051683, A159797, A162611, A162614, A162622.

Cf. A178883. - Alford Arnold, Sep 28 2010

a162608 n k = a162608_tabl !! n !! k
a162608_row n = a162608_tabl !! n
a162608_tabl = map fst $ iterate f ([1], 1) where
   f (row, n) = (row' ++ [head row' + last row'], n + 1) where
     row' = map (* n) row
-- Reinhard Zumkeller, Mar 09 2012

/* As triangle */ [[Factorial(n)*k: k in [1..n+1]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 04 2015

Table[k n!, {n, 0, 8}, {k, n + 1}] // Flatten (* Michael De Vlieger, Jul 03 2015 *)

From Robert Israel, Jul 03 2015: (Start)
T(n,k) = n!*k, k = 1 .. n+1.
T(n+1,k) = (n+1)*T(n,k).
T(n,k+1) = T(n,k)+T(n,1). (End)

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

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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A162611 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^2 - 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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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).

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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).

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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).

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A163284 Triangle read by rows in which row n lists n+1 terms, starting with n^4 and ending with n^5, such that the difference between successive terms is equal to n^4 - n^3.

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