A162611 -id:A162611 - cp's OEIS frontend

A163282 Triangle read by rows in which row n lists n+1 terms, starting with n^2 and ending with n^3, such that difference between successive terms is equal to n^2 - n.

0, 1, 1, 4, 6, 8, 9, 15, 21, 27, 16, 28, 40, 52, 64, 25, 45, 65, 85, 105, 125, 36, 66, 96, 126, 156, 186, 216, 49, 91, 133, 175, 217, 259, 301, 343, 64, 120, 176, 232, 288, 344, 400, 456, 512, 81, 153, 225, 297, 369, 441, 513, 585, 657, 729, 100, 190, 280, 370, 460, 550

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000290(n) and the last term of row n is A000578(n).

			Triangle begins:
    0;
    1,   1;
    4,   6,   8;
    9,  15,  21,  27;
   16,  28,  40,  52,  64;
   25,  45,  65,  85, 105, 125;
   36,  66,  96, 126, 156, 186, 216;
   49,  91, 133, 175, 217, 259, 301, 343;
   64, 120, 176, 232, 288, 344, 400, 456, 512;
   81, 153, 225, 297, 369, 441, 513, 585, 657, 729;
  100, 190, 280, 370, 460, 550, 640, 730, 820, 910, 1000;

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

Cf. A000290, A000578, A002378, A159797, A162611, A163283, A163284, A163285.

/* As triangle: */ [[n^2+k*(n^2-n): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 13 2016

T[n_, k_] := n^2 + k*(n^2 - n); Table[T[n, k], {n,0,10}, {k,0,n}] //Flatten (* G. C. Greubel, Dec 13 2016 *)
Join[{0,1},Table[Range[n^2,n^3,n^2-n],{n,10}]]//Flatten (* Harvey P. Dale, Sep 09 2019 *)

A163282(n,k)=n^2+k*(n^2-n) \\ Michael B. Porter, Feb 25 2010

T(n, k) = n^2 + k*(n^2 - n), for 0 <= k <= n, n>= 0. - G. C. Greubel, Dec 13 2016

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

A163283 Triangle read by rows in which row n lists n+1 terms, starting with n^3 and ending with n^4, such that the difference between successive terms is equal to n^3 - n^2.

Original entry on oeis.org

0, 1, 1, 8, 12, 16, 27, 45, 63, 81, 64, 112, 160, 208, 256, 125, 225, 325, 425, 525, 625, 216, 396, 576, 756, 936, 1116, 1296, 343, 637, 931, 1225, 1519, 1813, 2107, 2401, 512, 960, 1408, 1856, 2304, 2752, 3200, 3648, 4096, 729, 1377, 2025, 2673, 3321, 3969

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000578(n) and the last term of row n is A000583(n).

			Triangle begins:
0;
1,    1;
8,    12,   16;
27,   45,   63,   81;
64,   112,  160,  208,  256;
125,  225,  325,  425,  525,  625;
216,  396,  576,  756,  936,  1116, 1296;
343,  637,  931,  1225, 1519, 1813, 2107, 2401;
512,  960,  1408, 1856, 2304, 2752, 3200, 3648, 4096;
729,  1377, 2025, 2673, 3321, 3969, 4617, 5265, 5913, 6561;
1000, 1900, 2800, 3700, 4600, 5500, 6400, 7300, 8200, 9100, 10000;
...

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

Row sums: A099903.

Cf. A000578, A000583, A045991, A159797, A162611, A162614, A163282, A163284, A163285.

Table[n^3 + k*(n^3 - n^2), {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 13 2016 *)

A163283(n, k)=n^3 +k*(n^3 -n^2) \\ G. C. Greubel, Dec 13 2016

T(n, k) = n^3 + k*(n^3 - n^2), for 0 <= k <= n, n >= 0. - G. C. Greubel, Dec 13 2016

Edited by Omar E. Pol, Jul 25 2009

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).

A177058 a(n) = n^3 - 3n^2 + 3.

Original entry on oeis.org

3, 1, -1, 3, 19, 53, 111, 199, 323, 489, 703, 971, 1299, 1693, 2159, 2703, 3331, 4049, 4863, 5779, 6803, 7941, 9199, 10583, 12099, 13753, 15551, 17499, 19603, 21869, 24303, 26911, 29699, 32673, 35839, 39203, 42771, 46549, 50543, 54759, 59203

Offset: 0

Vincenzo Librandi, May 27 2010

For n>2, fourth diagonal of A162611.

B. Berselli, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A162611, A081438.

Table[n^3-3n^2+3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{3,1,-1,3},50] (* Harvey P. Dale, May 15 2020 *)

a(n)=n^3-3*n^2+3 \\ Charles R Greathouse IV, Jan 11 2012

From Bruno Berselli, Jun 04 2010: (Start)
G.f.: (3-11*x+13*x^2+x^3)/(1-x)^4.
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4) = 0, with n>3.
a(n)+a(n-1) = 2*A081438(n-3), with n>2. (End)
G.f.: 3+x+x^2*G(0) where G(k) = 1 - x*(k+1)*(k+1)*(k+4)/(1 - 1/(1 - (k+1)*(k+1)*(k+4)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012

cp's OEIS Frontend

A163282 Triangle read by rows in which row n lists n+1 terms, starting with n^2 and ending with n^3, such that difference between successive terms is equal to n^2 - n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A163283 Triangle read by rows in which row n lists n+1 terms, starting with n^3 and ending with n^4, such that the difference between successive terms is equal to n^3 - n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples