A163285 -id:A163285 - 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

A163275 a(n) = n^5*(n+1)^2/2.

Original entry on oeis.org

0, 2, 144, 1944, 12800, 56250, 190512, 537824, 1327104, 2952450, 6050000, 11595672, 21026304, 36386714, 60505200, 97200000, 151519232, 230016834, 341067024, 495219800, 705600000, 988352442, 1363135664, 1853666784

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163285.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A006002, A099903, A163102, A163274, A163276, A163277, A163285.

A163275 := proc(n) n^5*(n+1)^2/2 ; end proc: seq(A163275(n),n=0..60) ; # R. J. Mathar, Feb 05 2010

Table[(1/2)*n^5*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{8,-28,56, -70,56,-28,8,-1}, {0,2,144,1944,12800,56250,190512,537824}, 50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1+64*x+424*x^2+584*x^3+179*x^4+8*x^5)/(x-1)^8 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
G.f.: 2*x*(1 + 64*x + 424*x^2 + 584*x^3 + 179*x^4  +8*x^5)/(x-1)^8. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 12 -5*Pi^2/3 - 2*Pi^4/45 + 6*zeta(3) + 2*zeta(5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 20*log(2) + 9*zeta(3)/2 + 15*zeta(5)/8 - 12 - Pi^2/2 - 7*Pi^4/180. (End)

Extended by R. J. Mathar, Feb 05 2010

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.

Original entry on oeis.org

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

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

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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A163275 a(n) = n^5*(n+1)^2/2.

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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.

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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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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.

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