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

0,4

Comments

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

Links

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

Formula

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

Example

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;

Maple

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

Mathematica

Flatten[Table[NestList[#+n-1&, n, n], {n, 0, 12}]] (* Harvey P. Dale, Aug 04 2014 *)

Prog

(GNU bc) scale=0; for(n=0; n<76; n++){m=(sqrt(8*n+1)-1)/2; print m+(n-m*(m+1)/2)*(m-1), ", "}; print"\n" /* Carl R. White, Jul 24 2010 */

Crossrefs

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)

Sequence in context: A329895 A342552 A078908 * A152920 A361442 A288778

Adjacent sequences: A159794 A159795 A159796 * A159798 A159799 A159800

Keyword

easy,nonn,tabl

Author

Omar E. Pol, Jul 09 2009

Extensions

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

Status

approved