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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.
11

%I #14 Sep 08 2022 08:45:46

%S 0,1,1,2,17,32,3,83,163,243,4,259,514,769,1024,5,629,1253,1877,2501,

%T 3125,6,1301,2596,3891,5186,6481,7776,7,2407,4807,7207,9607,12007,

%U 14407,16807,8,4103,8198,12293,16388,20483,24578,28673,32768,9,6569,13129

%N 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.

%C Note that the last term of the n-th row is the 5th power of n, A000584(n).

%C See also the triangles of A162623 and A162624.

%H Harvey P. Dale, <a href="/A162622/b162622.txt">Table of n, a(n) for n = 0..1000</a>

%F Sum_{k=0..n} T(n,k) = n*(n+1)*(1+n^4)/2 (row sums). [_R. J. Mathar_, Jul 20 2009]

%e Triangle begins:

%e 0;

%e 1, 1;

%e 2, 17, 32;

%e 3, 83, 163, 243;

%e 4, 259, 514, 769, 1024;

%e 5, 629, 1253, 1877, 2501, 3125;

%e 6, 1301, 2596, 3891, 5186, 6481, 7776;

%e 7, 2407, 4807, 7207, 9607, 12007, 14407, 16807;

%e 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768;

%e 9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 59049; etc.

%p 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

%t Flatten[Table[NestList[#+n^4-1&,n,n],{n,0,9}]] (* _Harvey P. Dale_, Jun 23 2013 *)

%o (Magma) /* Triangle: */ [[n+k*(n^4-1): k in [0..n]]: n in [0..10]]; // _Bruno Berselli_, Dec 14 2012

%Y Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162623, A162624.

%K nonn,easy,tabl

%O 0,4

%A _Omar E. Pol_, Jul 15 2009

%E 7th and later rows from _R. J. Mathar_, Feb 11 2010