A345018 - OEIS

%I #31 Nov 05 2024 09:13:43

%S 1,2,3,4,5,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,

%T 19,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,

%U 28,29,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41

%N For each n, append to the sequence n^2 consecutive integers, starting from n.

%C Irregular triangle read by rows T(n,k) in which row n lists the integers from n to n + n^2 - 1, with n >= 1.

%H Paolo Xausa, <a href="/A345018/b345018.txt">Table of n, a(n) for n = 1..10416</a> (rows 1..31 of the triangle, flattened)

%F T(n,k) = n + k - 1, with n >= 1 and 1 <= k <= n^2.

%e Written as an irregular triangle T(n,k) the sequence begins:

%e   n\k|  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 ...

%e   ---+---------------------------------------------------------------

%e    1 |  1;

%e    2 |  2,  3,  4,  5;

%e    3 |  3,  4,  5,  6,  7,  8,  9, 10, 11;

%e    4 |  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19;

%e   ...

%p T:= n-> (t-> seq(n+i, i=0..t-1))(n^2):

%p seq(T(n), n=1..6);  # _Alois P. Heinz_, Nov 05 2024

%t Table[Range[n,n^2+n-1],{n,6}] (* _Paolo Xausa_, Sep 05 2023 *)

%o (PARI) row(n) = vector(n^2, k, n+k-1); \\ _Michel Marcus_, Jun 08 2021

%o (Python)

%o from sympy import integer_nthroot

%o def A345018(n): return n-1+(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k*(3-(k<<1))+5)//6 # _Chai Wah Wu_, Nov 05 2024

%Y Column 1: A000027.

%Y Right border: A028387.

%Y Row lengths: A000290.

%Y Row sums: A255499.

%Y Cf. A064866, A074279.

%K nonn,tabf

%O 1,2

%A _Paolo Xausa_, Jun 05 2021