login
Triangle read by rows, T(n,k) = 2*n, with n>=k>=0.
1

%I #40 Oct 09 2018 15:47:28

%S 0,2,2,4,4,4,6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,12,12,12,12,12,12,12,

%T 14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,

%U 18,18,18,18,20,20,20,20,20,20,20,20,20,20,20,22,22,22,22,22,22,22,22,22,22,22,22

%N Triangle read by rows, T(n,k) = 2*n, with n>=k>=0.

%C 2*n appears n+1 times in row n.

%F a(n) = 2*floor((sqrt(1+8*n)-1)/2).

%F a(n) = 2*A003056(n) = 2*(A002024(n+1)-1).

%e Triangle begins:

%e 0;

%e 2, 2;

%e 4, 4, 4;

%e 6, 6, 6, 6;

%e 8, 8, 8, 8, 8;

%e 10, 10, 10, 10, 10, 10;

%e 12, 12, 12, 12, 12, 12, 12;

%e 14, 14, 14, 14, 14, 14, 14, 14;

%e 16, 16, 16, 16, 16, 16, 16, 16, 16;

%e 18, 18, 18, 18, 18, 18, 18, 18, 18, 18;

%e 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20;

%e 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22;

%e ...

%t Table[ConstantArray[2 n, n + 1], {n, 0, 11}] // Flatten (* _Michael De Vlieger_, Dec 22 2016 *)

%t Table[#,#/2+1]&/@(2*Range[0,20])//Flatten(* _Harvey P. Dale_, Oct 09 2018 *)

%o (PARI) row(n) = vector(n, x, 2*(n-1))

%o trianglerows(n) = for(k=1, n, print(row(k)))

%o /* Print initial 12 rows of triangle as follows: */

%o trianglerows(12) \\ _Felix Fröhlich_, Dec 22 2016

%Y Twice A003056.

%Y Row sums give A046092.

%Y Where records occur gives A000217.

%Y Column 0, and the right border: A005843.

%Y Also row lengths of the following triangles: A237593, A262045, A262048, A279693.

%Y Cf. A002024.

%K nonn,tabl,easy

%O 0,2

%A _Omar E. Pol_, Dec 22 2016