A294317 - OEIS

%I #51 Sep 14 2024 11:47:00

%S 0,2,1,4,3,2,6,5,4,3,8,7,6,5,4,10,9,8,7,6,5,12,11,10,9,8,7,6,14,13,12,

%T 11,10,9,8,7,16,15,14,13,12,11,10,9,8,18,17,16,15,14,13,12,11,10,9,20,

%U 19,18,17,16,15,14,13,12,11,10

%N Triangle read by rows: T(n, k) = 2*n-k, k <= n.

%H Muniru A Asiru, <a href="/A294317/b294317.txt">Table of n, a(n) for n = 0..100000</a>

%H Thomas H. Sidebotham, <a href="https://books.google.com/books?id=VsAZa5PWLz8C&amp;pg=PA181">The A to Z of Mathematics: A Basic Guide</a>, John Wiley & Sons, p. 181, (2003) ISBN 9780471461630.

%F T(n, k) = 2*n - k, 0 <= k <= n.

%F T(n, 2*m) = A005843(n), n >= m >= 0 (even-indexed columns).

%F T(n, 2*m+1) = A005408(n), n >= m >= 0 (odd-indexed columns).

%F T(n, n-m) = A001477(n+m), n >= m >= 0 (diagonals m >= 0).

%F 2*A287326(n, k) = A287326(T(n, k),k) + A287326(T(n, k),0).

%F G.f.: x*(2 + y - 3*x*y)/((1 - x)^2*(1 - x*y)^2). - _Stefano Spezia_, Sep 14 2024

%e Triangle begins:

%e    0;

%e    2,  1;

%e    4,  3,  2;

%e    6,  5,  4,  3;

%e    8,  7,  6,  5,  4;

%e   10,  9,  8,  7,  6,  5;

%e   12, 11, 10,  9,  8,  7,  6;

%e   14, 13, 12, 11, 10,  9,  8,  7;

%e   16, 15, 14, 13, 12, 11, 10,  9,  8;

%e   18, 17, 16, 15, 14, 13, 12, 11, 10, 9;

%e   20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10;

%e   ...

%t Column[Table[2*n-k, {n, 0, 10}, {k, 0, n}], Center]

%o (Python)

%o def f(x):

%o     a=[]

%o     for k in range(x):

%o         for m in range (k+1):

%o             a.append(2*k-m)

%o     return a

%o print(f(10))

%o (GAP) A294317 := Flat(List([0..149],n->List([0..n],k->2*n-k))); # _Muniru A Asiru_, Dec 29 2017

%Y Cf. A001477, A005408, A005843, A287326.

%K nonn,tabl,easy

%O 0,2

%A _Kolosov Petro_, Oct 28 2017

%E Edited by _Wolfdieter Lang_, Dec 08 2017