A370291 - OEIS

%I #21 Feb 17 2024 04:04:27

%S 0,1,3,3,6,6,6,6,6,10,10,10,10,10,10,10,10,10,10,10,10,10,10,15,15,15,

%T 15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,

%U 15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,21,21,21,21,21

%N Triangular number T(n) = A000217(n) occurs C(n) = A000108(n) times.

%H Paolo Xausa, <a href="/A370291/b370291.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A000217(A072643(n)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1/2)^(n-1)/(2^n-1) = 0.86233289403022175171... . - _Amiram Eldar_, Feb 17 2024

%e Written as a triangle:

%e    0;

%e    1;

%e    3,  3;

%e    6,  6,  6,  6,  6;

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

%e   ...

%p T:= n-> n*(n+1)/2$binomial(2*n,n)/(n+1):

%p seq(T(n), n=0..5);  # _Alois P. Heinz_, Feb 16 2024

%t Flatten[Array[Table[PolygonalNumber[#], CatalanNumber[#]] &, 7, 0]]

%Y Cf. A000108, A000217, A072643.

%Y Row sums of A370221 (for n >= 1).

%Y Row sums as triangle give A002457(n-1) for n>=1.

%K nonn,easy,tabf

%O 0,3

%A _Paolo Xausa_, Feb 14 2024