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Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.
3

%I #24 Dec 09 2024 15:55:13

%S 1,1,1,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,

%T 1,2,1,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,

%U 1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,2,1,1,2,1,2,1,1

%N Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.

%H Alois P. Heinz, <a href="/A214856/b214856.txt">Table of n, a(n) for n = 0..10000</a>

%e 10, 15 are in interval ]9, 16] , a(4) = 2.

%o (PARI) a(n) = if (n, sum(i=(n-1)^2+1, n^2, ispolygonal(i, 3)), 1); \\ _Michel Marcus_, Nov 12 2022

%o (Python)

%o from math import isqrt

%o def A214856(n): return (isqrt((m:=n**2<<3)+8)+1>>1)-(isqrt(m-(n-1<<4))+1>>1) if n else 1 # _Chai Wah Wu_, Dec 09 2024

%Y Cf. A000217, A000290, A214848, A214857.

%K nonn,easy

%O 0,5

%A _Philippe Deléham_, Mar 08 2013