A377187 Triangle read by rows: T(n,k) = numerator((n^2 + k)/(n^2 - k)).

3, 11, 2, 9, 19, 5, 27, 14, 29, 3, 19, 13, 5, 41, 7, 51, 26, 53, 27, 55, 4, 33, 67, 17, 69, 35, 71, 9, 83, 14, 85, 43, 29, 44, 89, 5, 51, 103, 13, 21, 53, 107, 27, 109, 11, 123, 62, 125, 63, 127, 64, 129, 65, 131, 6, 73, 49, 37, 149, 25, 151, 19, 17, 77, 155, 13

Offset: 2

Author

Stefano Spezia, Oct 19 2024

Keywords

• nonn
• easy
• frac
• tabl

Examples

			The triangle begins as:
   3;
  11,   2;
   9,  19,  5;
  27,  14, 29,  3;
  19,  13,  5, 41,  7;
  51,  26, 53, 27, 55,   4;
  33,  67, 17, 69, 35,  71,  9;
  83,  14, 85, 43, 29,  44, 89,   5;
  51, 103, 13, 21, 53, 107, 27, 109, 11;
  ...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.

Crossrefs

Cf. A001113, A026741, A063656, A164900, A377188 (denominator).

Programs

Mathematica

T[n_,k_]:=Numerator[(n^2+k)/(n^2-k)]; Table[T[n,k],{n,2,12},{k,2,n}]//Flatten

Python

from math import isqrt, comb, gcd
def A377187(n): return (d:=(a:=(m:=isqrt(k:=n-1<<1))+(k>m*(m+1))+1)**2+(b:=n-comb(a-1,2)))//gcd(d,d-(b<<1)) # Chai Wah Wu, Nov 12 2024

Formula

Limit_{n->oo} Product_{k=1..n} T(n,k)/A377188(n,k) = e = A001113 (see Finch).

T(n,n) = A026741(n+1).

T(n,2) = A164900(n-1).