A266579 Numerator of the continued fraction [Sum_{k=0..n} k; 1, 2, 3,..., n].

2, 11, 67, 460, 3532, 30225, 286289, 2979896, 33852226, 417123475, 5543942107, 79086006756, 1205573749892, 19561113090785, 336643494142657, 6125614986385360, 117514626855080914, 2370682022353448571, 50173196512398036851, 1111614380526424428380

Offset: 1

Author

Ilya Gutkovskiy, May 07 2016

Keywords

• nonn
• frac

Examples

			2, 11/3, 67/10, 460/43, 3532/225, 30225/1393, 286289/9976, 2979896/81201, 33852226/740785, 417123475/7489051, 5543942107/83120346, 79086006756/1004933203, 1205573749892/13147251985, 19561113090785/185066460993,...
a(10) = 417123475 because 55+1/(1+1/(2+1/(3+1/(4+1/(5+1/(6+1/(7+1/(8+1/(9+1/10))))))))) = 417123475/7489051 and 1+2+3+4+5+6+7+8+9+10 = 55.
		

Links

• Ilya Gutkovskiy, Extended example
• Eric Weisstein's World of Mathematics, Continued Fraction Constant
• Eric Weisstein's World of Mathematics, Triangular Number

Crossrefs

Cf. A000217, A001040 (denominator, offset 2), A001053, A052119.

Programs

Mathematica

Table[Numerator[n ((n + 1)/2) + ContinuedFractionK[1, k, {k, n}]], {n, 20}]

Formula

a(n) = A001040(n+1)*A000217(n) + A001053(n+1).