Borwein*2, Exercise 3.7.10 (at least my old edition): "A remarkable elementary result is Sum[1/(Fibonacci[1 + 2*k] + Fibonacci[1 + 2*n]), {n, 0, Infinity}] == (Sqrt[5]*(1 + 2*k))/ (2*Fibonacci[1 + 2*k]) [. . .] A related formula is [...]" which is just the k=3 case of the first formula, but with a different, and correct rhs! I.e., the first formula seems completely wrong, Empirically, I get Sum[1/(Fibonacci[1 + 2*k] + Fibonacci[1 + 2*n]), {n, 0, Infinity}] == ((1 + GoldenRatio^(2 + 4*k))* (1 + 2*k))/(2*(-1 + GoldenRatio^(2 + 4*k))* Fibonacci[1 + 2*k]) The rhs might have a slick reexpression in fibonaccis. --rwg (Recall our old sum 1/fib(n) saga, where B&B seem to cop out and give only a Lambert series, but reading between the lines, there's a hairy closed form in terms of d/dq log qpochhammers.)