A172170 1 followed by the duplicated entries of A090368.

1, 1, 1, 3, 3, 5, 5, 7, 7, 3, 3, 11, 11, 13, 13, 3, 3, 17, 17, 19, 19, 3, 3, 23, 23, 5, 5, 3, 3, 29, 29, 31, 31, 3, 3, 5, 5, 37, 37, 3, 3, 41, 41, 43, 43, 3, 3, 47, 47, 7, 7, 3, 3, 53, 53, 5, 5, 3, 3, 59, 59, 61, 61, 3, 3, 5, 5, 67, 67, 3, 3, 71, 71, 73, 73, 3, 3, 7, 7, 79, 79, 3, 3, 83, 83, 5, 5

Offset: 0

Author

Paul Curtz, Jan 28 2010

Keywords

• nonn

Comments

We start from the expansion tan(x)+sec(x) = sum_{n>=1} A099612(n)/A099617(n) * x^n with Taylor coefficients 1, 1, 1/2, 1/3, 5/24, 2/15,...

The first differences of this sequence of fractions are 0, -1/2, -1/6, -1/8, -3/40, -7/144, -31/1008, -113/5760,... which is 0 followed by the negated ratios A034428(n)/(n+1)! = 0, -1/2, -1/6, -3/24, -9/120,....

(The factorial follows because A034428 is obtained by multiplying with 1-x to generate first differences of the o.g.f. and then moving on to the e.g.f.)

The common multiple to reduce numerator and denominator of A034428(n)/A000142(n+1) to the standard coprime representation is this sequence here.

Formula

a(2n+1)=a(2n+2) = A090368(n), n>=0.