A173294 Values of 16*n^2+24*n+7, n>=0, each duplicated.

7, 7, 47, 47, 119, 119, 223, 223, 359, 359, 527, 527, 727, 727, 959, 959, 1223, 1223, 1519, 1519, 1847, 1847, 2207, 2207, 2599, 2599, 3023, 3023, 3479, 3479, 3967, 3967, 4487, 4487, 5039, 5039, 5623, 5623, 6239, 6239, 6887, 6887, 7567, 7567, 8279, 8279, 9023, 9023, 9799, 9799, 10607, 10607, 11447, 11447, 12319, 12319, 13223, 13223, 14159, 14159, 15127, 15127, 16127

Offset

0,1

Comments

The Leibniz series for Pi/4 involves 1, -1/3, 1/5, -1/7, 1/9, -1/11, .. inverses of the odd numbers. The first differences of this sequence of fractions are -4/3, 8/15, -12/35, 16/63, -20/99, 24/143,... = (-1)^(n+1)*A008586(n+1)/A000466(n+1).

a(n) is the difference of the n-th denominator and numerator, A000466(n+1)+(-1)^n*A008586(n+1). (Note that A000466 is a bisection of A005563, which establishes a very distant relation between this sequence and the Lyman series.)

If one would add the n-th denominator and numerator, -1, 23, 23, 79, 79, 167, 167, 287, 287, 439,...(duplicated values of 16n^2+40n+23 and a -1) would result.

Links

Table of n, a(n) for n=0..62.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).

G..f: ( -7-26*x^2+x^4 ) / ( (1+x)^2*(x-1)^3 ).

a(2n) = a(2n+1) = 16n^2+24n+7.

Mathematica

With[{c=16n^2+24n+7}, Table[{c, c}, {n, 0, 40}]]//Flatten (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {7, 7, 47, 47, 119}, 80] (* Harvey P. Dale, Jan 16 2019 *)

Crossrefs

Sequence in context: A219399 A219447 A271064 * A165828 A161343 A038273

Adjacent sequences: A173291 A173292 A173293 * A173295 A173296 A173297

Keyword

nonn,easy

Author

Paul Curtz, Feb 15 2010

Status

approved