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

Author

Paul Curtz, Feb 15 2010

Keywords

• nonn
• easy

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

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

Programs

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 *)

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.