A173294 - OEIS

%I #10 Jan 16 2019 14:59:11

%S 7,7,47,47,119,119,223,223,359,359,527,527,727,727,959,959,1223,1223,

%T 1519,1519,1847,1847,2207,2207,2599,2599,3023,3023,3479,3479,3967,

%U 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

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

%C 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).

%C 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.)

%C 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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

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

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

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

%t 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 *)

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Feb 15 2010