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A226089 Denominators of the series a(n+1) = (a(n)+k)/(1+a(n)*k); where k=1/(n+1), a(1)=1/2. 2

%I

%S 2,7,11,8,11,29,37,23,28,67,79,46,53,121,137,77,86,191,211,116,127,

%T 277,301,163,176,379,407,218,233,497,529,281,298,631,667,352,371,781,

%U 821,431,452,947,991,518,541,1129,1177,613,638,1327,1379,716,743,1541

%N Denominators of the series a(n+1) = (a(n)+k)/(1+a(n)*k); where k=1/(n+1), a(1)=1/2.

%C The sequence shares numerators with the Harary numbers, A160050.

%C This is the sequence 1/2 + 1/3 + 1/4 +...+1/n using relativistic velocity addition, where the addition of velocities a and b = (a+b) / (1 + a*b/c^2). That is, for objects traveling at c/2 + c/3 + ... +c/n relative to each other, the n-th object has velocity A160050(n)/a(n)*c relative to a stationary observer.

%H Christian N. K. Anderson, <a href="/A226089/b226089.txt">Table of n, a(n) for n = 1..10000</a>

%H Christian N. K. Anderson, <a href="/A226089/a226089.gif">Ulam Spiral</a> of the first 4000 terms.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).

%F G.f.: -x*(x^8-2*x^7+4*x^6-6*x^5+7*x^4-3*x^3+2*x^2+x+2) / ((x-1)^3*(x^2+1)^3). - _Colin Barker_, Jul 18 2015

%e a(10) = a(9) + 1/11 using relativistic velocity addition. Since a(9) = 27/28, the sum is (27/28 + 1/11) / (1 + 27/28 * (1/11)) = (325 / 308) / (335/308) = 65/67.

%o (R) library(gmp); reladd<-function(x,y) (x+y)/(1+x*y)y=as.bigq(rep(1,100)); y[1]=y[1]/2; for(i in 2:100) y[i]=reladd(y[i-1],y[i]/(i+1)); denominator(y)

%o (PARI) Vec(-x*(x^8-2*x^7+4*x^6-6*x^5+7*x^4-3*x^3+2*x^2+x+2) / ((x-1)^3*(x^2+1)^3) + O(x^100)) \\ _Colin Barker_, Jul 18 2015

%Y Cf. A160050, A001844.

%K nonn,easy

%O 1,1

%A _Rick G. Rosner_, _Christian N. K. Anderson_, and _Kevin L. Schwartz_, May 25 2013

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)