A226089 - OEIS

%I #57 Nov 08 2024 10:28:06

%S 1,2,7,11,8,11,29,37,23,28,67,79,46,53,121,137,77,86,191,211,116,127,

%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 b(n+1) = (b(n)+k)/(1+b(n)*k); where k = 1/(n+1), b(1) = 0.

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

%C This is the sequence 0 + 1/2 + 1/3 + ... + 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 0 + 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

%F A160050(n)/a(n) = tanh(Sum_{k=2..n} arctanh(1/k)), a(n) = A160050(n) + (1,1,2,2). - _Thomas Ordowski_, Oct 23 2024

%F a(4k) = 4k^2 + 3k + 1, a(4k+1) = 4k^2 + 5k + 2, a(4k+2) = 8k^2 + 14k + 7, a(4k+3) = 8k^2 + 18k + 11. - _David Radcliffe_, Oct 25 2024

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

%t LinearRecurrence[{3, -6, 10, -12, 12, -10, 6, -3, 1}, {1, 2, 7, 11, 8, 11, 29, 37, 23}, 100] (* or *)

%t Module[{m = 1}, Denominator[NestList[(++m*# + 1)/(m + #) &, 0, 99]]] (* _Paolo Xausa_, Nov 06 2024 *)

%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 1,2

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

%E Edited by _Thomas Ordowski_, Oct 23 2024

%E Edited by _Paolo Xausa_, Nov 06 2024