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, 7, 11, 8, 11, 29, 37, 23, 28, 67, 79, 46, 53, 121, 137, 77, 86, 191, 211, 116, 127, 277, 301, 163, 176, 379, 407, 218, 233, 497, 529, 281, 298, 631, 667, 352, 371, 781, 821, 431, 452, 947, 991, 518, 541, 1129, 1177, 613, 638, 1327, 1379, 716, 743, 1541

Offset

1,1

Comments

The sequence shares numerators with the Harary numbers, A160050.

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.

Links

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Christian N. K. Anderson, Ulam Spiral of the first 4000 terms.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Formula

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

Example

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.

Prog

(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)
(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

Crossrefs

Cf. A160050, A001844.

Sequence in context: A020638 A091385 A053247 * A208846 A087723 A359167

Adjacent sequences: A226086 A226087 A226088 * A226090 A226091 A226092

Keyword

nonn,easy

Author

Rick G. Rosner, Christian N. K. Anderson, and Kevin L. Schwartz, May 25 2013

Status

approved