A226089 Denominators of the series b(n+1) = (b(n)+k)/(1+b(n)*k); where k = 1/(n+1), b(1) = 0.

1, 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

Author

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

Keywords

• nonn
• easy

Comments

The sequence shares numerators with the Harary numbers, A160050.

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.

Examples

			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.
		

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).

Crossrefs

Cf. A160050, A001844.

Programs

Mathematica

LinearRecurrence[{3, -6, 10, -12, 12, -10, 6, -3, 1}, {1, 2, 7, 11, 8, 11, 29, 37, 23}, 100] (* or *)
Module[{m = 1}, Denominator[NestList[(++m*# + 1)/(m + #) &, 0, 99]]] (* Paolo Xausa, Nov 06 2024 *)

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

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)

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

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

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

Extensions

Edited by Thomas Ordowski, Oct 23 2024

Edited by Paolo Xausa, Nov 06 2024