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

1,2

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.

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

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

Example

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.

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 *)

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

Extensions

Edited by Thomas Ordowski, Oct 23 2024

Edited by Paolo Xausa, Nov 06 2024

Status

approved