A259055 a(n) = 9*n^2 + 18*n + 7.

7, 34, 79, 142, 223, 322, 439, 574, 727, 898, 1087, 1294, 1519, 1762, 2023, 2302, 2599, 2914, 3247, 3598, 3967, 4354, 4759, 5182, 5623, 6082, 6559, 7054, 7567, 8098, 8647, 9214, 9799, 10402, 11023, 11662, 12319, 12994, 13687, 14398, 15127

Offset

0,1

Comments

a(n) gives twice the curvature of the n-th circle touching the two semicircles of the (2/3,1/3) arbelos and the (n-1)-th circle, with input circle of twice the curvature a(0) = A114949(1) = 7 (referring to the second circle of the counterclockwise Pappus chain).

Links

Table of n, a(n) for n=0..40.

Kival Ngaokrajang, Illustration of one half of the initial terms.

Eric Weisstein's World of Mathematics, Descartes Circle theorem.

Eric Weisstein's World of Mathematics, Pappus chain.

Wikipedia, Descartes' Theorem.

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

Formula

a(n) = 9*(n+1)^2 - 2, n >= 0.

O.g.f.: (-2*x^2+13*x+7)/(1-x)^3.

Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3, with a(0)=7, a(1)=34, and a(2)=79.

Descartes' three (actually five) circle theorem (see links) leads to a nonlinear recurrence for twice the curvatures: a(n) = 2*(3 + 3/2) + a(n-1) + 4*sqrt((3 + 3/2)*a(n-1)/2 + 9/2) = 9 + a(n-1) + 6*sqrt(a(n-1) + 2), with input a(0) = 7 = 2*A114949(1). This leads to a quadratic equation with the relevant solution a(n) = 9*n^2 + 18*n + 7.

E.g.f.: exp(x)*(9*x*(x + 3) + 7). - Elmo R. Oliveira, Oct 20 2024

Maple

A259055:=n->9*n^2+18*n+7: seq(A259055(n), n=0..100); # Wesley Ivan Hurt, Feb 04 2017

Mathematica

Table[9 n^2 + 18 n + 7, {n, 0, 40}] (* Michael De Vlieger, Jul 03 2015 *)
LinearRecurrence[{3, -3, 1}, {7, 34, 79}, 50] (* Harvey P. Dale, Sep 05 2018 *)

Prog

(PARI) a(n)=9*n^2+18*n+7 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A114949, A259054, A259555.

Sequence in context: A209814 A117663 A063166 * A195018 A024817 A201230

Adjacent sequences: A259052 A259053 A259054 * A259056 A259057 A259058

Keyword

nonn,easy

Author

Wolfdieter Lang and Kival Ngaokrajang, Jul 03 2015

Status

approved