A158187 a(n) = 10*n^2 + 1.

1, 11, 41, 91, 161, 251, 361, 491, 641, 811, 1001, 1211, 1441, 1691, 1961, 2251, 2561, 2891, 3241, 3611, 4001, 4411, 4841, 5291, 5761, 6251, 6761, 7291, 7841, 8411, 9001, 9611, 10241, 10891, 11561, 12251, 12961, 13691, 14441, 15211, 16001, 16811, 17641

Offset

0,2

Comments

Sequence found by reading the segment (1, 11) together with the line from 11, in the direction 11, 41, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

The identity (10n^2 + 1)^2 - (25n^2 + 5)*(2n)^2 = 1 can be written as a(n)^2 - A158445(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Jan 03 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

Formula

a(n) = A033583(n) + 1.

For n > 0: a(n) = A010010(n)/2.

From Vincenzo Librandi, Jan 03 2012: (Start)

G.f: x*(11 + 8*x + x^2)/(1-x)^3.

a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

From Amiram Eldar, Jul 15 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(10))*coth(Pi/sqrt(10)))/2.

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(10))*csch(Pi/sqrt(10)))/2. (End)

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(10))*sinh(Pi/sqrt(5)).

Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(10))*csch(Pi/sqrt(10)). (End)

E.g.f.: exp(x)*(1 + 10*x + 10*x^2). - Stefano Spezia, Feb 05 2021

Mathematica

Table[10*n^2+1, {n, 0, 50}] (* Vincenzo Librandi, Jan 03 2012 *)

Prog

(PARI) a(n)=10*n^2+1 \\ Charles R Greathouse IV, Oct 16 2015

Crossrefs

Cf. A158445, A005843. - Vincenzo Librandi, Mar 19 2009

Sequence in context: A360154 A217624 A097991 * A239462 A065145 A030685

Adjacent sequences: A158184 A158185 A158186 * A158188 A158189 A158190

Keyword

nonn,easy

Author

Reinhard Zumkeller, Mar 13 2009

Status

approved