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

Author

Reinhard Zumkeller, Mar 13 2009

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Mathematica

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

PARI

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

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