A140811 a(n) = 6*n^2 - 1.

-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615

Offset: 0

Author

Paul Curtz, Jul 16 2008

Keywords

• sign
• easy

Comments

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).

a(n) is the set of all k such that 6*k + 6 is a perfect square. - Gary Detlefs, Mar 04 2010

The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012

Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012

From Paul Curtz, Sep 17 2018: (Start)

Terms from center to right in the following spiral:

.

65--63--61--59

/ \

67 31--29--27 57

/ / \ \

69 33 9---7 25 55

/ / / \ \ \

71 35 11 -1===5==23==53==>

/ / / / / /

37 13 1---3 21 51

\ \ / /

39 15--17--19 49

\ /

41--43--45--47 (End)

References

• P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Leo Tavares, Illustration: Barred Stars.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).

Cf. A000012, A001318, A005408, A091137, A140825.

Cf. A003154, A032528, A033581, A016777, A017593.

Programs

Magma

[6*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jun 02 2011

Mathematica

LinearRecurrence[{3,-3,1},{-1,5,23},40] (* Vincenzo Librandi, Feb 05 2012 *)
CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

PARI

a(n)=6*n^2-1 \\ Charles R Greathouse IV, Jun 01 2011

Formula

a(n) = 2*a(n-1) - a(n-2) + 12.

First differences: a(n+1) - a(n) = A017593(n).

Second differences: A071593(n+1) - A071593(n) = 12.

G.f.: (1-8*x-5*x^2)/(x-1)^3. - Jaume Oliver Lafont, Aug 30 2009

From Vincenzo Librandi, Feb 05 2012: (Start)

a(n) = a(n-1) + 12*n - 6.

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

a(n) = A033581(n) - 1. - Omar E. Pol, Jul 18 2012

a(n) = A032528(2*n) - 1. - Adriano Caroli, Jul 21 2013

For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014

a(-n) = a(n). - Paul Curtz, Sep 17 2018

From Amiram Eldar, Feb 04 2021: (Start)

Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.

Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).

Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)

a(n) = A003154(n+1) - 2*A016777(n). - Leo Tavares, May 13 2022

E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - Elmo R. Oliveira, Jan 16 2025

Extensions

Edited and extended by R. J. Mathar, Aug 06 2008

Better description Ray Chandler, Feb 03 2009