A099761 a(n) = ( n*(n+2) )^2.

0, 9, 64, 225, 576, 1225, 2304, 3969, 6400, 9801, 14400, 20449, 28224, 38025, 50176, 65025, 82944, 104329, 129600, 159201, 193600, 233289, 278784, 330625, 389376, 455625, 529984, 613089, 705600, 808201, 921600, 1046529, 1183744, 1334025

Offset: 0

Author

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

Keywords

• nonn
• easy

Comments

Using four consecutive triangular numbers t1, t2, t3, t4, form a 2 X 2 determinant with the first row t1 and t2 and the second row t3 and t4. Squaring the determinant gives the numbers in this sequence. - J. M. Bergot, May 17 2012

Numbers k such that sqrt(1 + sqrt(k)) is an integer. - Jaroslav Krizek, Jan 23 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005563.

Programs

GAP

List([0..40], n-> (n*(n+2))^2); # G. C. Greubel, Sep 03 2019

Magma

[(n*(n+2))^2: n in [0..40]]; // G. C. Greubel, Sep 03 2019

Maple

A099761 := proc(n) n^2*(n+2)^2 ; end proc:
seq(A099761(n), n=0..40) ; # R. J. Mathar, Apr 02 2011

Mathematica

Table[1 -2m^2 +m^4, {m,40}] (* Artur Jasinski, Aug 15 2007 *)

PARI

vector(40, n, (n^2-1)^2) \\ G. C. Greubel, Sep 03 2019

Sage

[(n*(n+2))^2 for n in (0..40)] # G. C. Greubel, Sep 03 2019

Formula

G.f.: x*(9 +19*x -5*x^2 +x^3)/(1-x)^5. - R. J. Mathar, Apr 02 2011

a(n) = (A005563(n))^2. - Pedro Caceres, Aug 04 2019

E.g.f.: exp(x)*x*(9 + 23*x + 10*x^2 + x^3). - Stefano Spezia, Aug 05 2019

a(n) = (determinant [T(n-1) T(n) ; T(n+1) T(n+2)])^2 where T is A000217. - J. M. Bergot, May 17 2012 and Bernard Schott, Aug 06 2019

From Amiram Eldar, Jul 13 2020: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/12 - 11/16.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/24 - 5/16. (End)

Extensions

Deleted a trivial formula which was based on another offset - R. J. Mathar, Dec 16 2009