A152734 5 times pentagonal numbers: 5*n*(3*n-1)/2.

0, 5, 25, 60, 110, 175, 255, 350, 460, 585, 725, 880, 1050, 1235, 1435, 1650, 1880, 2125, 2385, 2660, 2950, 3255, 3575, 3910, 4260, 4625, 5005, 5400, 5810, 6235, 6675, 7130, 7600, 8085, 8585, 9100, 9630, 10175, 10735, 11310, 11900, 12505, 13125, 13760, 14410

Offset

0,2

Comments

a(n) can be represented as a figurate number using n concentric pentagons (see example). - Omar E. Pol, Aug 21 2011

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = 5*A000326(n).

a(n) = a(n-1)+15*n-10 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010

G.f.: 5*x*(1+2*x)/(1-x)^3. a(n) = 4*A000217(n)+A051865(n). - Bruno Berselli, Feb 11 2011

E.g.f.: (5/2)*(3*x^2 + 2*x)*exp(x). - G. C. Greubel, Jul 17 2017

From Amiram Eldar, Feb 26 2022: (Start)

Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/15.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi- 6*log(2))/15. (End)

Example

From Omar E. Pol, Aug 22 2011 (Start):
Illustration of initial terms as concentric pentagons (in a precise representation the pentagons should be strictly concentric):
.
.                                          o
.                                        o   o
.                                      o       o
.                o                   o     o     o
.              o   o               o     o   o     o
.            o       o           o     o       o     o
.  o       o     o     o       o     o     o     o     o
.o   o   o     o   o     o   o     o     o   o     o     o
. o o     o     o o     o     o     o     o o     o     o
.          o           o       o     o           o     o
.           o         o         o     o         o     o
.            o o o o o           o     o o o o o     o
.                                 o                 o
.                                  o               o
.                                   o o o o o o o o
.
.  5             25                        60
(End)

Maple

A152734:=n->5*n*(3*n-1)/2: seq(A152734(n), n=0..50); # Wesley Ivan Hurt, Sep 19 2014

Mathematica

Table[5 n (3 n - 1)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 19 2014 *)
5*PolygonalNumber[5, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 13 2020 *)

Prog

(Magma) [5*n*(3*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 19 2014
(PARI) a(n)=5*n*(3*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A000326, A033581, A085250, A152751, A194275.

Cf. sequences of the form n*(d*n+10-d)/2 indexed in A140090.

Sequence in context: A273357 A146412 A228169 * A080856 A060820 A182211

Adjacent sequences: A152731 A152732 A152733 * A152735 A152736 A152737

Keyword

easy,nonn

Author

Omar E. Pol, Dec 11 2008

Status

approved