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A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2. 115
0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of edges in the join of the complete graph and the cycle graph, both of order n, K_n * C_n. - Roberto E. Martinez II, Jan 07 2002

Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth (mail(AT)matroid.com), Aug 11 2002

a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 2004

Sum of the next n consecutive integers greater than n: a(n) = A014105(n)-A000217(n). - Reinhard Zumkeller, Aug 13 2005

The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2))^24 = q*(1 + Sum{n>=1} A033999(n)*(q^A000326(n)+q^A005449(n))^24. - Jonathan Vos Post, Mar 15 2006

a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006

Row sums of triangle A134403.

Bisection of A001318. - Omar E. Pol, Aug 22 2011

Sequence found by reading the line from 0 in the direction 0, 7,... and the line from 2, in the direction 2, 15,... in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - Omar E. Pol, Aug 04 2012

Partial sums give A006002. - Denis Borris, Jan 07 2013

A002260 is the following array A read by antidiagonals:

  0,  1,  2,  3,  4,  5,...

  0,  1,  2,  3,  4,  5,...

  0,  1,  2,  3,  4,  5,...

  0,  1,  2,  3,  4,  5,...

  0,  1,  2,  3,  4,  5,...

  0,  1,  2,  3,  4,  5,...

and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013

Sum of the numbers from n+1 to 2n, n>0. - Wesley Ivan Hurt, Apr 06 2016

REFERENCES

H. Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp. 61:29-68, 1993.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 1

L. Euler, Observatio de summis divisorum p. 8.

L. Euler, An observation on the sums of divisors, p. 8, arXiv:math/0411587 [math.HO], 2004.

L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.

Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067

D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217.

M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung

Index entries for two-way infinite sequences

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

FORMULA

a(n) = A110449(n, 1) for n>0.

G.f.: x*(2+x)/(1-x)^3. E.g.f.: exp(x)*(2*x + 3*x^2/2). a(n) = n*(3*n + 1)/2. a(-n) = A000326(n). - Michael Somos, Jul 18 2003

a(n) = right term of M^n * [1 0 0] where M = the 3 X 3 matrix [1 0 0 / 1 1 0 / 2 3 1]. M^n * [1 0 0] = [1 n a(n)]. E.g. a(4) = 26 since M^4 * [1 0 0] = [1 4 26] = [1 n a(n)]. - Gary W. Adamson, Dec 19 2004

a(n) = Sum_{j=1..n} n+j. - Zerinvary Lajos, Sep 12 2006

a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006

a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - Zerinvary Lajos, Jan 02 2007

a(n) = A000217(n) + A000290(n). - Zak Seidov, Apr 06 2008

a(n) = a(n-1) + 3*n-1 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010

a(n) = A129267(n+5,n). - Philippe Deléham, Dec 21 2011

a(n) = 2*A000217(n) + A000217(n-1). - Philippe Deléham, Mar 25 2013

a(n) = A130518(3n+1). - Philippe Deléham, Mar 26 2013

a(n) = 12/(n+2)!*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). [Vladimir Kruchinin, Jun 04 2013]

a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - Richard R. Forberg, Jun 22 2013

a(n) = Sum_{i=1..n} 2i - 1 + (i mod 2). - Wesley Ivan Hurt, Oct 11 2013

a(n) = (A000292(6*n+k+1)-A000292(k))/(6*n+1)-A000217(3*n+k+1), for any k >= 0. - Manfred Arens, Apr 26 2015

Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - Vaclav Kotesovec, Apr 27 2016

a(n) = A000217(2*n) - A000217(n). - Bruno Berselli, Sep 21 2016

EXAMPLE

From Omar E. Pol, Aug 22 2011 (Start):

Illustration of initial terms:

.                                               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

.

.    2     7        15         26           40

.

(End)

MAPLE

A005449:=n->n*(3*n + 1)/2; seq(A005449(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

MATHEMATICA

Table[n (3 n + 1)/2, {n, 0, 100}] (* Moshe Levin, Jan 31 2012 *)

PROG

(PARI) {a(n) = n * (3*n + 1) / 2} /* Michael Somos, Jul 18 2003 */

(MAGMA) [n*(3*n + 1) / 2: n in [0..40]]; // Vincenzo Librandi, May 02 2011

CROSSREFS

Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672-A140675, A151542.

Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488 (this sequence is the case k=3).

Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1).

Sequence in context: A194140 A029888 A194112 * A113422 A061802 A263603

Adjacent sequences:  A005446 A005447 A005448 * A005450 A005451 A005452

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 28 11:47 EDT 2017. Contains 288821 sequences.