0,2

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 (remartin(AT)fas.harvard.edu), 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 (reinhard.zumkeller(AT)gmail.com), Dec 03 2004

Also sum of next n consecutive numbers greater than n: a(n)=A014105(n)-A000217(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 13 2005

The modular form Delta(q) = q*(1 + SUM[1..infinity]((-1)^n)*((q^(n*(3*n-1)/2))+((q^(n*(3*n+1)/2))))) = q*Prod[1..infinity] (1-q^n)^24. Thus Delta(q) = q*(1 + SUM[1..infinity](A033999(n)*(q^A000326(n))+(q^A005449(n)))). - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 15 2006

a(n) = A126890(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

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

Row sums of triangle A134403.

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

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

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

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.

L. Euler, On the remarkable properties of the pentagonal numbers

M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung

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

G.f.: x(2+x)/(1-x)^3. E.g.f.: exp(x)(2x+3x^2/2). a(n)=n(3n+1)/2. a(-n)=A000326(n).

a(n) = right term of M^n * [1 0 0] where M = the 3X3 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 (qntmpkt(AT)yahoo.com), Dec 19 2004

a(n) = n*(3n+1)/2. G.f.: x(2+x)/(1-x)^3. E.g.f.: exp(x)(2x+3x^2/2). a(-n)=A000326(n). - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 15 2006

sum (n+j,j=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 12 2006

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

a(n)=3*n+a(n-1)-4 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

For n=2, a(2)=3*2+0-4=2; n=3, a(3)=3*3+2-4=7; n=4, a(4)=3*4+7-4=15 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

[seq(2*binomial(3*n, 4)/binomial(3*n, 2), n=1..49)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 02 2007

a:=n->sum(n/3, j=0..n): seq(a(3*n)/2, n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

a:=n->sum(k+sum(1, k=1..n), k=1..n):seq(a(n), n=0...48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 17 2008

seq(sum(binomial(n, m), m=1..2)+n^2, n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 17 2008

lst={}; Do[AppendTo[lst, n*(3*n+1)/2], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 06 2008]

(PARI) a(n)=n*(3*n+1)/2

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

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

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, A140673, A140674, A140675, A151542.

Sequence in context: A132746 A167543 A029888 this_sequence A113422 A061802 A003452

Adjacent sequences: A005446 A005447 A005448 this_sequence A005450 A005451 A005452

nonn,new

N. J. A. Sloane (njas(AT)research.att.com).