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, Aug 11 2002
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^a(n)))^24. -Jonathan Vos Post, Mar 15 2006
Row sums of triangleA134403.
Bisection ofA001318. -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 giveA006002. -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
FromKlaus Purath, May 13 2021: (Start)
This sequence andA000326 provide all integers m such that 24*m + 1 is a square. The union of the two sequences isA001318.
If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n - 1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)
a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. -Harry Richman, Jul 13 2021
Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. -Gary W. Adamson, Dec 17 2022
a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. -Torlach Rush, Feb 04 2024
REFERENCES
Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 78.
LINKS
Vincenzo Librandi,Table of n, a(n) for n = 0..2000
A. O. L. Atkin and F. Morain,Elliptic Curves and Primality Proving, Math. Comp., Vol. 61, No. 203 (1993), pp. 29-68.
Charles H. Conley and Valentin Ovsienko,Quiddities of polygon dissections and the Conway-Coxeter frieze equation, arXiv:2107.01234 [math.CO], 2021.
Leonhard Euler,De mirabilibus proprietatibus numerorum pentagonalium, Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 1780: I, pp. 56-75.
Leonhard Euler,Observatio de summis divisorum, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 5, pp. 59-74.
Leonhard Euler,An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
Leonhard Euler,On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
D. Suprijanto and I. W. Suwarno,Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2211-2217.
Martin Wohlgemuth,Pentagon, Kartenhaus und Summenzerlegung.
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) = 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) = 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) =A130518(3*n+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} (3*i - 1) for n >= 1. -Wesley Ivan Hurt, Oct 11 2013 [Corrected byRémi Guillaume, Oct 24 2024]
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
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. -Amiram Eldar, Jan 18 2021
FromKlaus Purath, May 13 2021: (Start)
Partial sums ofA016789 for n > 0.
a(n) = 3*n^2 -A000326(n).
a(n) =A000326(n) + n.
FromKlaus Purath, Jul 14 2021: (Start)
b^2 = 24*a(n) + 1 we get by b^2 = (a(n+1) - a(n-1))^2 = (a(2*n)/n)^2.
a(2*n) = n*(a(n+1) - a(n-1)), n > 0.
a(2*n+1) = n*(a(n+1) - a(n)). (End)
A generalization of Lajos' formula, dated Sep 12 2006, follows. Let SP(k,n) = the n-th second k-gonal number. Then SP(2k+1,n) = Sum_{j=1..n} (k-1)*n+j+k-2. -Charlie Marion, Jul 13 2024
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2) * binomial(3*n+k, 2*k). -Peter Bala, Nov 03 2024
For integer m, (6*m + 1)^2*a(n) + a(m) = a((6*m+1)*n + m). -Peter Bala, Jan 09 2025
EXAMPLE
FromOmar 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
MATHEMATICA
Table[n (3 n + 1)/2, {n, 0, 100}] (*Zak Seidov, 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
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequencesA000326, this sequence,A045943,A115067,A140090,A140091,A059845,A140672-A140675,A151542.
Cf. numbers of the form n*(n*k-k+4)/2 listed inA226488 (this sequence is the case k=3).
Cf. numbers of the form n*((2*k+1)*n+1)/2 listed inA022289 (this sequence is the case k=1).
KEYWORD
nonn,easy
AUTHOR
STATUS
approved