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A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2. 140
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, 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 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
From Klaus Purath, May 13 2021: (Start)
This sequence and A000326 provide all integer m such that 24*m + 1 is a square. The union of the two sequences is A001318.
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.
LINKS
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.
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) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 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(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} 2*i - 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
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - Amiram Eldar, Jan 18 2021
From Klaus Purath, May 13 2021: (Start)
Partial sums of A016789 for n > 0.
a(n) = 3*n^2 - A000326(n).
a(n) = A000326(n) + n.
a(n) = A002378(n+1) + A000217(n). (End)
From Klaus Purath, Jul 14 2021: (Start)
In addition to the May 13 2021 comment: The same square 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)
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}] (* 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 sequences A000326, this sequence, 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 * A293401 A323038 A333637
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:09 EDT 2024. Contains 370951 sequences. (Running on oeis4.)