

A002411


Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.
(Formerly M4116 N1709)


141



0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, 11368, 12615, 13950, 15376, 16896, 18513, 20230, 22050, 23976, 26011, 28158, 30420, 32800, 35301, 37926, 40678
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OFFSET

0,3


COMMENTS

a(n) = n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors.  R. H. Hardin, Feb 23 2002
a(n) = number of (n+6)bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0.  David Callan, Jul 15 2004
Also as a(n) = (1/6)*(3*n^3+3*n^2), n > 0: structured trigonal prism numbers (cf. A100177  structured prisms; A100145 for more on structured numbers).  James A. Record (james.record(AT)gmail.com), Nov 07 2004
If Y is a 3subset of an nset X then, for n >= 5, a(n4) is the number of 5subsets of X having at least two elements in common with Y.  Milan Janjic, Nov 23 2007
a(n1), n >= 2, is the number of ways to have n identical objects in m=2 of altogether n distinguishable boxes (n2 boxes stay empty).  Wolfdieter Lang, Nov 13 2007
a(n+1) is the convolution of (n+1) and (3n+1).  Paul Barry, Sep 18 2008
The number of 3character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.
a(n1):=N_1(n), n >= 1, is the number of edges of n planes in generic position in threedimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 199011, see the Arnold reference, p.506.  Wolfdieter Lang, May 27 2011
For n > 0, the digital roots of this sequence A010888(A002411(n)) form the purely periodic 9cycle {1,6,9,4,3,9,7,9,9}.
For n > 0, the units' digits of this sequence A010879(A002411(n)) form the purely periodic 20cycle {1,6,8,0,5,6,6,8,5,0,6,6,3,0,0,6,1,8,0,0}.
(End)
a(n) is the number of inequivalent ways to color a path graph having 3 nodes using at most n colors. Note, here there is no restriction on the color of adjacent nodes as in the above comment by R. H. Hardin (Feb 23 2002). Also, here the structures are counted up to graph isomorphism, where as in the above comment the "three points on a line" are considered to be embedded in the plane.  Geoffrey Critzer, Mar 20 2013
Latin Square Towers: Take a Latin square of order n, with symbols from 1 to n, and replace each symbol x with a tower of height x. Then the total number of unit cubes used is a(n).  Arun Giridhar, Mar 29 2015
This is the case k = n+4 of b(n,k) = n*((k2)*n(k4))/2, which is the nth kgonal number. Therefore, this is the 3rd upper diagonal of the array in A139600.  Luciano Ancora, Apr 11 2015
For n > 0, a(n) is the number of compositions of n+7 into n parts avoiding the part 2.  Milan Janjic, Jan 07 2016
For n > 0, a(2n+1) is the number of nonisomorphic 5C_msnakes, where m = 2n+1 or m = 2n (for n >= 2). A kC_nsnake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the blockcutpoint graph is a path.  Christian Barrientos, May 15 2019
For n >= 1, a(n1) is the number of 0° and 45°tilted squares that can be drawn by joining points in an n X n lattice.  Paolo Xausa, Apr 13 2021
a(n) is the number of all possible products of n rolls of a sixsided die. This can be easily seen by the recursive formula a(n) = a(n  1) + 2 * binomial(n, 2) + binomial(n + 1, 2).  Rafal Walczak, Jun 15 2024


REFERENCES

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 199011 (p. 75), pp. 503510. Numbers N_1.
Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., Vol. 60 (2001), pp. 8596.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see Vol. 2, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., Vol. 15 (1923), pp. 34. [Annotated scanned copy]


FORMULA

Average of n^2 and n^3.
G.f.: x*(1+2*x)/(1x)^4.  Simon Plouffe in his 1992 dissertation
a(n) = n*Sum_{k=0..n} (nk) = n*Sum_{k=0..n} k.  Paul Barry, Jul 21 2003
a(n) = n*A000217(n).  Xavier Acloque, Oct 27 2003
Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0, ...] = (1, 6, 18, 40, 75, ...).  Gary W. Adamson, Aug 10 2007
Sum_{j>=1} 1/a(j) = hypergeom([1, 1, 1], [2, 3], 1) = 2 + 2*zeta(2) = A195055  2.  Stephen Crowley, Jun 28 2009
a(n) = 4*a(n1)  6*a(n2) + 4*a(n3)  a(n4); a(0)=0, a(1)=1, a(2)=6, a(3)=18.  Harvey P. Dale, Oct 20 2011
a(n) = 3*a(n1)  3*a(n2) + a(n3) + 3.
a(n) = binomial(n+2,3) + 2*binomial(n+1,3).
(End)
a(n) = (24/(n+3)!)*Sum_{j=0..n} (1)^(nj)*binomial(n,j)*j^(n+3).  Vladimir Kruchinin, Jun 04 2013
For n >= 1, a(n) = (Sum_{i=1..n} i^2) + Sum_{i=0..n1} i^2*((i+n) mod 2).  Paolo Xausa, Apr 13 2021
Sum_{n>=1} (1)^(n+1)/a(n) = 2 + Pi^2/6  4*log(2).  Amiram Eldar, Jan 03 2022


EXAMPLE

a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!) + 2!/2!) = 6*(2+1) = 18 ways. The m=2 part partitions of 4, namely (1,3) and (2,2), specify the filling of each of the 6 possible twobox choices.  Wolfdieter Lang, Nov 13 2007


MAPLE

seq(n^2*(n+1)/2, n=0..40);


MATHEMATICA

Table[n^2 (n + 1)/2, {n, 0, 40}]
LinearRecurrence[{4, 6, 4, 1}, {0, 1, 6, 18}, 50] (* Harvey P. Dale, Oct 20 2011 *)
CoefficientList[Series[x (1 + 2 x) / (1  x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jan 08 2016 *)


PROG

(PARI) a(n)=n^2*(n+1)/2
(Haskell)
(PARI) concat(0, Vec(x*(1+2*x)/(1x)^4 + O(x^100))) \\ Altug Alkan, Jan 07 2016


CROSSREFS

a(n) = A093560(n+2, 3), (3, 1)Pascal column.
Cf. similar sequences listed in A237616.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



