

A002411


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


85



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
Kekulé numbers for certain benzenoids.  Emeric Deutsch, Nov 18 2005
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
a(n+1) = denom(2/((n+2)*(n+1)^2)).  Stephen Crowley, Jun 28 2009
The number of 3character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.
Partial sums are A001296 4dimensional pyramidal numbers: (3n+1)*C(n+2,3)/4.  Jonathan Vos Post, Mar 26 2011
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
Partial sums of pentagonal numbers A000326.  Reinhard Zumkeller, Jul 07 2012
From Ant King, Oct 23 2012: (Start)
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
After 0, row sums of the triangle in A101468.  Bruno Berselli, Feb 10 2014
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


REFERENCES

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 199011 (p. 75), pp. 503510. Numbers N_1.
A. 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.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.
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

T. D. Noe, Table of n, a(n) for n=0..1000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),34. [Annotated scanned copy]
S. M. Losanitsch, Die IsomerieArten bei den Homologen der ParaffinReihe, Chem. Ber. 30 (1897), 19171926.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number.
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for twoway infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

Average of n^2 and n^3.
a(n) = sum of n smallest multiples of n.  Amarnath Murthy, Sep 20 2002
G.f.: x*(1+2*x)/(1x)^4.  Paul Barry, Mar 21 2003
a(n) = Sum_{k=0..n} n*(nk).  Paul Barry, Jul 21 2003
a(n) = whole numbers * triangular numbers.  Xavier Acloque, Oct 27 2003
a(n) = (1/2)*Sum_{j=1..n}{i=1..n} (i+j) = (1/2)*(n^2+n^3) = (1/2)*A011379(n).  Alexander Adamchuk, Apr 13 2006
a(n) = Sum_{j=0..n} n*j.  Zerinvary Lajos, Sep 12 2006
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
G.f.: x*F(2,3;1;x).  Paul Barry, Sep 18 2008
Sum_{j>=1} 1/a(j) = hypergeom([1, 1, 1], [2, 3], 1) = 2+2*Zeta(2).  Stephen Crowley, Jun 28 2009
a(0)=0, a(1)=1, a(2)=6, a(3)=18, a(n)=4*a(n1)6*a(n2)+4*a(n3)a(n4).  Harvey P. Dale, Oct 20 2011
a(n) = n*binomial(n+1,2) = n*A000217(n).  Arkadiusz Wesolowski, Feb 10 2012
From Ant King, Oct 23 2012: (Start)
a(n) = 3*a(n1)  3*a(n2) + a(n3) + 3.
a(n) = (n+1)*(2*A000326(n)+n)/6 = A000292(n)+2*A000292(n1).
a(n) = A000330(n)+A000292(n1) = A000217(n)+3*A000292(n1).
a(n) = binomial(n+2,3) + 2*binomial(n+1,3).
(End)
a(n) = (A000330(n) + A002412(n))/2 = (A000292(n) + A002413(n))/2.  Omar E. Pol, Jan 11 2013
a(n) = 24/(n+3)!*Sum_{j=0..n} (1)^(nj)*binomial(n,j)*(j)^(n+3).  Vladimir Kruchinin, Jun 04 2013
Sum_{n>=1} a(n)/n! = 3.5*exp(1).  Richard R. Forberg, Jul 15 2013


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 two box choices.  Wolfdieter Lang, Nov 13 2007


MAPLE

seq(n^2*(n+1)/2, n=0..40);
A002411:=(1+2*z)/(z1)**4; # Simon Plouffe in his 1992 dissertation.


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 *)
Nest[Accumulate, Range[1, 140, 3], 2]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)


PROG

(PARI) a(n)=n^2*(n+1)/2
(Haskell)
a002411 n = n * a000217 n  Reinhard Zumkeller, Jul 07 2012
(MAGMA) [n^2*(n+1)/2: n in [0..40]]; // Wesley Ivan Hurt, May 25 2014


CROSSREFS

Cf. A001296, A011379, A015223, A015224, A014799, A014800, A127739, A132118, A139600.
A006002(n)=a(1n).
a(n)= A093560(n+2, 3), (3, 1)Pascal column.
A row or column of A132191.
Second column of triangle A103371.
Cf. similar sequences listed in A237616.
Sequence in context: A035489 A219143 A122061 * A023658 A059834 A015224
Adjacent sequences: A002408 A002409 A002410 * A002412 A002413 A002414


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane, Apr 30 1991


STATUS

approved



