A019298 Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).

0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186

Offset

0,3

Comments

Alternately add and subtract successively longer sets of integers: 0; 1 = 0+1; -4 = 1-2-3; 11 = -4+4+5+6; -23 = 11-7-8-9-10; 42 = -23+11+12+13+14+15; -69 = 42-16-17-18-19-20-21; ... then take absolute values. - Walter Carlini, Aug 28 2003

Number of 3 X 3 symmetric matrices with nonnegative integer entries, such that every row (and column) sum equals n-1.

Equals Sum_{0..n} of "three-quarter squares" sequence (A077043). - Philipp M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007

a(n) is the sum of the n-th row in A220075, n > 0. - Reinhard Zumkeller, Dec 03 2012

Sum of all the smallest parts in the partitions of 3n into three parts (see example). - Wesley Ivan Hurt, Jan 23 2014

For n > 0, a(n) is the number of (nonnegative integer) magic labelings of the prism graph Y_3 with magic sum n - 1. - L. Edson Jeffery, Sep 09 2017

Or number of magic labelings of LOOP X C_3 with magic sum n - 1, where LOOP is the 1-vertex, 1-loop-edge graph, as Y_k = I X C_k and LOOP X C_k have the same numbers of magic labelings when k is odd. - David J. Seal, Sep 13 2017

a(n) is the number of triples of integers in [1,n]^3 such that each pair has sum larger than n. - Bob Zwetsloot, Jul 23 2020

References

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_3(lambda).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.14(a), p. 452.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, January 1999, p. 13.

L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).

R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, Section 5, F_3(x).

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = floor((n^2+1)(2n+3)/8).

G.f.: x*(x^2+x+1)/((x+1)*(x-1)^4).

a(n) = floor((2n^3 + 3n^2 + 2n)/8); also nearest integer to ((n+1)^4 - n^4)/16.

a(n) = (4n^3 + 6n^2 + 4n+1 - (-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Mar 06 2004

a(n) = Sum_{i=1..n} i^2 - floor(i^2/4) = Sum_{i=1..n} i * (2n - 2i + 1 - floor((n - i + 1)/2) ). - Wesley Ivan Hurt, Jan 23 2014

E.g.f.: (1/16)*(-exp(-x) + exp(x)*(1 + 14*x + 18*x^2 + 4*x^3)). - Stefano Spezia, Nov 29 2019

a(2*n) = (1/2)*( n*(n + 1)^3 - (n - 1)*n^3 ); a(2*n-1) = (1/2)*( (n + 1)*n^3 - n*(n - 1)^3 ) (note: replacing the exponent 3 with 2 throughout gives the sequence of generalized pentagonal numbers A001318). - Peter Bala, Aug 11 2021

Example

Add last column for a(n) (n > 0).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           4           11          23          42      ..  a(n)

Maple

series(x*(x^2+x+1)/(x+1)/(x-1)^4, x, 80);

Mathematica

Table[ Ceiling[3*n^2/4], {n, 0, 37}] // Accumulate (* Jean-François Alcover, Dec 20 2012, after Philipp M. Buluschek's comment *)
CoefficientList[Series[x (x^2 + x + 1) / ((x + 1) (x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 28 2013 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 11, 23}, 38] (* L. Edson Jeffery, Sep 09 2017 *)

Prog

(PARI) a(n)=(n^2+1)*(2*n+3)\8 \\ Charles R Greathouse IV, Apr 04 2013
(Magma) [Floor((n^2+1)*(2*n+3)/8): n in [0..80]]; // Vincenzo Librandi, Jul 28 2013

Crossrefs

Cf. A053493, A077043 (first differences), A002717.

Cf. A061927, A244497, A292281, A244873, A289992 (# of magic labelings of prism graph Y_k = I X C_k, for k = 4,5,6,7,8, up to an offset).

Cf. A006325, A244879, A244880 (# of magic labelings of LOOP X C_k, for k = 4,6,8, up to an offset).

Sequence in context: A131177 A092498 A301165 * A237586 A173702 A244281

Adjacent sequences: A019295 A019296 A019297 * A019299 A019300 A019301

Keyword

nonn,easy,nice

Author

Eric E Blom (eblom(AT)REM.re.uokhsc.edu)

Extensions

Error in n=8 term corrected May 15 1997

Status

approved