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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). 34
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 (list; graph; refs; listen; history; text; internal format)
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) = sum of 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) = 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

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, 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).

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

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]

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

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

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-Fran├žois 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: A027378 A131177 A092498 * 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

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Last modified October 21 23:44 EDT 2017. Contains 293749 sequences.