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A019298 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). 20
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 G. Carlini (541carlini(AT)charter.net), 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

REFERENCES

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

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.

Index to sequences with 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

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 *)

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.

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 December 20 19:24 EST 2014. Contains 252289 sequences.