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

Author

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

Keywords

• nonn
• easy
• nice

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

Examples

			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)
		

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. J. Mathar, Illustrations of magic labelings (2025)
• 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).

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).

Programs

Magma

[Floor((n^2+1)*(2*n+3)/8): n in [0..80]]; // Vincenzo Librandi, Jul 28 2013

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 *)

PARI

a(n)=(n^2+1)*(2*n+3)\8 \\ Charles R Greathouse IV, Apr 04 2013

Python

def A019298(n): return n*(n*(2*n+3)+2)+3>>3 # Chai Wah Wu, Jun 07 2025

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

a(2n-1) = A213772(n). - R. J. Mathar, Mar 02 2025

(n-2)*a(n) -3*a(n-1) -(n+1)*a(n-2) +2*n-1 =0. - R. J. Mathar, Mar 09 2025

Extensions

Error in n=8 term corrected May 15 1997