A220075 -id:A220075 - cp's OEIS frontend

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

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

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

			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)

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.

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).

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).

[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);
```

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 *)

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

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

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

Error in n=8 term corrected May 15 1997

A220073 Mirror of the triangle A130517.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 1, 2, 4, 4, 2, 1, 3, 5, 5, 3, 1, 2, 4, 6, 6, 4, 2, 1, 3, 5, 7, 7, 5, 3, 1, 2, 4, 6, 8, 8, 6, 4, 2, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 12, 10, 8, 6, 4, 2

Offset: 1

Reinhard Zumkeller, Dec 03 2012

T(n,k) = A130517(n,n-k+1), 1 <= k <= n;
T(n,n) = T(n,1) + 1.
From Boris Putievskiy, Jan 15 2013: (Start)
General case see A187760. Let m be natural number. Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n < k.  Table T(n,k) read by antidiagonals. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m}  shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. (End)
First inverse function (numbers of rows) for pairing function A209293. - Boris Putievskiy, Jan 28 2013

			From _Boris Putievskiy_, Jan 15 2013: (Start)
The start of the sequence as table:
1..1..2..3..4..5..6..7...
2..1..1..2..3..4..5..6...
3..2..1..1..2..3..4..5...
4..3..2..1..1..2..3..4...
5..4..3..2..1..1..2..3...
6..5..4..3..2..1..1..2...
7..6..5..4..3..2..1..1...
8..7..6..5..4..3..2..1...
. . .
The start of the sequence as triangle array read by rows:
1,
1, 2,
2, 1, 3,
3, 1, 2, 4,
4, 2, 1, 3, 5,
5, 3, 1, 2, 4, 6,
6, 4, 2, 1, 3, 5, 7,
7, 5, 3, 1, 2, 4, 6, 8,
. . .
Row number r contains r numbers: r-1, r-3,...,1,...r-2,r.
(End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A028310 (left edge), A000027 (right edge), A000012 (central terms), A000217 (row sums), A220075 (partial sums in rows), A002260, A000027, A143182, A187760, A209293.

a220073 n k = a220073_tabl !! (n-1) !! (k-1)
a220073_row n = a220073_tabl !! (n-1)
a220073_tabl = map reverse a130517_tabl

max = 13;
row[n_] := Join[Range[n, 1, -1], Range[max - n + 1]];
T = Array[row, max];
Table[T[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)

T(1,1)=1, for n>1: T(n,k)=T(n-1,n-k+1), 1<=k

From Boris Putievskiy, Jan 15 2013: (Start)

For the general case

a(n) = |(t+1)^2 - 2n| + m*floor((t^2+3t+2-2n)/(t+1)),

where t = floor((-1+sqrt(8*n-7))/2).

For m = 2

a(n) = |(t+1)^2 - 2n| + floor((t^2+3t+2-2n)/(t+1)),

where t=floor((-1+sqrt(8*n-7))/2). (End)

A220053 Partial sums in rows of A130517, triangle read by rows.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 6, 7, 10, 5, 8, 9, 11, 15, 6, 10, 12, 13, 16, 21, 7, 12, 15, 16, 18, 22, 28, 8, 14, 18, 20, 21, 24, 29, 36, 9, 16, 21, 24, 25, 27, 31, 37, 45, 10, 18, 24, 28, 30, 31, 34, 39, 46, 55, 11, 20, 27, 32, 35, 36, 38, 42, 48, 56, 66, 12, 22, 30

Offset: 1

Reinhard Zumkeller, Dec 03 2012

			1;
2, 3;
3, 4, 6;
4, 6, 7, 10;
5, 8, 9, 11, 15;
6, 10, 12, 13, 16, 21;
7, 12, 15, 16, 18, 22, 28;
8, 14, 18, 20, 21, 24, 29, 36;
9, 16, 21, 24, 25, 27, 31, 37, 45;
10, 18, 24, 28, 30, 31, 34, 39, 46, 55;
11, 20, 27, 32, 35, 36, 38, 42, 48, 56, 66;

Reinhard Zumkeller, Table of n, a(n) for n = 1..7260

Cf. A000027 (left edge), A000217 (right edge), A000290 (central terms), A002717 (row sums); A220075.

a220053 n k = a220053_tabl !! (n-1) !! (k-1)
a220053_row n = a220053_tabl !! (n-1)
a220053_tabl = map (scanl1 (+)) a130517_tabl
-- Reinhard Zumkeller, Dec 03 2012

T[n_, 1] := n;
T[n_, n_] := n-1;
T[n_, k_] := Abs[2k - n - If[2k <= n+1, 2, 1]];
row[n_] := Table[T[n, k], {k, 1, n}] // Accumulate;
Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Sep 23 2021 *)

T(n,k) = Sum_{i=1..k} A130517(n, i).

Showing 1-3 of 3 results.

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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).

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A220073 Mirror of the triangle A130517.

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A220053 Partial sums in rows of A130517, triangle read by rows.

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