A019298 -id:A019298 - cp's OEIS frontend

A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.

1, 14, 85, 336, 1023, 2610, 5860, 11942, 22555, 40068, 67677, 109578, 171157, 259196, 382096, 550116, 775629, 1073394, 1460845, 1958396, 2589763, 3382302, 4367364, 5580666, 7062679, 8859032, 11020933, 13605606, 16676745, 20304984, 24568384, 29552936, 35353081, 42072246, 49823397, 58729608

Offset: 0

N. J. A. Sloane, Jul 07 2014

a(n) is the number of perimeter-magic hollow triangles of order 4 (4 elements per edge, 9 elements in total) with magic edge sum n+4. The triangles have 9 elements >=1, not necessarily distinct. Triangles obtained by rotations and flips (D_6 symmetry) are counted as being distinct. Associated triangles of order 3 are counted in A019298. - R. J. Mathar, Mar 05 2025

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Vec((1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)).
From Colin Barker, Jan 11 2017: (Start)
a(n) = (15*(127+(-1)^n) + 6432*n + 8936*n^2 + 6480*n^3 + 2570*n^4 + 528*n^5 + 44*n^6) / 1920.
a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8) for n>7.
(End)

A008764 Number of 3 X 3 symmetric stochastic matrices under row and column permutations.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 12, 16, 21, 27, 34, 42, 52, 62, 74, 88, 103, 119, 138, 158, 180, 204, 230, 258, 289, 321, 356, 394, 434, 476, 522, 570, 621, 675, 732, 792, 856, 922, 992, 1066, 1143, 1223, 1308, 1396, 1488, 1584, 1684, 1788, 1897, 2009, 2126, 2248, 2374, 2504

Offset: 0

N. J. A. Sloane

nonn

			There are 6 nonisomorphic symmetric 3 X 3 matrices with row and column sums 4:
[0 0 4] [0 1 3] [0 1 3] [0 2 2] [0 2 2] [1 1 2]
[0 4 0] [1 2 1] [1 3 0] [2 0 2] [2 1 1] [1 2 1]
[4 0 0] [3 1 0] [3 0 1] [2 2 0] [2 1 1] [2 1 1]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Cf. A019298.

a:=[1,1,2,4,6,8,12,16,21];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # G. C. Greubel, Sep 10 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 10 2019

seq(coeff(series((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Sep 10 2019

LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,4,6,8,12,16,21}, 60] (* G. C. Greubel, Sep 10 2019 *)

my(x='x+O('x^60)); Vec((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))) \\ G. C. Greubel, Sep 10 2019

def A008764_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
A008764_list(60) # G. C. Greubel, Sep 10 2019

Expansion of (1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = (1-x+x^2)/( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^4).

a(n) = A008762(n) - A008762(n-1) + A008762(n-2). - R. J. Mathar, Aug 06 2013

Better description and more terms from Vladeta Jovovic, Feb 06 2000

A130046 Hexagonal pyramid of Pascal numbers in 3 dimensions. The 3-dimensional sequence is split into slices of the pyramid which in turn consist of rows of the slice, each containing multiple columns of numbers and where each element of slice j is composed of the sum of the three elements above it in slice j-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 5, 2, 1, 5, 5, 1, 1, 2, 1, 1, 2, 1, 2, 8, 8, 2, 1, 8, 15, 8, 1, 2, 8, 8, 2, 1, 2, 1, 1, 2, 1, 3, 11, 11, 3, 3, 18, 31, 18, 3, 1, 11, 31, 31, 11, 1, 2, 11, 18, 11, 2, 1, 3, 3, 1, 1, 3, 3, 1, 3, 15, 24, 15, 3, 3, 24, 60, 60, 24, 3, 1, 15, 60, 93, 60, 15, 1, 3, 24

Offset: 0

Jeffrey C. Jacobs (darklord(AT)timehorse.com), May 03 2007

nonn

Successive slices [0,0], [1,0], [1,1], [2,1], [2,2], [3,2], [3,3], [4,3], [4,4], ... in table A109672; see also A046816 (slices [n,0]), A109673 (slices [n,n]), A109649 (slices [0,k]), A109390 (slices [n,1]), A109393 (slices [1,k]), A109494 (slices [n,2]), A109495 (slices [2,k]). - Philippe Deléham, May 03 2007

			Slice[0]:
...
Slice[1]:
1
Slice[2]:
.1
1.1
Slice[3]:
.1.1
1.3.1
.1.1
Slice[4]:
..1.1
.2.5.2
1.5.5.1
.1.2.1
Slice[5]:
....1..2..1
..2..8..8..2
.1..8.15..8..1
..2..8..8..2
....1..2..1
Slice[6]:
.....1..2..1
....3.11.11..3
..3.18.31.18..3
.1.11.31.31.11..1
..2.11.18.11..2
....1..3..3..1

Jeffrey C. Jacobs, Table of n, a(n) for n = 0..30324
Jeffrey C. Jacobs, Python program for this sequence

Cf. A077043, A019298.
Subsequence of A109672 table of slices.
Other tables of slices (see 2007 comment from Philippe Deléham): A046816, A109390, A109393, A109494, A109495, A109649, A109673.
Cf. A007318 (Pascal's triangle).

Let j be a given slice of the hexagonal pyramid. For j = 0, there are no elements.
For j > 0, let a[x] to a[x+y-1] represent the elements of the slice, where x is the (j-1)th element of A019298 and y is the j-th element of A077043. Each slice j consists of j rows of varying column length, numbered 0 to j-1.
The length of the first row of slice j is given by floor((j+1)/2) and the last row by floor(j/2)+1, where by convention the last row is always greater or equal in length to the first row.
The floor(j/2)th row is j columns in length and any row before it is given by the formula floor((j+1)/2) + row#. For rows after the floor(j/2)th row, the length is given by floor(j/2) + j - row#.
The elements a[x] to a[x+y-1] are thus layed out as a concatenated series of rows of varying column lengths as specified above.
Thus for a given slice j, the element at row row# and column col# is represented by a[x + floor((j+1)/2) * row# + row# * (row# - 1) / 2 + col# ] when row# <= floor(j/2) and by a[x + y - (floor(j/2) + 1) * (j - row#) - (j - row#) * (j - row# + 1) / 2 + col#] otherwise, where x and y are defined above and row# and col# start counting from 0.
The elements of a for a given slice j, row# and col#, represented by the coordinate pair (row#, col#), is given by the following recursive relation:
For j = 1, there is 1 element whose value is 1 at (0, 0). Call this Slice[1] whose first and only element forms a0 = 1.
For j > 1, each element (row#, col#) is given by the sum of the 3 elements above it in the pyramid. If the preceding slice does not contain one of the cells specified because the coordinates are invalid for that slice, the value is assumed to be 0.
The cells above can be found using the following formula for a given cell Slice[j](row#, col#):
If j is odd:
If row# > floor(j/2):
Sum:
Slice[j-1](row#, col#-1)
Slice[j-1](row#-1,col#)
Slice[j-1](row#-1,col#-1)
Otherwise:
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#-1,col#)
Slice[j-1](row#-1,col#-1)
Otherwise:
If row# > floor(j/2):
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#-1,col#)
Slice[j-1](row#,col#-1)
Otherwise:
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#,col#-1)
Slice[j-1](row#-1,col#-1)
Each slice is also a solution to the Prouhet-Tarry-Escott problem for a given n and k. The slices[n,k] in sequence A109672 map to the slices here by the relation k + n = j - 1, where k = n (j odd) or k = n + 1 (j even).
When j is even, k = n - 1 would also be a solution to the Pascal hexagonal pyramid, however here the k = n + 1 solution is chosen. For j even, the slices are also given by A109673.
Only 3 of the 6 hexagonal vertices have corresponding cells in the slice below them. Only every other vertex has a cell below it and all vertices with cells below them are always separated by 2 edges.
By convention, when constructing Slice[j] for j odd, the uppermost vertices of Slice[j-1] are chosen to have cells below them and for j even the 2 vertices adjacent to the uppermost vertices of Slice[j-1] are chosen.

A237669 Number of prime parts in the partitions of 3n into 3 parts.

Original entry on oeis.org

0, 5, 12, 17, 29, 35, 50, 59, 77, 87, 108, 120, 144, 156, 182, 198, 228, 243, 275, 292, 327, 346, 383, 402, 443, 465, 507, 531, 578, 601, 649, 674, 722, 748, 800, 829, 886, 915, 974, 1006, 1067, 1097, 1158, 1189, 1253, 1286, 1353, 1388, 1456, 1491, 1561

Offset: 1

Wesley Ivan Hurt, Feb 11 2014

			Count the primes in the partitions of 3n into 3 parts for a(n).
                                               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
---------------------------------------------------------------------
    0           5           12          17          29      ..  a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for sequences related to partitions

Cf. A010051, A019298, A236364, A236370, A236758, A236762, A237264.

Table[Sum[Sum[PrimePi[i] - PrimePi[i - 1], {i, n + Floor[j/2] + 1 - Floor[1/(j + 1)], n + 2 (j + 1)}], {j, 0, n - 2}] + Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, n}] + Sum[(PrimePi[n + i] - PrimePi[n + i - 1]) (n - 2 i), {i, Floor[(n - 1)/2]}] + Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 70}]
Table[Count[Flatten[IntegerPartitions[3 n,{3}]],?PrimeQ],{n,60}] (* _Harvey P. Dale, Oct 16 2016 *)

a(n) = A237264(n) + A236762(n) + A236758(n).

A270694 Alternating sum of centered heptagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -23, 51, -94, 157, -242, 354, -495, 670, -881, 1133, -1428, 1771, -2164, 2612, -3117, 3684, -4315, 5015, -5786, 6633, -7558, 8566, -9659, 10842, -12117, 13489, -14960, 16535, -18216, 20008, -21913, 23936, -26079, 28347, -30742, 33269

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

More generally, the ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers is -x*(1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^4).

Ilya Gutkovskiy, Table of n, a(n) for n = 0..500
OEIS Wiki, Centered pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A004126 (centered heptagonal pyramidal numbers).

Cf. A000330, A006323 (partial sums of centered heptagonal pyramidal numbers), A019298, A232599.

[((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270694:= n-> ((-1)^n*(2*n+1)*(14*n^2+14*n-9) + 9)/48; seq(A270694(n), n=0..40); # G. C. Greubel, Apr 02 2021

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -23, 51}, 39]
Table[((-1)^n (2 n + 1) (14 n^2 + 14 n - 9) + 9)/48, {n, 0, 38}]

my(x='x+O('x^50)); concat(0, Vec(-x*(1-5*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016

[((-1)^n*(2*n+1)*(14*n^2+14*n-9) +9)/48 for n in (0..40)] # G. C. Greubel, Apr 02 2021

G.f.: -x*(1 - 5*x + x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48.
E.g.f.: (1/48)*(9*exp(x) - (9 + 66*x - 126*x^2 + 28*x^3)*exp(-x)). - G. C. Greubel, Mar 28 2016

A270695 Alternating sum of centered octagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 9, -26, 58, -107, 179, -276, 404, -565, 765, -1006, 1294, -1631, 2023, -2472, 2984, -3561, 4209, -4930, 5730, -6611, 7579, -8636, 9788, -11037, 12389, -13846, 15414, -17095, 18895, -20816, 22864, -25041, 27353, -29802, 32394, -35131, 38019

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Centered pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000447 (centered octagonal pyramidal numbers).

Cf. A000330, A006324 (partial sums of centered octagonal pyramidal numbers), A019298, A232599.

[((-1)^n*(4*n^2 - 1)*(2*n + 3) + 3)/12 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270695:= n-> ((-1)^n*(4*n^2 -1)*(2*n+3) +3)/12: seq(A270695(n), n=0..40); # G. C. Greubel, Apr 02 2021

LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 9, -26, 58}, 39]
Table[((-1)^n (4 n^2 - 1) (2 n + 3) + 3)/12, {n, 0, 38}]

x='x+O('x^100); concat(0, Vec(-x*(1-6*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016

[((-1)^n*(4*n^2 -1)*(2*n+3) +3)/12 for n in (0..40)] # G. C. Greubel, Apr 02 2021

G.f.: -x*(1 - 6*x + x^2)/((1 - x)*(1 + x)^4).
E.g.f.: (1/12)*(3*exp(x) - (3 + 18*x - 36*x^2 + 8*x^3)*exp(-x)).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(4*n^2 - 1)*(2*n + 3) + 3)/12.

a(6)=179 inserted by Georg Fischer, Apr 03 2019

A319777 a(n) is the number of equivalence classes of triples of sets each with n or fewer elements where two triples are equivalent if the number of elements in all intersections is the same.

Original entry on oeis.org

1, 15, 100, 436, 1459, 4069, 9929, 21871, 44426, 84494, 152171, 261749, 432906, 692102, 1074198, 1624314, 2399943, 3473337, 4934182, 6892578, 9482341, 12864643, 17232007, 22812673, 29875352, 38734384, 49755317, 63360923, 80037668, 100342652, 124911036

Offset: 0

Peter Kagey, Sep 26 2018

nonn

A019298(n) is the analogous sequence if the three sets must each have exactly n elements.

			The triple (A, B, C) = ({1, 2}, {1, 2, 3}, {1, 4}) is equivalent to the triple (A', B', C') = ({1, 8}, {1, 4, 8}, {5, 8}) because all intersections of the sets in a triple are equal:
|A|         = |{1, 2}|    = 2 = |{1, 8}|    = |A'|
|B|         = |{1, 2, 3}| = 3 = |{1, 4, 8}| = |B'|
|C|         = |{1, 4}|    = 2 = |{5, 8}|    = |C'|
|A & B|     = |{1, 2}|    = 2 = |{1, 8}|    = |A' & B'|
|A & C|     = |{1}|       = 1 = |{8}|       = |A' & C'|
|B & C|     = |{1}|       = 1 = |{8}|       = |B' & C'|
|A & B & C| = |{1}|       = 1 = |{8}|       = |A' & B' & C'|

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Cf. A019298, A244865.

Cf. A000330(n-1) is analogous, but with pairs instead of triples.

List([0..30],n->Sum([0..n],k->(15*(127+(-1)^k)+6432*k+8936*k^2+6480*k^3+2570*k^4+528*k^5+44*k^6)/1920)); # Muniru A Asiru, Sep 28 2018

a:=n->add((15*(127+(-1)^k)+6432*k+8936*k^2+6480*k^3+2570*k^4+528*k^5+44*k^6)/1920,k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Sep 28 2018

a(n) = sum(k=0, n, (15*(127+(-1)^k) + 6432*k + 8936*k^2 + 6480*k^3 + 2570*k^4 + 528*k^5 + 44*k^6) / 1920); \\ Michel Marcus, Dec 27 2018

Vec((1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^8*(1 + x)) + O(x^40)) \\ Colin Barker, Dec 28 2018

a(n) = Sum_{k=0..n} A244865(k). [corrected by Michel Marcus, Dec 27 2018]
From Colin Barker, Dec 27 2018: (Start)
G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^8*(1 + x)).
a(n) = 7*a(n-1) - 20*a(n-2) + 28*a(n-3) - 14*a(n-4) - 14*a(n-5) + 28*a(n-6) - 20*a(n-7) + 7*a(n-8) - a(n-9) for n>8.
(End)

cp's OEIS Frontend

A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.

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A008764 Number of 3 X 3 symmetric stochastic matrices under row and column permutations.

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A237669 Number of prime parts in the partitions of 3n into 3 parts.

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Mathematica

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A270694 Alternating sum of centered heptagonal pyramidal numbers.

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Magma

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A270695 Alternating sum of centered octagonal pyramidal numbers.

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Magma

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A319777 a(n) is the number of equivalence classes of triples of sets each with n or fewer elements where two triples are equivalent if the number of elements in all intersections is the same.

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