A077043 -id:A077043 - cp's OEIS frontend

A212504 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>2z.

0, 0, 0, 7, 24, 76, 162, 333, 576, 976, 1500, 2275, 3240, 4572, 6174, 8281, 10752, 13888, 17496, 21951, 27000, 33100, 39930, 48037, 57024, 67536, 79092, 92443, 107016, 123676, 141750, 162225, 184320, 209152, 235824, 265591, 297432

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A211795.

A212504:=n->ceil(3*n^2/4)*floor((n-1)^2/4); seq(A212504(n), n=0..40); # Wesley Ivan Hurt, Jan 24 2014

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y > 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212504 *)

a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).
From Alois P. Heinz, May 31 2012: (Start)
a(n) = A077043(n) * A002620(n-1) = ceiling(n^2*3/4) * floor((n-1)^2/4).
G.f.: x^3*(7+10*x+14*x^2+4*x^3+x^4) / ((x+1)^3*(1-x)^5). (End)

A212505 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>=2z.

Original entry on oeis.org

0, 0, 3, 14, 48, 114, 243, 444, 768, 1220, 1875, 2730, 3888, 5334, 7203, 9464, 12288, 15624, 19683, 24390, 30000, 36410, 43923, 52404, 62208, 73164, 85683, 99554, 115248, 132510, 151875, 173040, 196608, 222224, 250563, 281214, 314928

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A211795.

a:= n-> floor(n^2/4)*ceil(n^2*3/4):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 13 2013

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y >= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212505 *)

a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).
From Alois P. Heinz, May 31 2012: (Start)
a(n) = A077043(n) * A002620(n) = floor(n^2/4)*ceiling(n^2*3/4).
G.f.: x^2*(x^2+2*x+3)*(3*x^2+2*x+1) / ((x+1)^3*(1-x)^5). (End)

A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.

Original entry on oeis.org

0, 1, 12, 56, 168, 399, 810, 1480, 2496, 3965, 6000, 8736, 12312, 16891, 22638, 29744, 38400, 48825, 61236, 75880, 93000, 112871, 135762, 161976, 191808, 225589, 263640, 306320, 353976, 406995, 465750, 530656, 602112, 680561, 766428

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A211795.

[(2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32: n in [0..34]]; // Bruno Berselli, May 31 2012

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y <= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212507 *)
CoefficientList[Series[x (1 + 2 x) (1 + 7 x + 7 x^2 + 3 x^3)/((1 + x)^2 (1 - x)^5), {x, 0, 34}], x] (* Bruno Berselli, May 31 2012 *)

a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).
G.f.: x*(1+2*x)*(1+7*x+7*x^2+3*x^3)/((1+x)^2*(1-x)^5). [Bruno Berselli, May 31 2012]
a(n) = (2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32. [Bruno Berselli, May 31 2012]
a(n) = A006578(n) * A077043(n). - Alois P. Heinz, May 31 2012

A256524 Number of partitions of 3n into at most 4 parts.

Original entry on oeis.org

1, 3, 9, 18, 34, 54, 84, 120, 169, 225, 297, 378, 478, 588, 720, 864, 1033, 1215, 1425, 1650, 1906, 2178, 2484, 2808, 3169, 3549, 3969, 4410, 4894, 5400, 5952, 6528, 7153, 7803, 8505, 9234, 10018, 10830, 11700, 12600, 13561, 14553, 15609, 16698, 17854, 19044

Offset: 0

Colin Barker, Apr 01 2015

			For n=1 the 3 partitions of 1*3 = 3 are [3], [1,2] and [1,1,1].

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

Cf. A001400, A077043 (3 parts), A256525 (5 parts), A256315 (6 parts).

LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 3, 9, 18, 34, 54, 84, 120}, 50] (* Jean-François Alcover, Apr 26 2017 *)

concat(1, vector(40, n, k=0; forpart(p=3*n, k++, , [1,4]); k))

Vec((x^2+x+1)*(2*x^2+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100))

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

a(n) = A001400(3n). - Alois P. Heinz, Apr 01 2015

A256525 Number of partitions of 3n into at most 5 parts.

Original entry on oeis.org

1, 3, 10, 23, 47, 84, 141, 221, 333, 480, 674, 918, 1226, 1602, 2062, 2611, 3266, 4033, 4932, 5969, 7166, 8529, 10083, 11835, 13811, 16019, 18487, 21224, 24260, 27604, 31289, 35324, 39744, 44559, 49806, 55496, 61667, 68331, 75529, 83273, 91606, 100540

Offset: 0

Colin Barker, Apr 01 2015

			For n=1 the 3 partitions of 1*3 = 3 are [3], [1,2] and [1,1,1].

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

Cf. A001401, A077043 (3 parts), A256524 (4 parts), A256315 (6 parts).

Table[Length[IntegerPartitions[3n,5]],{n,0,50}] (* Harvey P. Dale, Mar 08 2019 *)

concat(1, vector(40, n, k=0; forpart(p=3*n, k++, , [1,5]); k))

Vec(-(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100))

G.f.: -(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)).

a(n) = A001401(3n). - Alois P. Heinz, Apr 01 2015

A276382 a(1) = 1, and a(n) = a(n-1) + floor(3*n/2) + 1 for n >= 2.

Original entry on oeis.org

1, 5, 10, 17, 25, 35, 46, 59, 73, 89, 106, 125, 145, 167, 190, 215, 241, 269, 298, 329, 361, 395, 430, 467, 505, 545, 586, 629, 673, 719, 766, 815, 865, 917, 970, 1025, 1081, 1139, 1198, 1259, 1321, 1385, 1450, 1517, 1585, 1655, 1726, 1799, 1873, 1949, 2026

Offset: 1

Bob Selcoe, Sep 01 2016

Given 3 distinct numbers i, j and k whose prime signatures are exactly n 1's, then a(n) is the number of prime signatures for all permutations of i*j*k.
a(n) is the number of partitions of 3n such that there are no more than n-1 3's and no parts > 3.
a(3n+1) represents the number of prime signature sets whose members are excluded as terms in A026477, as a consequence of being products of three smaller terms whose prime signatures are exactly 3n+1 1's.
First differences are floor(3n/2) + 1 (A001651(n+1)); second differences are 1 when n is even and 2 when n is odd; third differences are 1 when n is even and -1 when n is odd.

			a(2)=5; the 5 prime signatures / partitions are: {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1} and {1,1,1,1,1,1}.
G.f. = x + 5*x^2 + 10*x^3 + 17*x^4 + 25*x^5 + 35*x^6 + 46*x^7 + ... - _Michael Somos_, Sep 08 2023

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

Cf. A000040 (prime numbers), A001651, A026477.

Equals one less than A331952(n+1), two less than A077043(n+1), and three less than A084684(n+1). - Greg Dresden, Feb 22 2020

[((-1)^n+12*n+6*n^2-9)/8: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[3 n/2] + 1 ; Array[a, 51] (* Michael De Vlieger, Sep 01 2016 *)
a[n_] := Floor[(3*(n+1)^2 - 7)/4]; (* Michael Somos, Sep 08 2023 *)

Vec(x*(1+3*x-x^3)/((1-x)^3*(1+x)) + O(x^60)) \\ Colin Barker, Sep 01 2016

{a(n) = (3*(n+1)^2 - 7)\4}; /* Michael Somos, Sep 09 2023 */

From Colin Barker, Sep 01 2016: (Start)
a(n) = ((-1)^n + 12*n + 6*n^2 - 9)/8 for n > 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.
G.f.: x*(1 + 3*x - x^3) / ((1-x)^3*(1+x)). (End)
a(n) = -1 + Sum_{k=1..n} floor((n+k+3)/2). - Wesley Ivan Hurt, Apr 01 2017
a(n) = a(-2-n) for all n in Z. - Michael Somos, Sep 08 2023
a(n) = floor((3*(n+1)^2 - 7)/4). - Michael Somos, Sep 09 2023

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

Original entry on oeis.org

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179

Offset: 0

Paul Curtz, Feb 11 2020

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:
                       17  17  17  17  17  18  18
                     16  11  11  11  11  12  12  18
                   16  11   6   6   7   7   7  12  18
                 16  10   6   3   3   3   3   7  12  18
               16  10   6   3   1   1   1   4   7  12  19
             16  10   6   2   0   0   0   1   4   8  13  19
               15  10   5   2   0   0   1   4   8  13  19
                 15  10   5   2   2   2   4   8  13  19
                   15   9   5   5   5   4   8  13  19
                     15   9   9   9   9   8  13  20
                       15  14  14  14  14  14  20
a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.
Difference table:
  0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)
  1, 2, 3, 5,  6,  7,  8,  9, 11, 12, ... = A047256(n+1)
  1, 1, 2, 1,  1,  1,  1,  2,  1,  1, ... = A130782.
There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

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

Cf. A002266, A008602, A047256, A130782, A330707.

Cf. A001859, A006578, A077043.

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)

concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

a(8+n) - a(8-n) = 20*n.

G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020

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.

A213801 Number of 3 X 3 0..n symmetric arrays with all rows summing to floor(n*3/2).

Original entry on oeis.org

4, 13, 29, 57, 96, 153, 226, 323, 440, 587, 759, 967, 1204, 1483, 1796, 2157, 2556, 3009, 3505, 4061, 4664, 5333, 6054, 6847, 7696, 8623, 9611, 10683, 11820, 13047, 14344, 15737, 17204, 18773, 20421, 22177, 24016, 25969, 28010, 30171, 32424, 34803, 37279

Offset: 1

R. H. Hardin, Jun 20 2012

nonn

Row 3 of A213800.

Sequence is difference between numbers of triangles, regardless of size, in A064412 (a family of ((3*n^2+3*n+2)/2)-iamonds, see also illustration of initial terms there) and a quantity A077043 of triangles of dimension 1. - Luce ETIENNE, Aug 23 2014

			Some solutions for n=4:
..1..3..2....2..4..0....0..4..2....1..2..3....1..1..4....4..0..2....2..2..2
..3..1..2....4..0..2....4..0..2....2..2..2....1..3..2....0..2..4....2..2..2
..2..2..2....0..2..4....2..2..2....3..2..1....4..2..0....2..4..0....2..2..2
a(2)=5-1=4, a(3)=14-1=13, a(210)=4118206-8269=4109937. - _Luce ETIENNE_, Aug 23 2014

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A064412, A077403.

Empirical: a(n) = 2*a(n-1) -2*a(n-3) +2*a(n-4) -2*a(n-5) +2*a(n-7) -a(n-8).
Empirical: G.f. -x*(-4-5*x-3*x^2-7*x^3-x^5-2*x^6+x^7) / ( (x^2+1)*(1+x)^2*(x-1)^4 ). - R. J. Mathar, Jul 04 2012
a(n) = (14*n^3+42*n^2+53*n+25+3*(n+1)*(-1)^n+2*((-1)^((2*n+1-(-1)^n)/4)-(-1)^((6*n+5-(-1)^n)/4)))/32. - Luce ETIENNE, Aug 23 2014
a(n) = A064412(n+1) - A077043((2*n+1-(-1)^n)/4). - Luce ETIENNE, Aug 23 2014

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

cp's OEIS Frontend

A212504 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>2z.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A212505 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>=2z.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A256524 Number of partitions of 3n into at most 4 parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A256525 Number of partitions of 3n into at most 5 parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A276382 a(1) = 1, and a(n) = a(n-1) + floor(3*n/2) + 1 for n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A332410 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.