cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

Original entry on oeis.org

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267
Offset: 0

Views

Author

Clark Kimberling, Jun 01 2012

Keywords

Comments

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
A: 2, 0, -2, 1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
B: 3, -2, -2, 3, -1;
C: 4, -6, 4, -1;
D: 1, 2, -2, -1, 1;
E: 2, 1, -4, 1, 2, -1;
F: 2, -1, 1, -2, 1;
G: 2, -1, 0, 1, -2, 1;
H: 2, -1, 2, -4, 2, -1, 2, -1;
I: 3, -3, 2, -3, 3, -1;
J: 4, -7, 8, -7, 4, -1.
...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties. Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

Examples

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

References

  • A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
  • P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Links

Crossrefs

Cf. A047241, A211795.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)
  • PARI
    a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A211790 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k

Original entry on oeis.org

1, 7, 1, 23, 7, 1, 54, 22, 7, 1, 105, 51, 22, 7, 1, 181, 97, 50, 22, 7, 1, 287, 166, 96, 50, 22, 7, 1, 428, 263, 163, 95, 50, 22, 7, 1, 609, 391, 255, 161, 95, 50, 22, 7, 1, 835, 554, 378, 253, 161, 95, 50, 22, 7, 1, 1111, 756, 534, 374, 252, 161, 95, 50, 22, 7
Offset: 1

Views

Author

Clark Kimberling, Apr 21 2012

Keywords

Comments

...
Let R be the array in A211790 and let R' be the array in A211793. Then R(k,n) + R'(k,n) = 3^(n-1). Moreover, (row k of R) =(row k of A211796) for k>2, by Fermat's last theorem; likewise, (row k of R')=(row k of A211799) for k>2.
...
Generalizations: Suppose that b,c,d are nonzero integers, and let U(k,n) be the number of ordered triples (w,x,y) with all terms in {1,...,n} and b*w*k c*x^k+d*y^k, where the relation is one of these: <, >=, <=, >. What additional assumptions force the limiting row sequence to be essentially one of these: A002412, A000330, A016061, A174723, A051925?
In the following guide to related arrays and sequences, U(k,n) denotes the number of (w,x,y) as described in the preceding paragraph:
first 3 rows limiting row sequence
A211790: w^k < x^k+y^k A004068, A211635, A211650 A002412
A211793: w^k >= x^k+y^k A000292, A211636, A211651 A000330
A211796: w^k <= x^k+y^k A002413, A211634, A211650 A002412
A211799: w^k > x^k+y^k A000292, A211637, A211651 A000330
A211802: 2w^k < x^k+y^k A182260, A211800, A211801 A016061
A211805: 2w^k >= x^k+y^k A055232, A211803, A211804 A000330
A211808: 2w^k <= x^k+y^k A055232, A211806, A211807 A174723
A182259: 2w^k > x^k+y^k A182260, A211810, A211811 A051925

Examples

			Northwest corner:
  1, 7, 23, 54, 105, 181, 287, 428, 609
  1, 7, 22, 51,  97, 166, 263, 391, 554
  1, 7, 22, 50,  96, 163, 255, 378, 534
  1, 7, 22, 50,  95, 161, 253, 374, 528
  1, 7, 22, 50,  95, 161, 252, 373, 527
For n=2 and k>=1, the 7 triples (w,x,y) are (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).

Crossrefs

Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.
Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259

Programs

  • Mathematica
    z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k < x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A004068 *)
Table[t[2, n], {n, 1, z}]  (* A211635 *)
Table[t[3, n], {n, 1, z}]  (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
Table[n (n + 1) (4 n - 1)/6,
  {n, 1, z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

R(k,n) = n(n-1)(4n+1)/6 for 1<=k<=n, and
R(k,n) = Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.

A213697 T(n,k)=Half the number of (n+1)X(n+1) symmetric 0..k arrays with no 2X2 subblock summing to 2k.

Original entry on oeis.org

3, 11, 13, 28, 192, 70, 56, 1320, 7827, 529, 99, 5470, 208072, 750261, 5148, 159, 17499, 2322599, 109820000, 168382284, 68798, 240, 45892, 16537086, 4286833981, 193946752094, 88681187619, 1220409, 344, 105856, 84059234, 83545858925
Offset: 1

Views

Author

R. H. Hardin Jun 18 2012

Keywords

Comments

Table starts
.....3..........11...............28..................56....................99
....13.........192.............1320................5470.................17499
....70........7827...........208072.............2322599..............16537086
...529......750261........109820000..........4286833981...........83545858925
..5148...168382284.....193946752094......34389632236877......2256314178023600
.68798.88681187619.1146407046198040.1199143411128185584.325754552781442854792

Examples

			Some solutions for n=4 k=4
..0..4..0..2..3....1..2..4..4..0....2..4..3..3..2....3..4..3..1..2
..4..2..0..3..3....2..1..3..4..1....4..2..3..3..1....4..2..2..1..0
..0..0..4..3..1....4..3..2..1..0....3..3..2..2..3....3..2..3..4..2
..2..3..3..2..0....4..4..1..0..0....3..3..2..3..2....1..1..4..2..4
..3..3..1..0..4....0..1..0..0..4....2..1..3..2..2....2..0..2..4..1

Links

Crossrefs

Row 1 is A182260(n+1)

A212782 T(n,k)=Half the number of 0..k arrays of length n+2 with second differences nonzero.

Original entry on oeis.org

3, 11, 5, 28, 27, 8, 56, 99, 66, 13, 99, 252, 350, 162, 21, 159, 546, 1134, 1238, 397, 34, 240, 1034, 3010, 5104, 4379, 973, 55, 344, 1803, 6724, 16594, 22972, 15490, 2385, 89, 475, 2925, 13544, 43727, 91482, 103391, 54793, 5846, 144, 635, 4517, 24870, 101743
Offset: 1

Views

Author

R. H. Hardin May 27 2012

Keywords

Comments

Table starts
...3....11......28.......56........99........159.........240.........344
...5....27......99......252.......546.......1034........1803........2925
...8....66.....350.....1134......3010.......6724.......13544.......24870
..13...162....1238.....5104.....16594......43727......101743......211461
..21...397....4379....22972.....91482.....284360......764297.....1797975
..34...973...15490...103391....504337....1849215.....5741427....15287517
..55..2385...54793...465336...2780392...12025596....43129809...129984123
..89..5846..193821..2094360..15328203...78203437...323992699..1105207105
.144.14329..685609..9426184..84503842..508563345..2433844978..9397168738
.233.35122.2425226.42424863.465866699.3307229006.18283132290.79900662826

Examples

			Some solutions for n=5 k=4
..2....2....2....1....2....4....1....3....3....4....3....2....1....1....0....4
..0....2....2....1....2....2....4....4....1....1....4....4....3....3....3....3
..3....3....0....2....1....2....2....0....0....3....2....1....0....3....2....3
..2....3....2....4....3....4....3....2....4....3....3....3....2....2....3....4
..3....1....2....4....4....1....0....0....3....1....2....4....1....0....0....1
..2....3....0....2....3....1....4....3....3....1....0....3....3....1....1....1
..4....3....4....2....1....2....0....0....1....0....2....1....3....1....1....4

Links

Crossrefs

Column 1 is A000045(n+3)
Row 1 is A182260(n+1)

A182259 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k<=x^k+y

Original entry on oeis.org

0, 3, 0, 11, 3, 0, 28, 11, 3, 0, 56, 28, 11, 3, 0, 99, 56, 26, 11, 3, 0, 159, 97, 52, 26, 11, 3, 0, 240, 153, 93, 50, 26, 11, 3, 0, 344, 230, 149, 85, 50, 26, 11, 3, 0, 475, 330, 222, 139, 85, 50, 26, 11, 3, 0, 635, 453, 314, 212, 133, 85, 50, 26, 11, 3, 0, 828
Offset: 1

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

Row 1: A182260
Row 2: A211810
Row 3: A211811
Limiting row sequence: A051925
Let R be the array in A211808 and let R' be the array in A182259. Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

Examples

			Northwest corner (with antidiagonals read from northeast to southwest):
0...3...11...28...56...99...159
0...3...11...28...56...97...153
0...3...11...26...52...93...149
0...3...11...26...50...85...139
0...3...11...26...50...85...133

Crossrefs

Cf. A211790.

Programs

  • Mathematica
    z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[2 w^k > x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A182260 *)
Table[t[2, n], {n, 1, z}]  (* A211810 *)
Table[t[3, n], {n, 1, z}]  (* A211811 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
    {n, 1, 12}, {k, 1, n}]] (* A182259 *)
Table[k (k - 1) (2 k + 5)/6,
    {k, 1, z}] (* row-limit sequence, A051925 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A211802 R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2*w^k < x^k + y^k; square array read by descending antidiagonals.

Original entry on oeis.org

0, 3, 0, 11, 3, 0, 28, 13, 3, 0, 56, 32, 13, 3, 0, 99, 64, 34, 13, 3, 0, 159, 113, 68, 34, 13, 3, 0, 240, 181, 117, 70, 34, 13, 3, 0, 344, 272, 187, 125, 70, 34, 13, 3, 0, 475, 388, 282, 197, 125, 70, 34, 13, 3, 0, 635, 535, 406, 292, 203, 125, 70, 34, 13, 3, 0
Offset: 1

Views

Author

Clark Kimberling, Apr 22 2012

Keywords

Comments

Row 1: A182260.
Row 2: A211800.
Row 3: A211801.
Limiting row sequence: A016061.
Let R be the array in this sequence and let R' be the array in A211805. Then R(k,n) + R'(k,n) = 3^(n-1).
See the Comments at A211790.

Examples

			Northwest corner:
  0   3  11  28  56  99 159 240
  0   3  13  32  64 113 181 272
  0   3  13  34  68 117 187 282
  0   3  13  34  70 125 197 292
  0   3  13  34  70 125 203 302

Crossrefs

Cf. A016061, A182260, A211790, A211800, A211801, A211802, A211805.

Programs

  • Mathematica
    z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[2 w^k < x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A182260 *)
Table[t[2, n], {n, 1, z}]  (* A211800 *)
Table[t[3, n], {n, 1, z}]  (* A211801 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12},
                {k, 1, n}]] (* this sequence *)
Table[k (k - 1) (4 k + 1)/6, {k, 1,
  z}] (* row-limit sequence, A016061 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Extensions

Definition corrected by Georg Fischer, Sep 10 2022

A008670 Molien series for Weyl group F_4.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 7, 9, 11, 12, 16, 18, 20, 24, 28, 30, 36, 40, 44, 50, 56, 60, 69, 75, 81, 90, 99, 105, 117, 126, 135, 147, 159, 168, 184, 196, 208, 224, 240, 252, 272, 288, 304, 324, 344, 360, 385, 405, 425, 450, 475, 495, 525, 550, 575, 605, 635, 660, 696, 726, 756
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of partitions of n into parts 1, 3, 4 and 6. - Ilya Gutkovskiy, May 24 2017

References

  • Coxeter and Moser, Generators and Relations for Discrete Groups, Table 10.
  • L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 28).

Links

Crossrefs

Cf. A002411, A006002, A010875, A011934, A027480, A055232, A182260.

Programs

  • Magma
    MolienSeries(CoxeterGroup("F4")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Magma
    R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 08 2019
  • Maple
    a:= proc(n) local m, r; m := iquo (n, 12, 'r'); r:= r+1; ([4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26][r]+ (6+r+4*m)*m)*m+ [1$3, 2, 3$2, 5, 6, 7, 9, 11, 12][r] end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 06 2008
  • Mathematica
    Take[CoefficientList[Series[1/((1-x^2)(1-x^6)(1-x^8)(1-x^12)),{x,0,130}], x], {1,-1,2}] (* or *) LinearRecurrence[ {1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1},{1,1,1,2,3,3,5,6,7,9,11,12,16,18},70] (* Harvey P. Dale, Feb 07 2012 *)
  • PARI
    my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))) \\ G. C. Greubel, Sep 08 2019
  • Sage
    def A008670_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6))).list()
A008670_list(70) # G. C. Greubel, Sep 08 2019

Formula

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^6)). [Corrected by Ralf Stephan, Apr 29 2014]
a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-11) + a(n-13) - a(n-14), with a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=3, a(6)=5, a(7)=6, a(8)=7, a(9)=9, a(10)=11, a(11)=12, a(12)=16, a(13)=18. - Harvey P. Dale, Feb 07 2012
a(n) ~ (1/432)*n^3. - Ralf Stephan, Apr 29 2014
a(n) = (120*floor(n/6)^3 + 60*(m+7)*floor(n/6)^2 + 2*(m^5-15*m^4+75*m^3-135*m^2+134*m+240)*floor(n/6) + 3*(m^5-15*m^4+75*m^3-135*m^2+84*m+70) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018
a(n) = 1 + floor((2*n^3 + 42*n^2 + n*(279 + 9*(-1)^n - 48*[(n mod 3)=2]))/864) where [] is the Iverson bracket. - Hoang Xuan Thanh, Jun 22 2025
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