A035599 -id:A035599 - cp's OEIS frontend

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22

Offset: 1

Thomas Wieder, Jul 30 2006, Aug 06 2006

			The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
The main diagonal seems to be A050146.
Square array A(n,k) begins:
   4,   8,   12,   16,    20,    24,     28,     32,      36, ...
   6,  18,   38,   66,   102,   146,    198,    258,     326, ...
   8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
  10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
  12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
  14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
  16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
  18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
  20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Other versions: A035607, A113413, A122542, A266213.

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.

A:= proc(m, n)  option remember;
      `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Apr 21 2012

A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.

Offset and typos corrected by Alois P. Heinz, Apr 21 2012

A008413 Coordination sequence for 5-dimensional cubic lattice.

Original entry on oeis.org

1, 10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, 14002, 20330, 28610, 39210, 52530, 69002, 89090, 113290, 142130, 176170, 216002, 262250, 315570, 376650, 446210, 525002, 613810, 713450, 824770, 948650, 1086002, 1237770, 1404930

Offset: 0

N. J. A. Sloane

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035599.
Row 5 of A035607, A266213.
Column 5 of A113413, A119800, A122542.

Maple
```
4/3*n^4+20/3*n^2+2;
```

LinearRecurrence[{5,-10,10,-5,1},{1,10,50,170,450,1002},40] (* Harvey P. Dale, May 02 2016 *)
{1}~Join~Table[4/3 n^4 + 20/3 n^2 + 2, {n, 32}] (* or *)
CoefficientList[Series[((1 + x)/(1 - x))^5, {x, 0, 32}], x] (* Michael De Vlieger, Oct 04 2016 *)

G.f.: ((1+x)/(1-x))^5.
a(n) = (4/3)*n^4 + (20/3)*n^2 + 2 for n > 0. - Michael De Vlieger, Oct 04 2016
n*a(n) = 10*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
From Shel Kaphan, Mar 03 2023: (Start)
a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=5, for n>=1.
a(n) = A035599(n)*5/n, for n>0. (End)

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

[( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

CoefficientList[Series[2*x*(1+x)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)

a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

A067969 Number of nodes in virtual, "optimal", chordal graphs of diameter 5, degree =n+1.

Original entry on oeis.org

11, 20, 61, 102, 231, 360, 681, 1002, 1683, 2364, 3653, 4942, 7183, 9424, 13073, 16722, 22363, 28004, 36365, 44726, 56695, 68664, 85305, 101946, 124515, 147084, 177045, 207006, 246047, 285088, 335137, 385186, 448427, 511668, 590557, 669446

Offset: 1

S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 11 2002

			a(5)=231 n=odd, t=3, a(5)=324/5+54+72+30+46/5+1=231 a(6)=360 n=even, t=3, a(6)=231+(24*16)/3+1=231+128+1=360

Concrete Mathematics - R. L. Graham, D. E. Knuth, O. Patashnik, 1994,Addison-Wesley Company, Inc.

Robert Israel, Table of n, a(n) for n = 1..10000
Ronald Cools, Ian H. Sloan, Minimial cubature formulae of trigonometric degree, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 5.
Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Cf. A001847 (bisection), A035599 (bisection).

for n from 1 to k do if ((n mod 2 ) = 1) then t := (n+1)/2; a[n] := 4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1; else t := n/2; a[n] := ((4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1)+((2*(t*(t+1)*(t^2+t+4))/3)+1)); fi; print(a[n]); od;

n - odd: t=(n+1)/2, a[n] := 4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1; n - even: t=n/2, a(n) := (4/15*t^5+2/3*t^4+8/3*t^3+10/3*t^2+46/15*t+1)+((2*(t*(t+1)*(t^2+t+4))/3)+1)
G.f.: x*(11-2*x-12*x^2+8*x^3+26*x^4-12*x^5-12*x^6+8*x^7+3*x^8-2*x^9)/ ((1+x)^4 * (x-1)^6) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]
(n+1)*a(n) -2*a(n-1) -18*a(n-2) -2*a(n-3) +(-n+1)*a(n-4)=0. - R. J. Mathar, Apr 07 2025

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

cp's OEIS Frontend

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

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A008413 Coordination sequence for 5-dimensional cubic lattice.

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A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

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A067969 Number of nodes in virtual, "optimal", chordal graphs of diameter 5, degree =n+1.

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