A103903
Square array T(n,k) read by antidiagonals: coordination sequence for lattice D_n.
Original entry on oeis.org
1, 1, 24, 1, 40, 144, 1, 60, 370, 456, 1, 84, 792, 1640, 1056, 1, 112, 1498, 4724, 4930, 2040, 1, 144, 2592, 11620, 18096, 11752, 3504, 1, 180, 4194, 25424, 55650, 52716, 24050, 5544, 1, 220, 6440, 50832, 149568, 195972, 127816, 44200, 8256, 1, 264
Offset: 4
1,24,144,456,1056,2040,3504,5544,8256,11736,
1,40,370,1640,4930,11752,24050,44200,75010,119720,
1,60,792,4724,18096,52716,127816,271908,524640,938652,
1,84,1498,11620,55650,195972,559258,1371316,2999682,6003956,
1,112,2592,25424,149568,629808,2100832,5910288,14610560,32641008,
1,144,4194,50832,361602,1801872,6976866,22413456,62407170,155242640,
- M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, Zeit. f. Kristallographie, 212 (1997), 253-256 and arXiv:cond-mat/9706122.
- 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.
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nmin = 4; nmax = 13; f[x_, n_] := ((1/2)*((-1+Sqrt[x])^(2n)+(1+Sqrt[x])^(2n))*(1-x)^n) / (-1+x)^(2n)-(2n*x*(1+x)^(n-2)) / (1-x)^n; t = Table[ CoefficientList[ Series[ f[x, n], {x, 0, nmax-nmin} ], x], {n, nmin, nmax} ]; Flatten[ Table[ t[[n-k+1, k]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after g.f. *)
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T(n,k)={polcoeff(sum(i=0, n, (binomial(2*n, 2*i) - 2*n*binomial(n-2, i-1))*x^i)/(1-x)^n + O(x*x^k), k)} \\ Andrew Howroyd, Jul 03 2018
A289156
Largest area of triangles with integer sides and area = n times perimeter.
Original entry on oeis.org
60, 1224, 8436, 34320, 103020, 254040, 546084, 1060896, 1907100, 3224040, 5185620, 8004144, 11934156, 17276280, 24381060, 33652800, 45553404, 60606216, 79399860, 102592080, 130913580, 165171864, 206255076, 255135840, 312875100, 380625960, 459637524, 551258736
Offset: 1
For n = 4, a(4) = 34320 means for the largest triangles (a,b,c) = (66,4225,4289), the area is 34320 which is 4 times the perimeter 8580.
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Table[4 n (2 n^2 + 1) (4 n^2 + 1), {n, 27}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {60, 1224, 8436, 34320, 103020, 254040}, 27] (* or *) Rest@ CoefficientList[Series[12 x (5 + 72 x + 166 x^2 + 72 x^3 + 5 x^4)/(1 - x)^6, {x, 0, 27}], x] (* Michael De Vlieger, Jul 03 2017 *)
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Vec(12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6 + O(x^30)) \\ Colin Barker, Jun 28 2017
A365356
Number of free face-connected 4-dimensional polyhypercubes with n cells.
Original entry on oeis.org
1, 1, 5, 36, 407, 7579
Offset: 1
See
A365366 for a table of similar sequences.
A323282
Coordination sequence for Leech lattice.
Original entry on oeis.org
1, 196560, 39462040800
Offset: 0
The graph G has degree 196560, so a(1)=196560.
Let L(i) denote the set of vectors in the Leech lattice having norm 2i, and let c(i) = |L(i)| = A008408(i). The first few sets L(i) are described on page 181 of the ATLAS and in Table 4.13 of the sphere packing book.
The vertices at edge-distance 2 from the vertex 0^24 consist of the sets L(3), L(4), L(5), L(6), and the doubles of L(2), so a(2) = c(2)+c(3)+...+c(6) = 39462040800.
To prove this, note that the group Aut(G) = Co_0 acts transitively on each of the sets L(2), L(3), L(4), L(5), and L(7) (cf. A323273). It is easy to find one representative from each set L(3), L(4), L(5) among the sums of pairs of minimal vectors. There are two orbits on L(6) and again it is easy to find a representative from each orbit, for example
[3, 3, 3, 3, 3, 3, 3, 3, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1]
and
[2, 2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2].
Among the sums of pairs of minimal vectors there are no vectors from L(7), so there is no contribution from c(7).
Finally, the only possibility for a vector of norm 16 is the double of a minimal vector, and there are c(2) = 196560 of these, all in one obit under the group.
So a(2) = c(3)+c(4)+c(5)+c(6)+c(2) = 39462040800.
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. 181.
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993; see page xxxvi and Tables 4.13 and 4.14.
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
A107506
Coordination sequence for D_11 lattice.
Original entry on oeis.org
1, 220, 9482, 166188, 1660450, 11260876, 57581898, 237782204, 831240322, 2543995388, 6986455498, 17539148684, 40829208354, 89125931884, 184069213834, 362272019868, 683489419778, 1242224123932, 2183844052874, 3726523300588, 6190586013986, 10037135290380
Offset: 0
- Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
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