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 1-3 of 3 results.

A110907 Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.

Original entry on oeis.org

1, 12, 50, 108, 194, 300, 434, 588, 770, 972, 1202, 1452, 1730, 2028, 2354, 2700, 3074, 3468, 3890, 4332, 4802, 5292, 5810, 6348, 6914, 7500, 8114, 8748, 9410, 10092, 10802, 11532, 12290, 13068, 13874, 14700, 15554, 16428, 17330, 18252, 19202
Offset: 0

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Author

N. J. A. Sloane, Apr 15 2008

Keywords

Comments

This lattice consists of all points (x,y,z) where x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
The sequence for the D_k lattice has the terms ((2*n+1)^k-(2*n-1)^k)/2, if k is even, and the terms ((2n+1)^k-(2*n-1)^k)/2+(-1)^n if k is odd (like here for k=3). The sequence for A_2 is A008458, for A_3 A010006, for A_4 the first differences of A083669. A_5 is 2+2*n^2*(25+44*n^2) if n>0, and 1 if n=0. - R. J. Mathar, Feb 09 2010

Examples

			a(0) = 1: 000
a(1) = 12: +-1 +-1 0, where the 0 can be in any of the three coordinates
a(2) = 50: +-2 0 0 (6), +-2 +-1 +-1 (24), +-2 +-2 0 (12), +-2 +-2 +-2 (8).
		

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

Crossrefs

Cf. A117216, A022144, A010014, A175112 (D_5), A175114 (D_6).

Programs

  • Maple
    A110907 := proc(n) a :=0 ; for x from -n to n do for y from -n to n do for z from -n to n do if type(x+y+z,'even') then m := max( abs(x),abs(y),abs(z)) ; if m = n then a := a+1 ; end if; end if; end do ; end do ; end do ; a ; end proc: seq(A110907(n),n=0..40) ; # R. J. Mathar, Feb 03 2010
  • Mathematica
    a[0] = 1; a[n_] := 1 + (-1)^n + 12*n^2;
    Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 16 2017, after R. J. Mathar *)

Formula

From R. J. Mathar, Feb 03 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = 1 + (-1)^n + 12*n^2, n>0.
G.f.: 1 - 2*x*(6 + 13*x + 4*x^2 + x^3)/((1+x)*(x-1)^3). (End)

Extensions

I would like to get analogous sequences for A_2, A_4, A_5, ..., D_4 (see A117216), D_5, ..., E_6, E_7, E_8.
Extended by R. J. Mathar, Feb 03 2010
Removed the "conjectured" attribute from formulas - R. J. Mathar, Feb 27 2010

A175114 First differences of A175113.

Original entry on oeis.org

1, 364, 7448, 51012, 206896, 620060, 1527624, 3281908, 6373472, 11454156, 19360120, 31134884, 48052368, 71639932, 103701416, 146340180, 201982144, 273398828, 363730392, 476508676, 615680240, 785629404, 991201288, 1237724852
Offset: 0

Views

Author

R. J. Mathar, Feb 13 2010

Keywords

Comments

Convolution of the finite sequence 1,358,5279,11764,5279,358,1 with A000389. Number of points in the standard root system of the D_6 lattice having L_infinity norm n.

Crossrefs

Programs

  • Magma
    I:=[1,364,7448,51012,206896,620060,1527624]; [n le 7 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Dec 20 2012
  • Mathematica
    CoefficientList[Series[(358 x + 5279 x^2 + 11764 x^3 + 5279 x^4 + 358 x^5 + 1+x^6)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 20 2012 *)

Formula

a(n)= 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), n>6.
a(n) = ((2*n+1)^6-(2*n-1)^6)/2 = 4*n*(12*n^2+1)*(4*n^2+3), n>0. - Bruno Berselli, Dec 27 2010
G.f.: (358*x+5279*x^2+11764*x^3+5279*x^4+358*x^5+1+x^6)/(x-1)^6. - R. J. Mathar, Jan 03 2011

A175111 a(n) = ((2*n+1)^5+(-1)^n)/2.

Original entry on oeis.org

1, 121, 1563, 8403, 29525, 80525, 185647, 379687, 709929, 1238049, 2042051, 3218171, 4882813, 7174453, 10255575, 14314575, 19567697, 26260937, 34671979, 45112099, 57928101, 73504221, 92264063, 114672503, 141237625, 172512625
Offset: 0

Views

Author

R. J. Mathar, Feb 13 2010

Keywords

Comments

Partial sums of A175112.
Convolution of the finite sequence 1,116,967,1672,967,116,1 with A001753.

Programs

  • Magma
    I:=[1, 121, 1563, 8403, 29525, 80525, 185647]; [n le 7 select I[n] else 5*Self(n-1) - 9*Self(n-2) + 5*Self(n-3) + 5*Self(n-4) - 9*Self(n-5) + 5*Self(n-6) - Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 19 2012
  • Mathematica
    CoefficientList[Series[(1 + 116*x + 967*x^2 + 1672*x^3 + 967*x^4 + 116*x^5 + x^6)/((1 + x)*(x - 1)^6), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
    LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,121,1563,8403,29525,80525,185647},50] (* Harvey P. Dale, May 30 2014 *)

Formula

a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7).
G.f: (1+116*x+967*x^2+1672*x^3+967*x^4+116*x^5+x^6)/((1+x)*(x-1)^6).
Showing 1-3 of 3 results.