A110907 -id:A110907 - cp's OEIS frontend

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

1, 40, 272, 888, 2080, 4040, 6960, 11032, 16448, 23400, 32080, 42680, 55392, 70408, 87920, 108120, 131200, 157352, 186768, 219640, 256160, 296520, 340912, 389528, 442560, 500200, 562640, 630072, 702688, 780680, 864240, 953560, 1048832, 1150248, 1258000, 1372280

Offset: 0

N. J. A. Sloane, Apr 15 2008

This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Feb 05 2010

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.
Index entries for sequences related to D_4 lattice.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A035878, A105374, A110907, A144965, A175110.

I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4,{x,0,40}],x]  (* Vincenzo Librandi, Jun 27 2012 *)

From R. J. Mathar, Feb 03 2010, Feb 13 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - Colin Barker, May 24 2012
E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - Elmo R. Oliveira, Aug 18 2025

a(2) corrected and sequence extended by R. J. Mathar, Feb 03 2010, Feb 13 2010

A175112 First differences of A175111.

Original entry on oeis.org

1, 120, 1442, 6840, 21122, 51000, 105122, 194040, 330242, 528120, 804002, 1176120, 1664642, 2291640, 3081122, 4059000, 5253122, 6693240, 8411042, 10440120, 12816002, 15576120, 18759842, 22408440, 26565122, 31275000, 36585122

Offset: 0

R. J. Mathar, Feb 13 2010

Convolution of the finite sequence 1,116,967,1672,967,116,1 with A001752. Number of points in the standard root system of the D_5 lattice having L_oo norm n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A110907, A117216, A175114.

I:=[1, 120, 1442, 6840, 21122, 51000, 105122]; [n le 7 select I[n] else 4*Self(n-1) - 5*Self(n-2) + 5*Self(n-4) - 4*Self(n-5) + Self(n-6): n in [1..40]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[(116*x + 967*x^2 + 1672*x^3 + 967*x^4 + 116*x^5 + x^6+1)/((1 + x)*(1 - x)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{4,-5,0,5,-4,1},{1,120,1442,6840,21122,51000,105122},30] (* Harvey P. Dale, Sep 12 2023 *)

a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6), n>6.
a(n) = ((2*n+1)^5-(2*n-1)^5)/2+(-1)^n, n>0.
G.f.: (116*x+967*x^2+1672*x^3+967*x^4+116*x^5+x^6+1)/((1+x)*(1-x)^5).

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

R. J. Mathar, Feb 13 2010

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A110907, A117216, A175112.

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

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 *)

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

A175109 a(n) = ((2*n+1)^3+(-1)^n)/2.

Original entry on oeis.org

1, 13, 63, 171, 365, 665, 1099, 1687, 2457, 3429, 4631, 6083, 7813, 9841, 12195, 14895, 17969, 21437, 25327, 29659, 34461, 39753, 45563, 51911, 58825, 66325, 74439, 83187, 92597, 102689, 113491, 125023, 137313, 150381, 164255, 178955

Offset: 0

R. J. Mathar, Feb 13 2010

Partial sums of A110907. Convolution of the finite sequence (1,10,26,10,1) with A002623.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002623, A110907.

I:=[1, 13, 63, 171, 365]; [n le 5 select I[n] else 3*Self(n-1) - 2*Self(n-2) - 2*Self(n-3) + 3*Self(n-4) - Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 19 2012

A175109:=n->((2*n+1)^3+(-1)^n)/2: seq(A175109(n), n=0..50); # Wesley Ivan Hurt, Apr 18 2017

CoefficientList[Series[(x^2 + 4*x + 1)*(x^2 + 6*x + 1)/((1 + x)*(x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).

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

A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset: 0

R. J. Mathar, Mar 02 2010

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:
.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1
.1,....3,......5,.......7,........9,.......11,........13,........15,........17
.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217
.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281
.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101
.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153
.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469
.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321
.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407
This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

			A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932

R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Cf. A008458 (row k=2), A010006 (row k=3), A110907.

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A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

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A175112 First differences of A175111.

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A175114 First differences of A175113.

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A175109 a(n) = ((2*n+1)^3+(-1)^n)/2.

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A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

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