A019560 -id:A019560 - cp's OEIS frontend

A057813 a(n) = (2n+1)(4n^2+4n+3)/3.

1, 11, 45, 119, 249, 451, 741, 1135, 1649, 2299, 3101, 4071, 5225, 6579, 8149, 9951, 12001, 14315, 16909, 19799, 23001, 26531, 30405, 34639, 39249, 44251, 49661, 55495, 61769, 68499, 75701, 83391, 91585, 100299, 109549, 119351, 129721, 140675, 152229, 164399

Offset: 0

N. J. A. Sloane, Nov 07 2000

For n>0, 30*a(n) is the sum of the ten distinct products of 2*n-1, 2*n+1, and 2*n+3. For example, when n = 1, we sum the ten distinct products of 1, 3, and 5: 1*1*1 + 1*1*3 + 1*1*5 + 1*3*3 + 1*3*5 + 1*5*5 + 3*3*3 + 3*3*5 + 3*5*5 + 5*5*5 = 330 = 30*11 = 30*a(1). - J. M. Bergot, Apr 06 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A019560, A336266.

[(2*n+1)*(4*n^2+4*n+3)/3 : n in [0..50]] // Wesley Ivan Hurt, Apr 22 2014

A057813:=n->(2*n + 1)*(4*n^2 + 4*n + 3)/3; seq(A057813(n), n=0..50); # Wesley Ivan Hurt, Apr 06 2014

Table[(2*n + 1)*(4*n^2 + 4*n + 3)/3, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)

P(x, y, z) = x^3 + x^2*y + x^2*z + x*y^2 + x*y*z + x*z^2 + y^3 + y^2*z + y*z^2 + z^3;
a(n) = P(2*n-1, 2*n+1, 2*n+3)/30; \\ Michel Marcus, Apr 22 2014

a(n) = 2*A050533(n) + 1. - N. J. A. Sloane, Sep 22 2004
G.f.: (1+7*x+7*x^2+x^3)/(1-x)^4. - Colin Barker, Mar 01 2012
G.f. for sequence with interpolated zeros: 1/(8*x)*sinh(8*arctanh(x)) = 1/(16*x)*( ((1 + x)/(1 - x))^4 - ((1 - x)/(1 + x))^4 ) = 1 + 11*x^2 + 45*x^4 + 119*x^6 + .... Cf. A019560. - Peter Bala, Apr 07 2017
E.g.f.: (3 + 30*x + 36*x^2 + 8*x^3)*exp(x)/3. - G. C. Greubel, Dec 01 2017
From Peter Bala, Mar 26 2024: (Start)
12*a(n) = (2*n + 1)*(a(n + 1) - a(n - 1)).
Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 3*Pi/16 - 1/2. Cf. A016754 and A336266. (End)

A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

Original entry on oeis.org

1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898

Offset: 0

N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

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

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Milan Janjic, Two Enumerative Functions
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 (3,-3,1).

Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513.

[1] cat [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

Join[{1},Table[16n^2+2,{n,50}]] (* Harvey P. Dale, Oct 15 2012 *)

A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.
G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
E.g.f.: (x*(x+1)*16+2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - Harvey P. Dale, Oct 15 2012
G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - Peter Bala, Apr 09 2017
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth( Pi*sqrt(2)/4) = 1.095237238050... - R. J. Mathar, May 07 2024
a(n) = 2*A081585(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069129(n)+A069129(n+1). - R. J. Mathar, May 07 2024

A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 66, 24, 1, 50, 192, 146, 32, 1, 72, 450, 608, 258, 40, 1, 98, 912, 1970, 1408, 402, 48, 1, 128, 1666, 5336, 5890, 2720, 578, 56, 1, 162, 2816, 12642, 20256, 14002, 4672, 786, 64, 1, 200, 4482, 27008, 59906, 58728, 28610, 7392, 1026, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    66,    146,     258,      402,       578, ... A010006;
  1,  32,   192,    608,    1408,     2720,      4672, ... A019560;
  1,  50,   450,   1970,    5890,    14002,     28610, ... A019561;
  1,  72,   912,   5336,   20256,    58728,    142000, ... A019562;
  1,  98,  1666,  12642,   59906,   209762,    596610, ... A019563;
  1, 128,  2816,  27008,  157184,   658048,   2187520, ... A019564;
  1, 162,  4482,  53154,  374274,  1854882,   7159170, ... A035746;
  1, 200,  6800,  97880,  822560,  4780008,  21278640, ... A035747;
  1, 242,  9922, 170610, 1690370, 11414898,  58227906, ... A035748;
  1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749;
  ...
Antidiagonals, T(n, k), begins as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   66,   24;
  1,  50,  192,  146,   32;
  1,  72,  450,  608,  258,   40;
  1,  98,  912, 1970, 1408,  402,  48;
  1, 128, 1666, 5336, 5890, 2720, 578, 56;

G. C. Greubel, Antidiagonals n = 2..50, flattened
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
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.

Rows include A022144, A010006, A019560, A019561, A019562, A019563, A019564, A035746, A035747, A035748, A035749, A035750 - A035787.
Columns include A001105, A035598, A035600, A035602, A035604, A035606.
Main diagonal is in A103885.
Cf. A103881, A103993, A108998.

A103884:= func< n,k | k eq 0 select 1 else 2*(&+[2^j*Binomial(n-k,j+1)*Binomial(2*k-1,j) : j in [0..2*k-1]]) >;
[A103884(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, May 23 2023

nmin = 2; nmax = 11; t[n_, 0]= 1; t[n_, k_]:= 2n*Hypergeometric2F1[1-2k, 1-n, 2, 2]; tnk= Table[ t[n, k], {n, nmin, nmax}, {k, 0, nmax-nmin}]; Flatten[ Table[ tnk[[ n-k+1, k ]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after formula *)

def A103884(n,k): return 1 if k==0 else 2*sum(2^j*binomial(n-k,j+1)*binomial(2*k-1,j) for j in range(2*k))
flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, May 23 2023

A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).
G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.
T(n, k) = A(n-k, k) (antidiagonals).
T(n, n-2) = A022144(n-2).
T(n, k) = 2*(n-k)*Hypergeometric2F1([1+k-n, 1-2*k], [2], 2), T(n, 0) = 1, for n >= 2, 0 <= k <= n-2. - G. C. Greubel, May 23 2023
From  Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Chebyshev_T(n, (1 + x)/(1 - x)), where Chebyshev_T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
T(n+1,k) = T(n+1,k-1)  + 2*T(n,k) + 2*T(n,k-1) + T(n-1,k) - T(n-1,k-1). (End)

Definition clarified by N. J. A. Sloane, May 25 2023

A142993 Crystal ball sequence for the lattice C_4.

Original entry on oeis.org

1, 33, 225, 833, 2241, 4961, 9633, 17025, 28033, 43681, 65121, 93633, 130625, 177633, 236321, 308481, 396033, 501025, 625633, 772161, 943041, 1140833, 1368225, 1628033, 1923201

Offset: 0

Peter Bala, Jul 18 2008

The lattice C_4 consists of all integer lattice points v = (a,b,c,d) in Z^4 such that a + b + c + d is even, equipped with the taxicab type norm ||v|| = (1/2) * (|a| + |b| + |c| + |d|). The crystal ball sequence of C_4 gives the number of lattice points v in C_4 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

			a(1) = 33. The origin has norm 0. The 32 lattice points in Z^4 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 4 and (+- e_i +- e_j), 1 <= i < j <= 4, where e_1, e_2, e_3 and e_4 denotes the standard basis of Z^4. These 32 vectors form a root system of type C_4. Hence sequence begins 1, 1 + 32 = 33, ... .

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Peter Bala, The C_4 lattice and a continued fraction for log(2)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019560, A063496, A142992, A142994.

a := n -> (2*n+1)^2*(4*n^2+4*n+3)/3: seq(a(n), n = 0..24)

Partial sums of A019560. a(n) = (2*n+1)^2*(4*n^2+4*n+3)/3 = Sum_{k = 0..4} C(8,2k)*C(n+k,4) = Sum_{k = 0..4} C(8,2k+1)*C(n+k+1/2,4). O.g.f.: (1+28*x+70*x^2+28*x^3+x^4)/(1-x)^5 = (1/(1-x)) * T(4,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind. 2*log(2) = 17/12 - Sum_{n >= 1} 1/(n*a(n-1)*a(n)).
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(33 - 3/(41 - 60/(57 - 315/(81 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*4^2))))).
E.g.f.: exp(x)*(1 + 32*x + 160*x^2/2! + 256*x^3/3! + 128*x^4/4!).
Note that T(8, i*sqrt(x)) = 1 + 32*x + 160*x^2 + 256*x^3 + 128*x^4. See A008310. (End)

A019561 Coordination sequence for C_5 lattice.

Original entry on oeis.org

1, 50, 450, 1970, 5890, 14002, 28610, 52530, 89090, 142130, 216002, 315570, 446210, 613810, 824770, 1086002, 1404930, 1789490, 2248130, 2789810, 3424002, 4160690, 5010370, 5984050, 7093250, 8350002

Offset: 0

mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A103884 (row 5). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

Cf. A008413, A137513.

LinearRecurrence[{5,-10,10,-5,1},{1,50,450,1970,5890,14002},30] (* Harvey P. Dale, Nov 21 2021 *)

G.f.: (1+45*x+210*x^2+210*x^3+45*x^4+x^5)/(1-x)^5 = 1+2*x*(5+10*x+x^2)^2/(1-x)^5.
G.f. for sequence with interpolated zeros: cosh(10*arctanh(x)) = 1/2*( ((1 + x)/(1 - x))^5 + ((1 - x)/(1 + x))^5 ) = 1 + 50*x^2 + 450*x^4 + 1970*x^6 + .... - Peter Bala, Apr 09 2017
a(n) = A008413(2*n). - Seiichi Manyama, Jun 08 2018

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A057813 a(n) = (2*n+1)*(4*n^2+4*n+3)/3.

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A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

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A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

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A142993 Crystal ball sequence for the lattice C_4.

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A019561 Coordination sequence for C_5 lattice.

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A057813 a(n) = (2n+1)(4n^2+4n+3)/3.