A005901 -id:A005901 - cp's OEIS frontend

A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 42, 18, 2, 1, 30, 110, 92, 24, 2, 1, 42, 240, 340, 162, 30, 2, 1, 56, 462, 1010, 780, 252, 36, 2, 1, 72, 812, 2562, 2970, 1500, 362, 42, 2, 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2, 1, 110, 2070, 11832, 26474, 27174, 14240, 4060, 642, 54, 2, 1, 132, 3080, 22530, 66222, 91112, 65226, 26070, 6040, 812, 60, 2

Offset: 1

Ralf Stephan, Feb 20 2005

T(n,k) is the number of integer sequences of length n+1 with sum zero and sum of absolute values 2k. - R. H. Hardin, Feb 23 2009

			Array begins:
  1,   2,     2,      2,       2,        2,         2,          2, ... A040000;
  1,   6,    12,     18,      24,       30,        36,         42, ... A008458;
  1,  12,    42,     92,     162,      252,       362,        492, ... A005901;
  1,  20,   110,    340,     780,     1500,      2570,       4060, ... A008383;
  1,  30,   240,   1010,    2970,     7002,     14240,      26070, ... A008385;
  1,  42,   462,   2562,    9492,    27174,     65226,     137886, ... A008387;
  1,  56,   812,   5768,   26474,    91112,    256508,     623576, ... A008389;
  1,  72,  1332,  11832,   66222,   271224,    889716,    2476296, ... A008391;
  1,  90,  2070,  22530,  151560,   731502,   2777370,    8809110, ... A008393;
  1, 110,  3080,  40370,  322190,  1815506,   7925720,   28512110, ... A008395;
  1, 132,  4422,  68772,  643632,  4197468,  20934474,   85014204, ... A035837;
  1, 156,  6162, 112268, 1219374,  9129276,  51697802,  235895244, ... A035838;
  1, 182,  8372, 176722, 2206932, 18827718, 120353324,  614266354, ... A035839;
  1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
  ...
Antidiagonals:
  1;
  1,  2;
  1,  6,    2;
  1, 12,   12,    2;
  1, 20,   42,   18,    2;
  1, 30,  110,   92,   24,    2;
  1, 42,  240,  340,  162,   30,    2;
  1, 56,  462, 1010,  780,  252,   36,   2;
  1, 72,  812, 2562, 2970, 1500,  362,  42,  2;
  1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48,  2;

Muniru A Asiru, Table of n, a(n) for n = 1..5050 (antidiagonals 1 to 100, 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).
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, Preserving Privacy and Fidelity via Ehrhart Theory, July 2017.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.
Columns include A002376, A001621.
Main diagonal is in A103882.
Cf. A047085, A103884, A103903, A103998, A143007.

T:=Flat(List([1..12],n->Concatenation([1],List([1..n-1],k->Sum([1..n],i->Binomial(n-k+1,i)*Binomial(k-1,i-1)*Binomial(n-i,k)))))); # Muniru A Asiru, Oct 14 2018

A103881:= func< n,k | k le 0 select 1 else (&+[Binomial(n-k+1, j)*Binomial(k-1, j-1)*Binomial(n-j, k): j in [1..n-k]]) >;
[A103881(n,k): k in [0..n-1], n in [1..15]]; // G. C. Greubel, Oct 16 2018; May 24 2023

T:=proc(n,k) option remember; local i;
if k=0 then 1 else
add( binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k),i=1..n); fi;
end:
g:=n->[seq(T(n-i,i),i=0..n-1)]:
for n from 1 to 14 do lprint(op(g(n))); od:

T[n_, k_]:= (n+1)*(n+k-1)!*HypergeometricPFQ[{1-k,1-n,-n}, {2,-n-k+1}, 1]/(k!*(n-1)!); T[, 0]=1; Flatten[Table[T[n-k, k], {n,12}, {k,0,n-1}]] (* _Jean-François Alcover, Dec 27 2012 *)

A103881(n,k) = if(k==0, 1, sum(j=1, n-k, binomial(n-k+1, j)*binomial(k-1, j-1)*binomial(n-j, k)));
for(n=1, 15, for(k=0, n-1, print1(A103881(n,k), ", "))) \\ G. C. Greubel, Oct 16 2018; May 24 2023

def A103881(n,k): return 1 if k==0 else (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1).simplify()
flatten([[A103881(n,k) for k in range(n)] for n in range(1,16)]) # G. C. Greubel, May 24 2023

T(n,k) = Sum_{i=1..n} C(n+1, i)*C(k-1, i-1)*C(n-i+k, k), T(n,0)=1.
G.f. of n-th row: (Sum_{i=0..n} C(n, i)^2*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).
T(n, k) = (n+1)*binomial(n+k-1,k)*hypergeometric([-n,1-n,1-k], [2,1-n-k], 1), with T(n, k) = 1 (array).
t(n, k) = (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1), with t(n, 0) = 1 (antidiagonals).
Sum_{k=0..n-1} t(n, k) = A047085(n). (End)
From Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).
(n+1)*T(n+1,k) = (n+1)*T(n+1,k-1) + (2*n+1)*(T(n,k) + T(n,k-1)) - n*(T(n-1,k) - T(n-1,k-1)). (End)

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

A004015 Theta series of face-centered cubic (f.c.c.) lattice.

Original entry on oeis.org

1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 12*x + 6*x^2 + 24*x^3 + 12*x^4 + 24*x^5 + 8*x^6 + 48*x^7 + 6*x^8 + ...
G.f. = 1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
From _Michael Somos_, Jan 05 2012: (Start)
a(2) = 6 since (1, -1, -1) is a solution to x^2 + y^2 + z^2 + x*y + x*z + y*z = 2 and the other 5 solutions are permutations and negations of this one.
a(2) = 6 since (1, 1, -1, -1) is a solution to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 4 and the other 5 solutions are permutations of this one. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 263.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Charles L. Hohn, a(0) to a(15), animated
G. Nebe and N. J. A. Sloane, Home page for this lattice
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to f.c.c. lattice

Cf. A004013, A005875, A005901, A045828. A055039 gives the positions of the 0's in this sequence.

Cf. A000007, A000122, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_4, ...)

L := Lattice("A",3); A := ThetaSeries(L, 140); A; /* Michael Somos, Nov 13 2014 */

A := Basis( ModularForms( Gamma1(8), 3/2), 70); A[1] + 12*A[2] + 6*A[3] + 24*A[4]; /* Michael Somos, Sep 08 2018 */

maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd); th4 := series(subs(q=-q, th3),q,maxd); series((1/2)*(th3^3+th4^3),q,200);

a[n_] := SquaresR[3, 2n]; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 + EllipticTheta[ 4, 0, q]^3) / 2, {q, 0, 2 n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^3 + 12 q QPochhammer[ q^4]^3 QPochhammer[ q^8]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
SquaresR[3,2*Range[0,70]] (* Harvey P. Dale, Jun 01 2015 *)

{a(n) = if( n<0, 0, n*=2; polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))}; /* Michael Somos, May 17 2008 */

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 1; 1, 2, 1; 1, 1, 2], n, 1)[n])}; /* Michael Somos, Jan 02 2012 */

from math import prod, isqrt
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1
def A004015(n): return A004018(m:=n<<1)+(sum(A004018(m-k**2) for k in range(1,isqrt(m)+1))<<1) # Chai Wah Wu, Feb 24 2025

Expansion of phi(q^2)^3 + 12 * q * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
Expansion of (phi(q)^3 + phi(-q)^3) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of b(q) * phi(q^18) + c(q^3) * phi(q^2) in powers of q^3 where b(), c() are cubic AGM theta functions and phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004013.
a(n) = A005875(2*n).
G.f.: Sum_{i, j, k in Z} x^( i*i + j*j + k*k + i*j + i*k + j*k ). - Michael Somos, Jan 02 2012
From Michael Somos, Jan 05 2012: (Start)
Number of integer solutions to x^2 + y^2 + z^2 + x*y + x*z + y*z = n.
Number of integer solutions to x + y + z even and x^2 + y^2 + z^2 = 2 * n.
Number of integer solutions to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 2 * n. (End)
a(2*n) = A005875(n). a(2*n+1) = 12 * A045828(n). - Michael Somos, Dec 28 2017

A008354 a(n) = (5n^2 + 1)n^2 / 6.

Original entry on oeis.org

0, 1, 14, 69, 216, 525, 1086, 2009, 3424, 5481, 8350, 12221, 17304, 23829, 32046, 42225, 54656, 69649, 87534, 108661, 133400, 162141, 195294, 233289, 276576, 325625, 380926, 442989, 512344, 589541, 675150, 769761, 873984, 988449, 1113806, 1250725, 1399896

Offset: 0

N. J. A. Sloane and J. H. Conway

Partial sums of A005902. - Jonathan Vos Post, Mar 14 2006

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A005901, A005902.

List([0..30], n -> (5*n^2+1)*n^2/6); # Muniru A Asiru, Feb 12 2018

a:= n-> 5*n^4/6 + n^2/6: seq(a(n), n=0..45);

Table[n^2 (5 n^2 + 1)/6, {n, 0, 30}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 14, 69, 216}, 30] (* Harvey P. Dale, Feb 12 2015 *)

From R. J. Mathar, Aug 10 2008: (Start)
O.g.f.: x*(1 + x)*(x^2 + 8*x + 1)/(1 - x)^5.
a(n) = n*A004068(n). (End)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4, a(0)=0, a(1)=1, a(2)=14, a(3)=69, a(4)=216. - Harvey P. Dale, Feb 12 2015

Definition corrected by R. J. Mathar, Aug 10 2008

A008383 Coordination sequence for A_4 lattice.

Original entry on oeis.org

1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620, 20260, 25740, 32130, 39500, 47920, 57460, 68190, 80180, 93500, 108220, 124410, 142140, 161480, 182500, 205270, 229860, 256340, 284780, 315250, 347820, 382560, 419540, 458830, 500500, 544620

Offset: 0

N. J. A. Sloane and J. H. Conway

M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

T. D. Noe, Table of n, a(n) for n = 0..1000
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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005901, A008384, A008385.

[n eq 0 select 1 else 5*n*(7*n^2+5)/3: n in [0..45]]; // G. C. Greubel, May 25 2023

a:= n-> `if`(n=0, 1, 35/3*n^3+25/3*n): seq (a(n), n=0..50);

CoefficientList[Series[(1+16x+36x^2+16x^3+x^4)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)
Join[{1}, LinearRecurrence[{4, -6, 4, -1}, {20, 110, 340, 780}, 40]] (* Jean-François Alcover, Jan 07 2019 *)

[5*n*(7*n^2+5)/3+int(n==0) for n in range(46)] # G. C. Greubel, May 25 2023

a(n) = 5*n*(7*n^2 + 5)/3, a(0) = 1.
G.f.: (1+16*x+36*x^2+16*x^3+x^4)/(1-x)^4 = 1+10*x*(2+3*x+2*x^2)/(x-1)^4. - Colin Barker, Apr 13 2012
E.g.f.: (1/3)*(3 + 5*x*(12 + 21*x + 7*x^2)*exp(x)). - G. C. Greubel, May 25 2023

A093485 a(n) = (27n^2 + 9n + 2)/2.

Original entry on oeis.org

1, 19, 64, 136, 235, 361, 514, 694, 901, 1135, 1396, 1684, 1999, 2341, 2710, 3106, 3529, 3979, 4456, 4960, 5491, 6049, 6634, 7246, 7885, 8551, 9244, 9964, 10711, 11485, 12286, 13114, 13969, 14851, 15760, 16696, 17659, 18649, 19666, 20710, 21781

Offset: 0

Michael Joseph Halm, May 13 2004

Dodecahedral gnomon numbers: first differences of dodecahedral numbers.
The sequence is related to other gnomon numbers of polyhedra, known by other more familiar names: triangular numbers (tetrahedral gnomon numbers), hexagonal numbers (cubic gnomon numbers), square pyramidal numbers (octahedral gnomon numbers).
A124388 = first differences; second differences = 27. - Reinhard Zumkeller, Oct 30 2006
Sums of the triangular numbers from A000217(3*n-1) to A000217(3*n+1), with A000217(-1) = 0. - Bruno Berselli, Sep 04 2018

			a(1) = 19 because (1+1)*(3*(1+1)-1)*(3*(1+1)-2)/2-1*(3*1-1)*(3*1-2)/2 = 2*(6-1)*(6-2)/2 - 1*(3-1)*(3-2)/2 = 20-1 = 19.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000330, A003215, A005901, A006656.

a093485 n = (9 * n * (3 * n + 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013

[(27*n^2 + 9*n + 2)/2 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

a(n)=(27*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

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

G.f.: (1 + 16*x + 10*x^2)/(1 - x)^3. - Colin Barker, Mar 28 2012

New definition from Ralf Stephan, Dec 01 2004

Name corrected and initial term added by Arkadiusz Wesolowski, Aug 15 2011

A071336 Number of vertices of Goldberg-Casper-Klug pseudo-icosahedra.

Original entry on oeis.org

12, 32, 42, 72, 92, 122, 132, 162, 192, 212, 252, 272, 282, 312, 362, 372, 392, 432, 482, 492, 522, 572, 612, 632, 642, 672, 732, 752, 762, 792, 812, 842, 912, 932, 972, 1002, 1032, 1082, 1092, 1112, 1122, 1172, 1212, 1242, 1272, 1292, 1332, 1392, 1442, 1472

Offset: 1

Lekraj Beedassy, Oct 30 2003

nonn

Refers to the capsomere count in virus architectural structures. Since the Loeschian numbers (A003136) of the form x^2 + xy + y^2 reduce to squares (A000290) when either x or y equals 0, a(n) contains A005901 which is 10*n^2 + 2.

I. Stewart, "Game, Set and Math", Chapter 6, Table 6.1 pp. 81 Penguin London 1991.

Michael De Vlieger, Table of n, a(n) for n = 1..1200
K. Urner, Synergetica:A case of Deja Vu

Cf. A003136, A005901.

10 # + 2 & /@ Rest@ Flatten[Position[CoefficientList[QPochhammer[q]^3/QPochhammer[q^3]/3 + O[q]^150, q], n_ /; n != 0] - 1] (* Michael De Vlieger, Jun 17 2020, after Jean-François Alcover at A003136 *)

a(n) = 10*A003136(n+1) + 2. [corrected by Georg Fischer, Jun 17 2020]

A093500 a(n) = (15n^2 + 5n + 2)/2.

Original entry on oeis.org

11, 36, 76, 131, 201, 286, 386, 501, 631, 776, 936, 1111, 1301, 1506, 1726, 1961, 2211, 2476, 2756, 3051, 3361, 3686, 4026, 4381, 4751, 5136, 5536, 5951, 6381, 6826, 7286, 7761, 8251, 8756, 9276, 9811, 10361, 10926, 11506, 12101, 12711, 13336, 13976, 14631, 15301

Offset: 1

Michael Joseph Halm, May 13 2004

Icosahedral gnomic numbers: first differences of icosahedral numbers.

The sequence is related to other gnomons of polyhedra, known by other more familiar names: triangular (tetrahedral gnomic), hex (cubic gnomic), square pyramidal numbers (octahedral gnomic).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000330, A003215, A005901, A006564.

[(15*n^2+5*n+2)/2: n in [1..50]]; // Vincenzo Librandi, Aug 16 2011

Table[(15n^2+5n+2)/2,{n,50}] (* Harvey P. Dale, Jun 28 2014 *)

a(n)=(15*n^2+5*n+2)/2 \\ Charles R Greathouse IV, Jun 16 2017

From Colin Barker, Apr 30 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (6*x^2 + 8*x + 1)/(1-x)^3. (End)
E.g.f.: exp(x)*(1 + 10*x + 15*x^2/2). - Elmo R. Oliveira, Oct 21 2024

New definition from Ralf Stephan, Dec 01 2004

Name corrected by Arkadiusz Wesolowski, Aug 15 2011

A344126 Coordination sequence for the hypertriangular lattice.

Original entry on oeis.org

1, 6, 24, 48, 86, 138, 192, 260, 348, 432, 530, 654, 768, 896, 1056, 1200, 1358, 1554, 1728, 1916, 2148, 2352, 2570, 2838, 3072, 3320, 3624, 3888, 4166, 4506, 4800, 5108, 5484, 5808, 6146, 6558, 6912, 7280, 7728, 8112, 8510, 8994, 9408, 9836, 10356, 10800

Offset: 0

Sean A. Irvine, May 09 2021

Reticular Chemistry Structure Resource (RCSR), The lcy net
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A046945, A344039, A344070, A005901, A038620.

G.f.: (1+5*x+18*x^2+22*x^3+28*x^4+16*x^5+7*x^6-3*x^7+2*x^8) / ((x^2+x+1)^2 * (1-x)^3).

A010022 a(0) = 1, a(n) = 40*n^2 + 2 for n>0.

Original entry on oeis.org

1, 42, 162, 362, 642, 1002, 1442, 1962, 2562, 3242, 4002, 4842, 5762, 6762, 7842, 9002, 10242, 11562, 12962, 14442, 16002, 17642, 19362, 21162, 23042, 25002, 27042, 29162, 31362, 33642, 36002, 38442, 40962, 43562, 46242, 49002, 51842, 54762, 57762, 60842

Offset: 0

N. J. A. Sloane

First bisection of A005901. - Bruno Berselli, Feb 07 2012

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

Cf. A206399.

[1] cat [40*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 40 Range[39]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {42, 162, 362}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+38*x+x^2)/(1-x)^3; a(n) = A008253(4n). - Bruno Berselli, Feb 07 2012
E.g.f.: (x*(x+1)*40+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(5)/40*Pi*coth(Pi*sqrt(5)/10) = 1.03983104279172.. - R. J. Mathar, May 07 2024
a(n) = 2*A158493(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A195317(n)+A195317(n+1) = 2+10*A016742(n), n>0. - R. J. Mathar, May 07 2024

A122973 Number of vertices on the surface of an icosahedron.

Original entry on oeis.org

1, 12, 42, 162, 642, 2562, 10242, 40962, 163842, 655362, 2621442, 10485762, 41943042, 167772162, 671088642, 2684354562, 10737418242, 42949672962, 171798691842, 687194767362, 2748779069442, 10995116277762, 43980465111042, 175921860444162, 703687441776642

Offset: 0

Alden Chew (aldenc98(AT)yahoo.com), Oct 27 2006

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A005901.

Join[{1},LinearRecurrence[{5,-4},{12,42},30]] (* or *) CoefficientList[ Series[1-6x (-2+3x)/((4x-1)(x-1)),{x,0,30}],x] (* Harvey P. Dale, May 05 2011 *)

a(n) = 1 if n=0, 10*4^(n-1)+2 otherwise.
G.f.: 1-6*x*(-2+3*x) / ( (4*x-1)*(x-1) ). - R. J. Mathar, Feb 02 2011
For a(n)>1: a(1)=12, a(2)=42, a(n) = 5*a(n-1)-4*a(n-2). [Harvey P. Dale, May 05 2011]

cp's OEIS Frontend

A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

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A004015 Theta series of face-centered cubic (f.c.c.) lattice.

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A008354 a(n) = (5*n^2 + 1)*n^2 / 6.

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A008383 Coordination sequence for A_4 lattice.

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A093485 a(n) = (27*n^2 + 9*n + 2)/2.

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A071336 Number of vertices of Goldberg-Casper-Klug pseudo-icosahedra.

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A008354 a(n) = (5n^2 + 1)n^2 / 6.

A093485 a(n) = (27n^2 + 9n + 2)/2.

A093500 a(n) = (15n^2 + 5n + 2)/2.