A097869 -id:A097869 - cp's OEIS frontend

A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

0, 2, 12, 38, 88, 170, 292, 462, 688, 978, 1340, 1782, 2312, 2938, 3668, 4510, 5472, 6562, 7788, 9158, 10680, 12362, 14212, 16238, 18448, 20850, 23452, 26262, 29288, 32538, 36020, 39742, 43712, 47938, 52428, 57190, 62232, 67562

Offset: 0

N. J. A. Sloane

Sums of the first n terms > 0 of A001105 in palindromic arrangement. a(n) = Sum_{i=1 .. n} A001105(i) + Sum_{i=1 .. n-1} A001105(i), e.g. a(3) = 38 = 2 + 8 + 18 + 8 + 2; a(4) = 88 = 2 + 8 + 18 + 32 + 18 + 8 + 2. - Klaus Purath, Jun 19 2020

Apart from multiples of 3, all divisors of n are also divisors of a(n), i.e. if n is not divisible by 3, a(n) is divisible by n. All divisors d of a(n) for d !== 0 (mod) 3 are also divisors of a(abs(n-d)) and a(n+d). For all n congruent to 0,2,7 (mod 9) a(n) is divisible by 3. If n is divisible by 3^k, a(n) is divisible by 3^(k-1). - Klaus Purath, Jul 24 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII. Coordination sequences, Proc. Roy. Soc. Lond. A 458 (1996) 2369-2389.
Milzn Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
R. J. Mathar, Bivariate generating functions for non-attacking Wazirs on rectangular boards, viXra:2404.0122 (2024) page 14
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 (4,-6,4,-1).

Partial sums of A069894.
Cf. A001105, A005900, A084570, A097869.
Cf. A001844, A001845, A005899.
Column 3 of A035607, A266213, A343599.
Row 3 of A113413, A119800, A122542.

[(4*n^3 + 2*n)/3: n in [0..40]]; // Vincenzo Librandi, Sep 19 2011

f := proc(n,m) local i; sum( 2^i*binomial(n,i)*binomial(m-1,i-1),i=1..min(n,m)); end; # n=dimension, m=norm

Table[(4n^3+2n)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,12,38},41] (* Harvey P. Dale, Sep 18 2011 *)

a(n) = (4*n^3 + 2*n)/3.
a(n) = 2*A005900(n). - R. J. Mathar, Dec 05 2009
a(0)=0, a(1)=2, a(2)=12, a(3)=38, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: (2*x*(x+1)^2)/(x-1)^4. - Harvey P. Dale, Sep 18 2011
a(n) = -a(-n), a(n+1) = A097869(4n+3) = A084570(2n+1). - Bruno Berselli, Sep 20 2011
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=3. Also, a(n) = A001845(n) - A001844(n). - Shel Kaphan, Feb 26 2023
a(n) = A005899(n)*n/3. - Shel Kaphan, Feb 26 2023
a(n) = A006331(n)+A006331(n-1). - R. J. Mathar, Aug 12 2025

A097870 Molien series for group of order 4608 acting on joint weight enumerators of a pair of binary doubly-even self-dual codes.

Original entry on oeis.org

1, 2, 4, 10, 17, 27, 45, 66, 92, 130, 173, 223, 289, 362, 444, 546, 657, 779, 925, 1082, 1252, 1450, 1661, 1887, 2145, 2418, 2708, 3034, 3377, 3739, 4141, 4562, 5004, 5490, 5997, 6527, 7105, 7706, 8332, 9010, 9713, 10443, 11229, 12042, 12884, 13786, 14717, 15679

Offset: 0

N. J. A. Sloane, Sep 02 2004

This is the Molien series for the group of order 128 discussed in A097869 extended by the extra generator diag{1,1,i,i}. This group was not considered in the reference cited.

The first g.f. inserts zeros between each pair of terms; the second g.f. does not. - Colin Barker, Feb 12 2015

Colin Barker, Table of n, a(n) for n = 0..1000
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A097869.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2 )); // G. C. Greubel, Feb 05 2020

m:=50; S:=series((1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2, x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 05 2020

CoefficientList[Series[(1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2, {x,0,50}], x] (* G. C. Greubel, Feb 05 2020 *)
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{1,2,4,10,17,27,45,66},50] (* Harvey P. Dale, Jun 11 2022 *)

Vec((x+1)*(x^2-x+1)*(x^4+x^3+x^2+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Feb 12 2015

def A097870_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^3)*(1+x^2+x^3+x^4)/((1-x)*(1-x^3))^2 ).list()
A097870_list(50) # G. C. Greubel, Feb 05 2020

G.f.: (1 + x^2 + 2*x^3 + x^4 + x^5 + x^6 + x^7)/(1 - 2*x + x^2 - 2*x^3 +
   4*x^4 - 2*x^5 + x^6 - 2*x^7 + x^8).
G.f.: (1+x)*(1-x+x^2)*(1+x^2+x^3+x^4) / ((1-x)^4*(1+x+x^2)^2). - Colin Barker, Feb 12 2015

cp's OEIS Frontend

A035597 Number of points of L1 norm 3 in cubic lattice Z^n.

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