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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

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.

Programs

Magma

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

Maple

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

Mathematica

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 *)

Formula

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