cp's OEIS Frontend

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

%I A035597 #82 Aug 12 2025 10:13:31
%S A035597 0,2,12,38,88,170,292,462,688,978,1340,1782,2312,2938,3668,4510,5472,
%T A035597 6562,7788,9158,10680,12362,14212,16238,18448,20850,23452,26262,29288,
%U A035597 32538,36020,39742,43712,47938,52428,57190,62232,67562
%N A035597 Number of points of L1 norm 3 in cubic lattice Z^n.
%C A035597 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
%C A035597 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
%H A035597 Vincenzo Librandi, <a href="/A035597/b035597.txt">Table of n, a(n) for n = 0..10000</a>
%H A035597 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A035597 J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1098/rspa.1997.0126">Low-dimensional lattices. VII. Coordination sequences</a>, Proc. Roy. Soc. Lond. A 458 (1996) 2369-2389.
%H A035597 Milzn Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.2.
%H A035597 Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
%H A035597 Milan Janjic and B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H A035597 R. J. Mathar, <a href="https://vixra.org/abs/2404.0122">Bivariate generating functions for non-attacking Wazirs on rectangular boards</a>, viXra:2404.0122 (2024) page 14
%H A035597 Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
%H A035597 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A035597 a(n) = (4*n^3 + 2*n)/3.
%F A035597 a(n) = 2*A005900(n). - _R. J. Mathar_, Dec 05 2009
%F A035597 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
%F A035597 a(n) = -a(-n), a(n+1) = A097869(4n+3) = A084570(2n+1). - _Bruno Berselli_, Sep 20 2011
%F A035597 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
%F A035597 a(n) = A005899(n)*n/3. - _Shel Kaphan_, Feb 26 2023
%F A035597 a(n) = A006331(n)+A006331(n-1). - _R. J. Mathar_, Aug 12 2025
%p A035597 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
%t A035597 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 *)
%o A035597 (Magma) [(4*n^3 + 2*n)/3: n in [0..40]]; // _Vincenzo Librandi_, Sep 19 2011
%Y A035597 Partial sums of A069894.
%Y A035597 Cf. A001105, A005900, A084570, A097869.
%Y A035597 Cf. A001844, A001845, A005899.
%Y A035597 Column 3 of A035607, A266213, A343599.
%Y A035597 Row 3 of A113413, A119800, A122542.
%K A035597 nonn,easy
%O A035597 0,2
%A A035597 _N. J. A. Sloane_