A035607 -id:A035607 - cp's OEIS frontend

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22

Offset: 1

Thomas Wieder, Jul 30 2006, Aug 06 2006

			The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
The main diagonal seems to be A050146.
Square array A(n,k) begins:
   4,   8,   12,   16,    20,    24,     28,     32,      36, ...
   6,  18,   38,   66,   102,   146,    198,    258,     326, ...
   8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
  10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
  12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
  14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
  16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
  18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
  20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
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.

Other versions: A035607, A113413, A122542, A266213.

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.

A:= proc(m, n)  option remember;
      `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Apr 21 2012

A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.

Offset and typos corrected by Alois P. Heinz, Apr 21 2012

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

Original entry on oeis.org

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

A050146 a(n) = T(n,n), array T as in A050143.

Original entry on oeis.org

1, 1, 4, 18, 88, 450, 2364, 12642, 68464, 374274, 2060980, 11414898, 63521352, 354870594, 1989102444, 11180805570, 63001648608, 355761664002, 2012724468324, 11406058224594, 64734486343480, 367891005738690, 2093292414443164, 11923933134635298, 67990160422313808

Offset: 0

Clark Kimberling

nonn

Also main diagonal of array : m(i,1)=1, i>=1; m(1,j)=2, j>1; m(i,j)=m(i,j-1)+m(i-1,j-1)+m(i-1,j): 1 2 2 2 ... / 1 4 8 12 ... / 1 6 18 38 ... / 1 8 32 88 ... / - Benoit Cloitre, Aug 05 2002
a(n) is also the number of order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008
Define a finite triangle T(r,c) with T(r,0) = binomial(n,r) for 0<=r<=n, and the other terms recursively with T(r,c) = T(r,c-1) + 2*T(r-1,c-1). The sum of the last terms in each row is Sum_{r=0..n} T(r,r)=a(n+1). For n=4 the triangle is 1; 4 6; 6 14 26; 4 16 44 96; 1 9 41 129 321 with the sum of the last terms being 1 + 6 + 26 + 96 + 321 = 450 = a(5). - J. M. Bergot, Jan 29 2013
It may be better to define a(0) = 0 for formulas without exceptions. - Michael Somos, Nov 25 2016
a(n) is the number of points at L1 distance n-1 from any point in Z^n, for n>=1. - Shel Kaphan, Mar 24 2023

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 450*x^5 + 2364*x^6 + 12642*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Cf. A002003, A006318, A050151, A008288, A002002, A026002.

-1-diagonal of A266213 for n>=1.

a050146 n = if n == 0 then 1 else a035607 (2 * n - 2) (n - 1)
-- Reinhard Zumkeller, Nov 05 2013, Jul 20 2013

Flatten[{1,RecurrenceTable[{(n-3)*(n-1)*a[n-2]-3*(n-2)*(2*n-3)*a[n-1]+(n-2)*(n-1)*a[n]==0,a[1]==1,a[2]==4},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_] := If[ n == 0, 1, Sum[ Binomial[n, k] Binomial[n + k - 2, k - 1], {k, n}]]; (* Michael Somos, Nov 25 2016 *)
a[ n_] := If[ n == 0, 1, n Hypergeometric2F1[1 - n, n, 2, -1]]; (* Michael Somos, Nov 25 2016 *)

taylor(-(x^4+sqrt(x^2-6*x+1)*(x^3-5*x^2+5*x+1)-8*x^3+16*x^2-6*x+1)/(x^3+sqrt(x^2-6*x+1)*(x^2-4*x-1)-7*x^2+7*x-1),x,0,10); /* Vladimir Kruchinin, Nov 25 2016 */

a(n)=if(n==0, 1, sum(k=1,n, binomial(n, k)*binomial(n+k-2, k-1)) ); \\ Joerg Arndt, May 04 2013

A050146 = lambda n : n*hypergeometric([1-n, n], [2], -1) if n>0 else 1
[round(A050146(n).n(100)) for n in (0..24)] # Peter Luschny, Sep 17 2014

From Vladeta Jovovic, Mar 31 2004: (Start)
Coefficient of x^(n-1) in expansion of ((1+x)/(1-x))^n, n > 0.
a(n) = Sum_{k=1..n} binomial(n, k)*binomial(n+k-2, k-1), n > 0. (End)
D-finite with recurrence (n-1)*(n-2)*a(n) = 3*(2*n-3)*(n-2)*a(n-1) - (n-1)*(n-3)*a(n-2) for n > 2. - Vladeta Jovovic, Jul 16 2004
a(n+1) = Jacobi_P(n, 1, -1, 3); a(n+1) = Sum{k=0..n} C(n+1, k)*C(n-1, n-k)*2^k. - Paul Barry, Jan 23 2006
a(n) = n*A006318(n-1) - Abdullahi Umar, Aug 25 2008
a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
a(n+1) = A035607(2*n,n). - Reinhard Zumkeller, Jul 20 2013
a(n) = n*hypergeometric([1-n, n], [2], -1) for n >= 1. - Peter Luschny, Sep 17 2014
O.g.f.: -(x^4 + sqrt(x^2 - 6*x + 1)*(x^3 - 5*x^2 + 5*x + 1) - 8*x^3 + 16*x^2 - 6*x + 1)/(x^3 + sqrt(x^2 - 6*x + 1)*(x^2 - 4*x - 1)- 7*x^2 + 7*x - 1). - Vladimir Kruchinin, Nov 25 2016
0 = a(n)*(a(n+1) - 18*a(n+2) + 65*a(n+3) - 12*a(n+4)) + a(n+1)*(54*a(n+2) - 408*a(n+3) + 81*a(n+4)) + a(n+2)*(72*a(n+2) + 334*a(n+3) - 90*a(n+4)) + a(n+3)*(-24*a(n+3) + 9*a(n+4)) for all integer n if a(0) = 0 and a(n) = -2*A050151(-n) for n < 0. - Michael Somos, Nov 25 2016
O.g.f: (2 - x + x*(3 - x)/sqrt(x^2 - 6*x + 1))/2. - Petros Hadjicostas, Feb 14 2021
a(n) = A002002(n) - A026002(n-1) for n>=2. - Shel Kaphan, Mar 24 2023

A070550 a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.

Original entry on oeis.org

1, 2, 2, 3, 6, 10, 15, 24, 40, 65, 104, 168, 273, 442, 714, 1155, 1870, 3026, 4895, 7920, 12816, 20737, 33552, 54288, 87841, 142130, 229970, 372099, 602070, 974170, 1576239, 2550408, 4126648, 6677057, 10803704, 17480760, 28284465, 45765226

Offset: 0

Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002

Shares some properties with Fibonacci sequence.
The sum of any two alternating terms (terms separated by one other term) produces a Fibonacci number (e.g., 2+6=8, 3+10=13, 24+65=89). The product of any two consecutive or alternating Fibonacci terms produces a term from this sequence (e.g., 5*8 = 40, 13*5 = 65, 21*8 = 168).
In Penney's game (see A171861), the number of ways that HTH beats HHH on flip 3,4,5,... - Ed Pegg Jr, Dec 02 2010
The Ca2 sums (see A180662 for the definition of these sums) of triangle A035607 equal the terms of this sequence. - Johannes W. Meijer, Aug 05 2011

			G.f.: 1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 24*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
David Applegate, Marc LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq., Vol. 14 (2011), Article # 11.9.8.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Bisections: A001654, A059929.

Cf. A049853, A290565.

a070550 n = a070550_list !! n
a070550_list = 1 : 2 : 2 : 3 :
   zipWith (+) a070550_list
               (zipWith (+) (tail a070550_list) (drop 3 a070550_list))
-- Reinhard Zumkeller, Aug 06 2011

with(combinat): A070550 := proc(n): fibonacci(floor(n/2)+1) * fibonacci(ceil(n/2)+2) end: seq(A070550(n),n=0..37); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1, 0, 1, 1}, {1, 2, 2, 3}, 40] (* Jean-François Alcover, Jan 27 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,a+b+d}; NestList[nxt,{1,2,2,3},40][[;;,1]] (* Harvey P. Dale, Jul 16 2024 *)

A070550(n) = fibonacci(n\2+1)*fibonacci((n+5)\2) \\ M. F. Hasler, Aug 06 2011

x='x+O('x^100); Vec((1+x)/(1-x-x^3-x^4)) \\ Altug Alkan, Dec 24 2015

a(n) = F(floor(n/2)+1)*F(ceiling(n/2)+2), with F(n) = A000045(n). - Ralf Stephan, Apr 14 2004
G.f.: (1+x)/(1-x-x^3-x^4) = (1+x)/((1+x^2)*(1-x-x^2))
a(n) = A126116(n+4) - F(n+3). - Johannes W. Meijer, Aug 05 2011
a(n) = (1+3*i)/10*(-i)^n + (1-3*i)/10*(i)^n + (2+sqrt(5))/5*((1+sqrt(5))/2)^n + (2-sqrt(5))/5*((1-sqrt(5))/2)^n, where i = sqrt(-1). - Sergei N. Gladkovskii, Jul 16 2013
a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - Michael Somos, Jan 19 2014
Sum_{n>=1} 1/a(n) = A290565. - Amiram Eldar, Feb 17 2021

A008413 Coordination sequence for 5-dimensional cubic lattice.

Original entry on oeis.org

1, 10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, 14002, 20330, 28610, 39210, 52530, 69002, 89090, 113290, 142130, 176170, 216002, 262250, 315570, 376650, 446210, 525002, 613810, 713450, 824770, 948650, 1086002, 1237770, 1404930

Offset: 0

N. J. A. Sloane

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A035599.
Row 5 of A035607, A266213.
Column 5 of A113413, A119800, A122542.

Maple
```
4/3*n^4+20/3*n^2+2;
```

LinearRecurrence[{5,-10,10,-5,1},{1,10,50,170,450,1002},40] (* Harvey P. Dale, May 02 2016 *)
{1}~Join~Table[4/3 n^4 + 20/3 n^2 + 2, {n, 32}] (* or *)
CoefficientList[Series[((1 + x)/(1 - x))^5, {x, 0, 32}], x] (* Michael De Vlieger, Oct 04 2016 *)

G.f.: ((1+x)/(1-x))^5.
a(n) = (4/3)*n^4 + (20/3)*n^2 + 2 for n > 0. - Michael De Vlieger, Oct 04 2016
n*a(n) = 10*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
From Shel Kaphan, Mar 03 2023: (Start)
a(n) = 2*d*Hypergeometric2F1(1-d, 1-n, 2, 2) where d=5, for n>=1.
a(n) = A035599(n)*5/n, for n>0. (End)

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
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 (5,-10,10,-5,1).

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

[( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012

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

CoefficientList[Series[2*x*(1+x)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)

a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

A080245 Inverse of coordination sequence array A113413.

Original entry on oeis.org

1, -2, 1, 6, -4, 1, -22, 16, -6, 1, 90, -68, 30, -8, 1, -394, 304, -146, 48, -10, 1, 1806, -1412, 714, -264, 70, -12, 1, -8558, 6752, -3534, 1408, -430, 96, -14, 1, 41586, -33028, 17718, -7432, 2490, -652, 126, -16, 1

Offset: 0

Paul Barry, Feb 13 2003

Formal inverse of A035607 when written as lower triangular matrix 1 2 1 2 4 1 ...

			Rows are {1}, {-2, 1}, {6, -4, 1}, {-22, 16, -6, 1}, ....
From _Paul Barry_, Apr 28 2009: (Start)
Triangle begins
  1,
  -2, 1,
  6, -4, 1,
  -22, 16, -6, 1,
  90, -68, 30, -8, 1,
  -394, 304, -146, 48, -10, 1,
  1806, -1412, 714, -264, 70, -12, 1
Production matrix is
  -2, 1,
  2, -2, 1,
  -2, 2, -2, 1,
  2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 1,
  2, -2, 2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 2, -2, 1 (End)

Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 7.

Row sums are signed little Schroeder numbers A080243. Diagonal sums are given by A080244.
Cf. A035607, A080243, A080244, A006603, A001003.
Cf. A000007 A084938.
Essentially same triangle as A033877 but with rows read in reversed order.

Essentially the same as the triangle T(n, k), for n>0 and k>0, given by [0, -2, -1, -2, -1, -2, -1, -2, ...] DELTA A000007. Triangle (unsigned) given by [0, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.

Riordan array ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2).

A126116 a(n) = a(n-1) + a(n-3) + a(n-4), with a(0)=a(1)=a(2)=a(3)=1.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 11, 19, 31, 49, 79, 129, 209, 337, 545, 883, 1429, 2311, 3739, 6051, 9791, 15841, 25631, 41473, 67105, 108577, 175681, 284259, 459941, 744199, 1204139, 1948339, 3152479, 5100817, 8253295, 13354113, 21607409, 34961521

Offset: 0

Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

nonn

This sequence has the same growth rate as the Fibonacci sequence, since x^4 - x^3 - x - 1 has the real roots phi and -1/phi.

The Ca1 sums, see A180662 for the definition of these sums, of triangle A035607 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 05 2011

			G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 11*x^7 + 19*x^8 + 31*x^9 + ...

S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002

Seiichi Manyama, Table of n, a(n) for n = 0..4786
K. T. Atanassov, D. R. Deford, A. G. Shannon, Pulsated Fibonacci recurrences, Fibonacci Quarterly, Vol. 52, No. 5, Dec. 2014, pp. 22-27.
Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. Fibonacci numbers A000045; Lucas numbers A000032; tribonacci numbers A000213; tetranacci numbers A000288; pentanacci numbers A000322; hexanacci numbers A000383; 7th-order Fibonacci numbers A060455; octanacci numbers A079262; 9th-order Fibonacci sequence A127193; 10th-order Fibonacci sequence A127194; 11th-order Fibonacci sequence A127624, A128429.

a:=[1,1,1,1];; for n in [5..50] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jul 15 2019

[n le 4 select 1 else Self(n-1) + Self(n-3) + Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2015

# From R. J. Mathar, Jul 22 2010: (Start)
A010684 := proc(n) 1+2*(n mod 2) ; end proc:
A000032 := proc(n) coeftayl((2-x)/(1-x-x^2),x=0,n) ; end proc:
A126116 := proc(n) ((-1)^floor(n/2)*A010684(n)+2*A000032(n))/5 ; end proc: seq(A126116(n),n=0..80) ; # (End)
with(combinat): A126116 := proc(n): fibonacci(n-1) + fibonacci(floor((n-4)/2)+1)* fibonacci(ceil((n-4)/2)+2) end: seq(A126116(n), n=0..38); # Johannes W. Meijer, Aug 05 2011

LinearRecurrence[{1,0,1,1},{1,1,1,1},50] (* Harvey P. Dale, Nov 08 2011 *)

Vec((x-1)*(1+x+x^2)/((x^2+x-1)*(x^2+1)) + O(x^50)) \\ Altug Alkan, Dec 25 2015

((1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2))).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 15 2019

From R. J. Mathar, Jul 22 2010: (Start)
G.f.: (1-x)*(1+x+x^2)/((1-x-x^2)*(1+x^2)).
a(n) = ( (-1)^floor(n/2) * A010684(n) + 2*A000032(n))/5.
a(2*n) = A061646(n). (End)
From Johannes W. Meijer, Aug 05 2011: (Start)
a(n) = F(n-1) + A070550(n-4) with F(n) = A000045(n).
a(n) = F(n-1) + F(floor((n-4)/2) + 1)*F(ceiling((n-4)/2) + 2). (End)
a(n) = (1/5)*((sqrt(5)-1)*(1/2*(1+sqrt(5)))^n - (1+sqrt(5))*(1/2*(1-sqrt(5)))^n + sin((Pi*n)/2) - 3*cos((Pi*n)/2)). - Harvey P. Dale, Nov 08 2011
(-1)^n * a(-n) = a(n) = F(n) - A070550(n - 6). - Michael Somos, Feb 05 2012
a(n)^2 + 3*a(n-2)^2 + 6*a(n-5)^2 + 3*a(n-7)^2 = a(n-8)^2 + 3*a(n-6)^2 + 6*a(n-3)^2 + 3*a(n-1)^2. - Greg Dresden, Jul 07 2021
a(n) = A293411(n)-A293411(n-1). - R. J. Mathar, Jul 20 2025

Edited by Don Reble, Mar 09 2007

A371967 Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 8, 2, 1, 9, 24, 22, 6, 1, 1, 12, 49, 84, 61, 18, 2, 1, 15, 83, 215, 276, 174, 53, 9, 1, 1, 18, 126, 442, 840, 880, 504, 158, 28, 2, 1, 21, 178, 792, 2023, 3063, 2763, 1478, 472, 93, 12, 1, 1, 24, 239, 1292, 4176, 8406, 10692, 8604, 4374, 1416, 297, 38, 2, 1, 27, 309

Offset: 0

R. J. Mathar, Apr 14 2024

			The triangle starts with r>=0 rows and w>=0 wazirs as
  1 ;
  1 3 1 ;
  1 6 8 2  ;
  1 9 24 22 6 1 ;
  1 12 49 84 61 18 2  ;
  1 15 83 215 276 174 53 9 1 ;
  1 18 126 442 840 880 504 158 28 2  ;
  1 21 178 792 2023 3063 2763 1478 472 93 12 1 ;
  1 24 239 1292 4176 8406 10692 8604 4374 1416 297 38 2  ;
  1 27 309 1969 7731 19591 32716 36257 26674 13035 4264 945 142 15 1 ;
  ...

Alois P. Heinz, Rows r = 0..165 (first 17 rows from R. J. Mathar)
R. J. Mathar, Bivariate Generating Functions for Non-attacking Wazirs on Rectangular Boards
Jacob A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869 [math.CO] (2014) Table 2.
Wikipedia, Wazir (chess)

Cf. A051736 (row sums), A035607 (on 2Xr board), A011973 (on 1Xr board), A232833 (on rXr board).
T(n,n) gives A371978.
Row maxima give A371979.
Cf. A007494.

b:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
      x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 14 2024

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[BitAnd[j, l] > 0, 0, Expand[b[n - 1, j]*x^DigitCount[j, 2, 1]]], {j, {0, 1, 2, 4, 5}}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 05 2024, after Alois P. Heinz *)

T(r,0) = 1.
T(r,1) = 3*r.
T(r,2) = A064225(r-1).
T(r,3) = A172229(r).
T(r,4) = 27*r^4/8 -117*r^3/4 +829*r^2/8 -715*r/4 +126. [Siehler Table 3]
T(3,w) = A232833(3,w).
G.f.: (1+x*y) *(1 +x*y +x*y^2 -x^2*y^3)/(1 -x -x*y -x^2*y^3 -2*x^2*y -3*x^2*y^2 -x^3*y^2 +x^3*y^4 +x^4*y^4). - R. J. Mathar, Apr 21 2024

A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 20, 102, 360, 1002, 2364, 4942, 9424, 16722, 28004, 44726, 68664, 101946, 147084, 207006, 285088, 385186, 511668, 669446, 864008, 1101450, 1388508, 1732590, 2141808, 2625010, 3191812, 3852630, 4618712, 5502170, 6516012

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
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 (6,-15,20,-15,6,-1).

Cf. A035596-A035606.

Column 5 of A035607, A266213. Row 5 of A113413, A119800, A122542.

[(4*n^4+20*n^2+6)*n/15: n in [0..30]]; // Vincenzo Librandi, Apr 23 2012

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

CoefficientList[Series[2*x*(1+x)^4/(1-x)^6,{x,0,33}],x] (* Vincenzo Librandi, Apr 23 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,20,102,360,1002},40] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=(4*n^5+20*n^3+6*n)/15 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = (4*n^4+20*n^2+6)*n/15. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^4/(1-x)^6. - Colin Barker, Mar 19 2012
a(n) = 2*A069038(n). - R. J. Mathar, Dec 10 2013
From Shel Kaphan, Mar 01 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=5.
a(n) = A001847(n) - A001846(n).
a(n) = A008413(n)*n/5. (End)

cp's OEIS Frontend

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

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

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A050146 a(n) = T(n,n), array T as in A050143.

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A070550 a(n) = a(n-1) + a(n-3) + a(n-4), starting with a(0..3) = 1, 2, 2, 3.

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A008413 Coordination sequence for 5-dimensional cubic lattice.

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A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

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A080245 Inverse of coordination sequence array A113413.

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