A122542 -id:A122542 - 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

A104934 Expansion of (1-x)/(1 - 3x - 2x^2).

Original entry on oeis.org

1, 2, 8, 28, 100, 356, 1268, 4516, 16084, 57284, 204020, 726628, 2587924, 9217028, 32826932, 116914852, 416398420, 1483024964, 5281871732, 18811665124, 66998738836, 238619546756, 849856117940, 3026807447332, 10780134577876, 38394018628292, 136742325040628, 487015012378468, 1734529687216660

Offset: 0

Creighton Dement, Mar 29 2005

A floretion-generated sequence relating A007482, A007483, A007484. Inverse is A046717. Inverse of Fibonacci(3n+1), A033887. Binomial transform is A052984. Inverse binomial transform is A006131. Note: the conjectured relation 2*a(n) = A007482(n) + A007483(n-1) is a result of the FAMP identity dia[I] + dia[J] + dia[K] = jes + fam
Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]tesseq[A*B] with A = - .25'i + .25'j + .25'k - .25i' + .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' - .25e and B = + 'i + i' + 'ji' + 'ki' + e
a(n) is also the number of ways to build a (2 x 2 x n)-tower using (2 X 1 X 1)-bricks (see Exercise 3.15 in Aigner's book). - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
a(n) is the number of compositions of n when there are 2 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 1,272, 12, 18, 24, ... - R. J. Mathar, Aug 10 2012

M. Aigner, A Course in Enumeration, Springer, 2007, p.103.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
Index entries for linear recurrences with constant coefficients, signature (3,2)

Cf. A006131, A007482 (partial sums), A007483, A007484, A033887, A046717, A052984, A055099, A104935, A122542.

# Following the Pari implementation.
function a(n)
   F = BigInt[0 1; 2 3]
   Fn = F^n * [1; 2]
   Fn[1, 1]
end # Peter Luschny, Jan 06 2019

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 - x)/(1 - 3*x - 2*x^2)); // Vincenzo Librandi, Jul 13 2018

a := proc(n) option remember; `if`(n < 2, [1, 2][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..28); # Peter Luschny, Jan 06 2019

LinearRecurrence[{3, 2}, {1, 2}, 40] (* Vincenzo Librandi, Jul 13 2018 *)
CoefficientList[Series[(1-x)/(1-3x-2x^2),{x,0,40}],x] (* Harvey P. Dale, May 02 2019 *)

a(n)=([0,1; 2,3]^n*[1;2])[1,1] \\ Charles R Greathouse IV, Jun 20 2015

[(i*sqrt(2))^(n-1)*(i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) - chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # G. C. Greubel, Jun 27 2021

Define A007483(-1) = 1. Then 2*a(n) = A007482(n) + A007483(n-1) (conjecture);
  a(n+2) = 4*A007484(n) (thus 8*A007484(n) = A007482(n+2) + A007483(n+1));
  a(n+1) = 2*A055099(n);
  a(n+2) - a(n+1) - a(n) = A007484(n+1) - A007484(n).
a(0)=1, a(1)=2, a(n) = 3*a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Sep 19 2006
a(n) = Sum_{k=0..n} 2^k*A122542(n,k). - Philippe Deléham, Oct 08 2006
a(n) = ((17+sqrt(17))/34)*((3+sqrt(17))/2)^n + ((17-sqrt(17))/34)*((3-sqrt(17))/2)^n. - Richard Choulet, Nov 19 2008
a(n) = 2*a(n-1) + 4*Sum_{k=0..n-2} a(k) for n > 0. - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
G.f.: (1-x)/(1-3*x-2*x^2). - M. F. Hasler, Jul 12 2018
a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) - ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - G. C. Greubel, Jun 27 2021
E.g.f.: exp(3*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, May 24 2024

A123164 Row sums of A123160.

Original entry on oeis.org

1, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718, 161439727075246592

Offset: 0

Roger L. Bagula, Oct 02 2006

nonn

Coefficient of x^n in ((1 + x)/(1 - x))^n. - Paul Barry, Jan 20 2008
a(n) is also the number of order-preserving partial transformations (of an n-element chain). Equivalently, it is the order of the semigroup (monoid) of order-preserving partial transformations (of an n-element chain), PO sub n. - Abdullahi Umar, Aug 25 2008
Hankel transform is A180966. - Paul Barry, Sep 29 2010

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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
A. Laradji and A. Umar, Asymptotic results for semigroups of order-preserving partial transformations Comm. Algebra 34 (2006), 1071-1075. [From _Abdullahi Umar_, Oct 11 2008]
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.

Essentially identical to A002003.

Cf. A006318, A122542, A180966.

[1] cat [n le 2 select 2*4^(n-1) else (4*(3*(n-1)^2-1)*Self(n-1) - (2*n-1)*(n-2)*Self(n-2))/((2*n-3)*(n)): n in [1..30]]; // G. C. Greubel, Jul 19 2023

a[n_]:= a[n]= Sum[Binomial[n+k-1,k]*Binomial[n,k], {k,0,n}];
Table[a[n], {n,0,30}]

def A123164(n): return sum(binomial(n,j)*binomial(n+j-1,j) for j in range(n+1))
[A123164(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

a(n) = A122542(2*n,n). - Philippe Deléham, May 28 2007
a(n) = Sum_{k=0..n} C(n, k)*C(n+k-1, k). - Paul Barry, Aug 22 2007
(2*n-1)*(n+1)*a(n+1) = 4*(3*n^2-1)*a(n) - (2*n+1)*(n-1)*a(n-1) for n >= 1 with a(0) = 1 and a(1) = 2. - Abdullahi Umar, Aug 25 2008
a(n) = Jacobi_P(n, 0, -1, 3). - Paul Barry, Sep 27 2009
G.f.: (1 + x + sqrt(1 - 6*x + x^2))/(2*sqrt(1 - 6*x + x^2)). - Paul Barry, Sep 29 2010
From Abdullahi Umar, Oct 11 2008: (Start)
a(n+1) - a(n) = (2*n + 1)*A006318 (n >= 0);
2*a(n) = (n + 1)*A006318(n) - (n - 1)*A006318(n-1) (n > 0). (End)
a(n) = Hypergeometric2F1([-n, n], [1], -1). - Peter Luschny, Aug 02 2014
a(n) ~ (1 + sqrt(2))^(2*n) / (2^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021
From Peter Bala, Oct 07 2021: (Start)
a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*C(n, k)*C(3*n-2*k-1, n-2*k).
a(p) == 2 (mod p^3) for prime p >= 5.
Conjecture: a(n*p^k) == a(n*p^(k-1)) mod( p^(3*k) ) for prime p >= 5 and all positive integers n and k. (End)

Edited by N. J. A. Sloane, Oct 04 2006

Offset changed (a(0)=1) by Michael Somos, Feb 07 2011

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)

A059283 Triangle T(n,k) (0<= k <=n) read by rows. Left edge is 1, 0, 0, ... Otherwise each entry is sum of entry to left, entries immediately above it to left and right and entry directly above it 2 rows back.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 2, 8, 11, 0, 2, 14, 36, 47, 0, 2, 20, 78, 172, 219, 0, 2, 26, 138, 424, 862, 1081, 0, 2, 32, 216, 856, 2314, 4476, 5557, 0, 2, 38, 312, 1522, 5116, 12768, 23882, 29439, 0, 2, 44, 426, 2476, 9970, 30168, 71294, 130172, 159611, 0, 2, 50, 558

Offset: 0

N. J. A. Sloane, Jan 24 2001

			1; 0,1; 0,2,3; 0,2,8,11; 0,2,14,36,47; ... [36 = 14 + 8 + 11 + 3 for example].

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Right edge is A059284. Cf. A059226.

Cf. A224729 (central terms), A122542.

a059283 n k = a059283_tabl !! n !! k
a059283_row n = a059283_tabl !! n
a059283_tabl = [1] : [0,1] : f [1] [0,1] where
   f us vs = ws : f vs ws where
     ws = scanl1 (+) $ zipWith (+)
                       ([0]++us++[0]) $ zipWith (+) ([0]++vs) (vs++[0])
-- Reinhard Zumkeller, Apr 17 2013

t[0, 0] = 1; t[, 0] = 0; t[n, k_] /; 0 <= k <= n := t[n, k] = t[n, k-1] + t[n-1, k-1] + t[n-1, k] + t[n-2, k-1]; t[, ] = 0; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2013 *)

T(0, 0)=1; T(n, 0)=0, n>0; T(n, k)=T(n, k-1)+T(n-1, k-1)+T(n-1, k)+T(n-2, k-1), n, k>0

G.f. for T(n, k): ((1+2*w+w^2)*z^2+(-1-2*w-w^2)*z-w*(-3*w^2-6*w+1)^(1/2)+2*w)/(1+w)^2/((1+w)*z^2+(w-1)*z+w) (expand first as series in z, then in w).

More terms from Larry Reeves (larryr(AT)acm.org), Jan 25 2001

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

A086901 a(1) = a(2) = 1; for n>2, a(n) = 4a(n-1) + 3a(n-2).

Original entry on oeis.org

1, 1, 7, 31, 145, 673, 3127, 14527, 67489, 313537, 1456615, 6767071, 31438129, 146053729, 678529303, 3152278399, 14644701505, 68035641217, 316076669383, 1468413601183, 6821884412881, 31692778455073, 147236767058935

Offset: 1

Rick Powers (rick.powers(AT)mnsu.edu), Sep 18 2003

			a(3) = 4*1 + 3*1 = 7;
a(4) = 4*7 + 3*1 = 31.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.

Cf. A015530, A102900, A122542.

a086901 n = a086901_list !! (n-1)
a086901_list = 1 : 1 : zipWith (+)
               (map (* 3) a086901_list) (map (* 4) $ tail a086901_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 28 2024

a[n_]:=(MatrixPower[{{3,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,4},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{4,3}, {1,1}, 41] (* G. C. Greubel, Oct 28 2024 *)

A086901(n)=if(n<3,1,4*A086901(n-1)+3*A086901(n-2)) \\ Michael B. Porter, Apr 04 2010

A086901=BinaryRecurrenceSequence(4,3,1,1)
[A086901(n) for n in range(41)] # G. C. Greubel, Oct 28 2024

a(n) = ((c + 5)*b^n - (b + 5)*c^n)/14, where b = 2 + sqrt(7), c = 2 - sqrt(7).
From Ralf Stephan, Feb 01 2004: (Start)
G.f.: x*(1-3*x)/(1 - 4*x - 3*x^2).
a(n) = A015530(n) - 3*A015530(n-1) = 1 + 6*Sum_{k=0..n-2} A015530(k). (End)
a(n+1) = Sum_{k=0..n} 3^(n-k)*A122542(n,k), n>=0. - Philippe Deléham, Oct 27 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 3,3]^(n-1). - Gary W. Adamson, Mar 02 2008
G.f.: G(0)*(1-3*x)/(2-4*x), where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
E.g.f.: exp(2*x)*( cosh(sqrt(7)*x) - (1/sqrt(7))*sinh(sqrt(7)*x) ). - G. C. Greubel, Oct 28 2024

More terms from Ray Chandler, Sep 19 2003

A122690 a(n) = 5a(n-1) + 4a(n-2) with a(0)=1, a(1)=4.

Original entry on oeis.org

1, 4, 24, 136, 776, 4424, 25224, 143816, 819976, 4675144, 26655624, 151978696, 866515976, 4940494664, 28168537224, 160604664776, 915697472776, 5220906022984, 29767320006024, 169720224122056, 967670400634376

Offset: 0

Philippe Deléham, Sep 22 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,4).

Cf. A015537.

a:=[1,4];; for n in [3..35] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x)/(1-5*x-4*x^2) )); // G. C. Greubel, Dec 26 2019

seq(coeff(series((1-x)/(1-5*x-4*x^2), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 26 2019

LinearRecurrence[{5,4},{1,4},35] (* Harvey P. Dale, Apr 06 2012 *)
Table[2^(n-1)*(2*Fibonacci[n+1, 5/2] - Fibonacci[n, 5/2]), {n,0,35}] (* G. C. Greubel, Dec 26 2019 *)

Vec((1-x)/(1-5*x-4*x^2)+O(x^35)) \\ Charles R Greathouse IV, Jan 17 2012

def A122690_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-5*x-4*x^2) ).list()
A122690_list(35) # G. C. Greubel, Dec 26 2019

a(n) = Sum_{k=0..n} 4^k*A122542(n,k).
G.f.: (1-x)/(1-5*x-4*x^2).
Lim_{n -> infinity} a(n+1)/a(n)-> (5+sqrt(41))/2 = 5.701562118716.
a(n) = 2^(n-1)*(2*Fibonacci(n+1, 5/2) - Fibonacci(n, 5/2)) = -(2/I)^(n-1)*( 2*I*ChebyshevU(n, 5*I/4) + ChebyshevU(n-1, 5*I/4)). - G. C. Greubel, Dec 26 2019

Corrected by T. D. Noe, Nov 07 2006

A123270 a(0)=1, a(1)=1, a(n) = 5a(n-1) + 4a(n-2).

Original entry on oeis.org

1, 1, 9, 49, 281, 1601, 9129, 52049, 296761, 1692001, 9647049, 55003249, 313604441, 1788035201, 10194593769, 58125109649, 331403923321, 1889520055201, 10773215969289, 61424160067249, 350213664213401, 1996764961336001

Offset: 0

Philippe Deléham, Oct 09 2006

nonn

First differences give {0, 8, 40, 232, 1320, 7528, 42920, ...} = 8*A015537(n) = 8 * {0, 1, 5, 29, 165, 941, 5365, ...}. - Alexander Adamchuk, Nov 03 2006

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina, Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (5, 4).

Cf. A015537.

Cf. A095344, A090390.

a123270 n = a123270_list !! n
a123270_list = 1 : 1 : zipWith (-) (map (* 5) $
   zipWith (+) (tail a123270_list) a123270_list) a123270_list
-- Reinhard Zumkeller, Aug 16 2013

LinearRecurrence[{5,4},{1,1},30] (* Harvey P. Dale, Jul 25 2011 *)

a(n) = Sum_{k=0..n} 4^(n-k)*A122542(n,k).
G.f.: (1-4*x)/(1-5*x-4*x^2).
a(n) = 1 + 8*Sum_{k=0..n} A015537(k). - Alexander Adamchuk, Nov 03 2006

A008416 Coordination sequence for 8-dimensional cubic lattice.

Original entry on oeis.org

1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120, 425000576

Offset: 0

N. J. A. Sloane

Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

Cf. A113413, A122542.

CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* Michael De Vlieger, Dec 18 2017 *)

G.f.: ((1+x)/(1-x))^8.
a(n) = A008415(n) + A008415(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017
n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

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A123270 a(0)=1, a(1)=1, a(n) = 5a(n-1) + 4a(n-2).