A122542 -id:A122542 - cp's OEIS frontend

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

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)

A091928 a(0)=1, a(1)=5; a(n) = 6a(n-1) + 5a(n-2) for n > 1.

Original entry on oeis.org

1, 5, 35, 235, 1585, 10685, 72035, 485635, 3273985, 22072085, 148802435, 1003175035, 6763062385, 45594249485, 307380808835, 2072256100435, 13970440646785, 94183924382885, 634955749531235, 4280654119101835

Offset: 0

Paul Barry, Feb 13 2004

Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=sum of first row of B^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A015551.

[n le 2 select 5^(n-1) else 6*Self(n-1) +5*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 27 2024

LinearRecurrence[{6,5},{1,5},30] (* Harvey P. Dale, Apr 09 2022 *)

A091928= BinaryRecurrenceSequence(6,5,1,5)
[A091928(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

G.f.: (1-x)/(1-6*x-5*x^2).
a(n) = (1/2 +1/sqrt(14))*(3 +sqrt(14))^n + (1/2 -1/sqrt(14))*(3 -sqrt(14))^n.
From Philippe Deléham, Sep 22 2006: (Start)
a(n) = Sum_{k=0..n} 5^k*A122542(n,k).
Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(14) = 6.741657386773... . (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

A122558 a(0)=1, a(1)=3, a(n) = 4a(n-1) + 3a(n-2) for n > 1.

Original entry on oeis.org

1, 3, 15, 69, 321, 1491, 6927, 32181, 149505, 694563, 3226767, 14990757, 69643329, 323545587, 1503112335, 6983086101, 32441681409, 150715983939, 700188979983, 3252903871749, 15112182426945, 70207441323027, 326166312572943

Offset: 0

Philippe Deléham, Sep 20 2006, Sep 22 2006

nonn

a(n) is the number of compositions of n when there are 3 types of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,3).

Cf. A122542.

[n le 2 select 3^(n-1) else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 27 2024

LinearRecurrence[{4,3},{1,3},30] (* Harvey P. Dale, Mar 18 2023 *)

Vec((1-x)/(1-4*x-3*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2022

A122558= BinaryRecurrenceSequence(4,3,1,3)
[A122558(n) for n in range(41)] # G. C. Greubel, Oct 27 2024

G.f.: (1-x)/(1-4*x-3*x^2).
a(n) = Sum_{k=0..n} 3^k*A122542(n,k).
Limit_{n->infinity} a(n+1)/a(n) = 2 + sqrt(7) = 4.645751311064....
a(n) = ((7+sqrt(7))/14)*(2+sqrt(7))^n + ((7-sqrt(7))/14)*(2-sqrt(7))^n. - Richard Choulet, Nov 20 2008

Corrected by T. D. Noe, Nov 07 2006

A123362 a(0) = 1, a(1) = 1, a(n) = 6a(n-1) + 5a(n-2) for n > 1.

Original entry on oeis.org

1, 1, 11, 71, 481, 3241, 21851, 147311, 993121, 6695281, 45137291, 304300151, 2051487361, 13830424921, 93239986331, 628592042591, 4237752187201, 28569473336161, 192605600952971, 1298480972398631, 8753913839156641

Offset: 0

Philippe Deléham, Oct 12 2006

Hankel transform is [1, 10, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

G. C. Greubel, 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 (6,5).

LinearRecurrence[{6, 5}, {1, 1}, 50] (* G. C. Greubel, Oct 12 2017 *)

x='x+O('x^50); Vec((1-5*x)/(1 - 6*x - 5*x^2)) \\ G. C. Greubel, Oct 12 2017

a(n) = Sum_{k = 0..n} 5^(n - k)*A122542(n, k).

G.f.: (1 - 5*x)/(1 - 6*x - 5*x^2).

A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 2, 16, 38, 1, 1, 2, 24, 146, 192, 1, 1, 2, 32, 326, 1408, 1002, 1, 1, 2, 40, 578, 4672, 14002, 5336, 1, 1, 2, 48, 902, 11008, 69002, 142000, 28814, 1, 1, 2, 56, 1298, 21440, 216002, 1038984, 1459810, 157184, 1, 1, 2, 64, 1766, 36992, 525002, 4320608, 15856206, 15158272, 864146, 1

Offset: 0

Seiichi Manyama, Jul 24 2020

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     2,     2,      2,      2, ...
  1,    8,    16,    24,     32,     40, ...
  1,   38,   146,   326,    578,    902, ...
  1,  192,  1408,  4672,  11008,  21440, ...
  1, 1002, 14002, 69002, 216002, 525002, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Column k=0-3 give A000012, A123164, A103885, A333715.
Main diagonal gives A336522.
Cf. A122542, A266213, A336534.

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n + j - 1, n - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 24 2020 *)

T(n,k) = (1/k) * [x^n] ( (1 + x)/(1 - x) )^(k*n) for k > 0 and n > 0.
T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j-1,n-1).
T(n,k) = (1/k) * Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j-1,j) for k > 0 and n > 0.
T(n,k) = Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n-1,j-1) for n > 0.
T(n,k) = binomial(k*n-1, n-1)*hypergeom([-n, k*n], [1+(k-1)*n], -1) for k > 0. - Stefano Spezia, Aug 09 2025

A210636 Riordan array ((1-x)/(1-2x-x^2), x(1+x)/(1-2*x-x^2)).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 17, 40, 32, 10, 1, 41, 117, 124, 60, 13, 1, 99, 332, 437, 286, 97, 16, 1, 239, 921, 1447, 1193, 553, 143, 19, 1, 577, 2512, 4584, 4556, 2682, 952, 198, 22, 1, 1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1

Offset: 0

Philippe Deléham, Mar 26 2012

Triangle T(n,k), 0<=k<=n, read by rows, given by (1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Product of A122542 and A007318 (Pascal's triangle) as lower triangular matrices .

			Triangle begins :
1
1, 1
3, 4, 1
7, 13, 7, 1
17, 40, 32, 10, 1
41, 117, 124, 60, 13, 1
99, 332, 437, 286, 97, 16, 1
239, 921, 1447, 1193, 553, 143, 19, 1
577, 2512, 4584, 4556, 2682, 952, 198, 22, 1
1393, 6761, 14048, 16336, 11666, 5282, 1510, 262, 25, 1

Cf. Columns :A001333, A119915, Diagonals : A000012, A016777, Antidiagonal sums : A077995

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-x)/(1-2*x-y*x-x^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A001333(n), A104934(n), A122958(n), A122690(n), A091928(n) for x = -1, 0, 1, 2, 3, 4 respectively.

cp's OEIS Frontend

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

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A091928 a(0)=1, a(1)=5; a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.

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A122558 a(0)=1, a(1)=3, a(n) = 4*a(n-1) + 3*a(n-2) for n > 1.

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A123362 a(0) = 1, a(1) = 1, a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.

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A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

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A210636 Riordan array ((1-x)/(1-2*x-x^2), x*(1+x)/(1-2*x-x^2)).

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A091928 a(0)=1, a(1)=5; a(n) = 6a(n-1) + 5a(n-2) for n > 1.

A122558 a(0)=1, a(1)=3, a(n) = 4a(n-1) + 3a(n-2) for n > 1.

A123362 a(0) = 1, a(1) = 1, a(n) = 6a(n-1) + 5a(n-2) for n > 1.

A210636 Riordan array ((1-x)/(1-2x-x^2), x(1+x)/(1-2*x-x^2)).