A034356 -id:A034356 - cp's OEIS frontend

A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 16, 22, 16, 6, 1, 1, 7, 23, 43, 43, 23, 7, 1, 1, 8, 32, 77, 106, 77, 32, 8, 1, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 1, 11, 71, 333, 1060, 1988, 1988

Offset: 0

N. J. A. Sloane, Nov 21 2002

"The familiar appearance of the first few rows [...] provides a good example of the perils of too hasty extrapolation in mathematics." - Slepian.

The difference between this triangle and the one for which it can be so easily mistaken is A250002. - Tilman Piesk, Nov 10 2014.

			     k    0   1   2   3    4    5    6    7    8   9  10  11        sum
   n
   0      1                                                           1
   1      1   1                                                       2
   2      1   2   1                                                   4
   3      1   3   3   1                                               8
   4      1   4   6   4    1                                         16
   5      1   5  10  10    5    1                                    32
   6      1   6  16  22   16    6    1                               68
   7      1   7  23  43   43   23    7    1                         148
   8      1   8  32  77  106   77   32    8    1                    342
   9      1   9  43 131  240  240  131   43    9   1                848
  10      1  10  56 213  516  705  516  213   56  10   1           2297
  11      1  11  71 333 1060 1988 1988 1060  333  71  11   1       6928

M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994

Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Apparently, the notation for T(n,k) is W_{nk2}; see p. 197.]
Petros Hadjicostas, Generating function for column k=4.
Petros Hadjicostas, Generating function for column k=5.
Petros Hadjicostas, Generating function for column k=6.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]
Index entries for sequences related to binary linear codes

Cf. A006116, A022166, A076766 (row sums).
A034356 gives same table but with the k=0 column omitted.
Columns include A000012 (k=0), A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A076831col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = (f1 - f2)/(1-x)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives
print(A076831col(4, 30)) # Petros Hadjicostas, Sep 30 2019

From Petros Hadjicostas, Sep 30 2019: (Start)
T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.
G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).
G.f. for column k=3: -(x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^8).
G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. (See also some of the links above.)
(End)

A034363 Triangle of number of linear [ n,k ] ternary codes (n >= 1, k >= 1) without 0 columns.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 8, 4, 1, 1, 8, 19, 15, 5, 1, 1, 10, 39, 50, 24, 6, 1, 1, 14, 78, 168, 118, 37, 7, 1

Offset: 1

N. J. A. Sloane

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254, A034356, A034363-A034374.

A034374 Triangle of number of indecomposable projective linear [ n,k ] GF(5) codes (n >= 1, k >= 1) without 0 columns.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 10, 7, 1, 0, 0, 21, 46, 10, 1, 0, 0, 42, 436, 193, 17, 1

Offset: 1

N. J. A. Sloane

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

H. Fripertinger, Isometry Classes of Codes

Cf. A034253, A034254, A034356, A034363-A034374.

A014631 Numbers in order in which they appear in Pascal's triangle.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 15, 20, 7, 21, 35, 8, 28, 56, 70, 9, 36, 84, 126, 45, 120, 210, 252, 11, 55, 165, 330, 462, 12, 66, 220, 495, 792, 924, 13, 78, 286, 715, 1287, 1716, 14, 91, 364, 1001, 2002, 3003, 3432, 105, 455, 1365, 5005, 6435, 16, 560, 1820, 4368, 8008, 11440

Offset: 1

Mohammad K. Azarian

A permutation of the natural numbers. - Robert G. Wilson v, Jun 12 2014

In Pascal's triangle a(n) occurs the first time in row A265912(n). - Reinhard Zumkeller, Dec 18 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
F. Richman, Combinations and Permutations
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences that are permutations of the natural numbers

Cf. A034356, A074909, A119629.

Cf. A034868, A119629 (inverse), A265912.

import Data.List (nub)
a014631 n = a014631_list !! (n-1)
a014631_list = 1 : (nub $ concatMap tail a034868_tabf)
-- Reinhard Zumkeller, Dec 19 2015

lst = {1}; t = Flatten[Table[Binomial[n, m], {n, 16}, {m, Floor[n/2]}]]; Do[ If[ !MemberQ[lst, t[[n]]], AppendTo[lst, t[[n]] ]], {n, Length@t}]; lst (* Robert G. Wilson v *)
DeleteDuplicates[Flatten[Table[Binomial[n,m],{n,20},{m,0,Floor[n/2]}]]] (* Harvey P. Dale, Apr 08 2013 *)

from itertools import count, islice
def A014631_gen(): # generator of terms
    s, c =(1,), set()
    for i in count(0):
        for d in s:
            if d not in c:
                yield d
                c.add(d)
        s=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ())
A014631_list = list(islice(A014631_gen(),30)) # Chai Wah Wu, Oct 17 2023

More terms from Erich Friedman

Offset changed by Reinhard Zumkeller, Dec 18 2015

A034198 Number of binary codes (not necessarily linear) of length n with 3 words.

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 23, 32, 43, 56, 71, 89, 109, 132, 158, 187, 219, 255, 294, 337, 384, 435, 490, 550, 614, 683, 757, 836, 920, 1010, 1105, 1206, 1313, 1426, 1545, 1671, 1803, 1942, 2088, 2241, 2401, 2569, 2744, 2927, 3118, 3317, 3524, 3740

Offset: 1

N. J. A. Sloane

nonn

Number of distinct triangles on vertices of n-dimensional cube.
Also, a(n) is the number of orbits of C_2^2 subgroups of C_2^n under automorphisms of C_2^n.
Also, a(n) is the number of faithful representations of C_2^2 of dimension n up to equivalence by automorphisms of (C_2^2).
Also, a([n/2]) is equal to the number of partitions mu such that there exists a C_2^2 subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles (see below example). - John M. Campbell, Jan 22 2016

			Let t denote the trivial representation and u_1, u_2, u_3 the three nontrivial irreducible representations of C_2^2 (so the u_i are all equivalent up to automorphisms of C_2^2). Then the a(4) = 6 faithful representations of dimension 4 are:
  2t+u_1+u_2; t+2u_1+u_2; t+u_1+u_2+u_3;
  3u_1+u_2;   2u_1+2u_2;  2u_1+u_2+u_3.
From _John M. Campbell_, Jan 22 2016: (Start)
Letting n=8, there are a([n/2])=a(4)=6 partitions mu such that there exists a Klein four-subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles, as indicated below:
{2, 1, 1} <-> {(12)(34), (12), (34), id}
{3, 2, 1} <-> {(12)(34)(56), (34)(56), (12), id}
{2, 2, 2} <-> {(12)(34), (34)(56), (56)(12), id}
{4, 3, 1} <-> {(12)(34)(56)(78), (34)(56)(78), (12), id}
{4, 2, 2} <-> {(12)(34)(56)(78), (56)(78), (12)(34), id}
{3, 3, 2} <-> {(12)(34)(56), (34)(56)(78), (12)(78), id}
(End)

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.
J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
H. Fripertinger, Isometry Classes of Codes
H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A034188.

Column k=2 of both A034356 and A076831 (which are the same except for column k=0).

[Floor(n*(2*n^2+21*n-6)/72): n in [1..50]]; // Vincenzo Librandi, Sep 18 2016

A034198 := n -> iquo(n*(2*n^2+21*n-6), 72):
seq(A034198(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[n (2n^2+21*n-6)/72],{n,50}] (* Harvey P. Dale, Dec 25 2011 *)
LinearRecurrence[ {2,0,-1,-1,0,2,-1},{0,1,3,6,10,16,23},50] (* Harvey P. Dale, Dec 25 2011 *)

a(n) = floor(n*(2*n^2 + 21*n - 6)/72).
G.f.: (-x^5 + x^3 + x^2)/((1 - x)^2*(1 - x^2)*(1 - x^3)) = 1/((1 - x)^2*(1 - x^2)*(1 - x^3)) - 1/(1 - x)^2.
a(1) = 0, a(2) = 1, a(3) = 3, a(4) = 6, a(5) = 10, a(6) = 16, a(7) = 23, and a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7) for n >= 8. [Harvey P. Dale, Dec 25 2011]
From Irena Swanson, Feb 11 2024: (Start)
The roots of the characteristic polynomial corresponding to the above recurrence are 1, 1, 1, 1, -1, -1/2 - sqrt(-3)/2 and -1/2 + sqrt(-3)/2. The corresponding closed form is:
a(n) = -25/144 - n/12 + 7n^2/24 + n^3/36 + (-1)^n/16 + (1/18 + sqrt(-3)/54)(-1/2 - sqrt(-3)/2)^n + (1/18 - sqrt(-3)/54)(-1/2 + sqrt(-3)/2)^n for n >= 1. (End)

Additional comments from Max Alekseyev, Jul 09 2006

Additional comments from Andrew Rupinski, Jan 20 2010

A034328 Triangle read by rows: T(n,k) = number of loopless, regular k X n-matrix matroids of dimension k (or n-matroids of rank k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 6, 12, 11, 5, 1, 1, 7, 20, 26, 17, 6, 1, 1, 9, 33, 58, 52, 25, 7, 1, 1, 11, 52, 121, 146, 95, 35, 8, 1, 1, 13, 78, 240, 388, 334, 162, 47, 9, 1, 1, 15, 113, 454, 975, 1123, 710, 262, 61, 10, 1, 1, 18, 163, 835, 2365

Offset: 0

N. J. A. Sloane

			1; 1,1; 1,2,1; 1,3,3,1; 1,4,6,4,1; 1,6,12,11,5,1; 1,7,20,26,17,6,1; ...

Computed by Harald Fripertinger (fripert(AT)kfunigraz.ac.at).

Cf. A034356, A034327.

Description corrected by Harald Fripertinger, Nov 14 2007

A034358 Number of binary [ n,4 ] codes.

Original entry on oeis.org

0, 0, 0, 1, 5, 16, 43, 106, 240, 516, 1060, 2108, 4064, 7641, 14036, 25253, 44560, 77245, 131658, 220883, 365027, 594674, 955649, 1515908, 2374875, 3676632, 5627587, 8520689, 12767557, 18941641, 27834607, 40530902, 58503994, 83741461, 118904892, 167534794, 234309554, 325373538, 448747606

Offset: 1

N. J. A. Sloane

nonn

Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=4.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [Here a(n) = W_{n,4,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = W_{n,4,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
M. Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
M. Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
Index entries for sequences related to binary linear codes

Cf. A034253, A034254.
Column k=4 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034345.

# Fripertinger's method to find the g.f. of column k >= 2 of A076831 or A034356 (for small k):
def A076831col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = (f1 - f2)/(1-x)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 (this sequence) gives
print(A076831col(4, 30)) # Petros Hadjicostas, Oct 07 2019

More terms from Petros Hadjicostas, Oct 07 2019

A034357 Number of binary [ n,3 ] codes.

Original entry on oeis.org

0, 0, 1, 4, 10, 22, 43, 77, 131, 213, 333, 507, 751, 1088, 1546, 2159, 2967, 4023, 5384, 7122, 9322, 12081, 15512, 19752, 24950, 31283, 38953, 48188, 59244, 72419, 88037, 106469, 128129, 153476, 183019, 217331, 257033

Offset: 1

N. J. A. Sloane

nonn

Also, a(n) is the number of orbits of C_2^3 subgroups of C_2^n under automorphisms of C_2^n. Also, a(n) is the number of faithful representations of C_2^3 of dimension n up to equivalence by automorphisms of (C_2^3). - Andrew Rupinski, Jan 20 2011

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=3.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,3,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,3,2}; see p. 197.]

Cf. A034198, A034253, A034254.
Column k=3 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034344.

G.f.: (-x^15+2*x^14-x^13+x^12+x^9-x^7+x^4+x^3)/((1-x)^3*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^7)).

A034359 Number of binary [ n,5 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 23, 77, 240, 705, 1988, 5468, 14724, 39006, 101818, 261924, 663748, 1655781, 4062110, 9793065, 23186825, 53896597, 122975627, 275449464, 605794093, 1308633243, 2777847319, 5797093774, 11900199553, 24042491094, 47833081481, 93765335118, 181200186060, 345389067067, 649704599010

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=5.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,5,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,5,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).

Cf. A034253, A034254.
Column k=5 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034346.

More terms from Joerg Arndt, Oct 09 2019

A034360 Number of binary [ n,6 ] codes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 32, 131, 516, 1988, 7664, 29765, 117169, 467266, 1880517, 7588675, 30491836, 121191234, 473940269, 1816579108, 6806904522, 24897540538, 88831250408, 309108741706, 1049278764758, 3476233500031, 11246972937210, 35561409388625, 109967835029368, 332834886787933, 986732945823099

Offset: 1

N. J. A. Sloane

nonn

H. Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [See column k=6.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have a(n) = W_{n,6,2}; see p. 4 of the preprint.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [We have a(n) = W_{n,6,2}; see p. 197.]
Petros Hadjicostas, Generating function for a(n).

Cf. A034253, A034254.
Column k=6 of both A034356 and A076831 (which are the same except for column k=0).
First differences give A034347.

More terms from Joerg Arndt, Oct 09 2019

cp's OEIS Frontend

A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

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A034363 Triangle of number of linear [ n,k ] ternary codes (n >= 1, k >= 1) without 0 columns.

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A034374 Triangle of number of indecomposable projective linear [ n,k ] GF(5) codes (n >= 1, k >= 1) without 0 columns.

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A014631 Numbers in order in which they appear in Pascal's triangle.

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Haskell

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A034198 Number of binary codes (not necessarily linear) of length n with 3 words.

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Magma

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A034328 Triangle read by rows: T(n,k) = number of loopless, regular k X n-matrix matroids of dimension k (or n-matroids of rank k).

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A034358 Number of binary [ n,4 ] codes.

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Sage

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A034357 Number of binary [ n,3 ] codes.

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A034359 Number of binary [ n,5 ] codes.