A050186 -id:A050186 - cp's OEIS frontend

A051199 T(2n+6,n), array T as in A050186; a count of aperiodic binary words.

0, 8, 40, 216, 980, 4368, 18468, 77520, 319440, 1307448, 5310448, 21474180, 86488020, 347373600, 1391956192, 5567901768, 22239898848, 88732378800, 353696824152, 1408831480056, 5608231863080, 22314239255088

Offset: 0

Clark Kimberling

nonn

A050187 a(n) = n * floor((n-1)/2).

Original entry on oeis.org

0, 0, 0, 3, 4, 10, 12, 21, 24, 36, 40, 55, 60, 78, 84, 105, 112, 136, 144, 171, 180, 210, 220, 253, 264, 300, 312, 351, 364, 406, 420, 465, 480, 528, 544, 595, 612, 666, 684, 741, 760, 820, 840, 903, 924, 990, 1012, 1081, 1104, 1176

Offset: 0

Clark Kimberling, Dec 11 1999

T(n,2), array T as in A050186; a count of aperiodic binary words.
The Row2 triangle sums A159797 lead to the sequence given above for n >= 1 with a(1)=0. For the definitions of the Row2 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011
The number of chords joining n equally distributed points on a circle with a length less than the diameter. - Wesley Ivan Hurt, Nov 23 2013
a(n) is the maximum possible length of a circuit in the complete graph on n vertices. - Geoffrey Critzer, May 23 2014
For n > 0, a(n) is half the sum of the perimeters of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L. For example, a(14) = 84; the rectangles are 1 X 13, 2 X 12, 3 X 11, 4 X 10, 5 X 9, 6 X 8 (the 7 X 7 rectangle is not considered since we have W < L). The sum of the perimeters gives 28 + 28 + 28 + 28 + 28 + 28 = 168, half of which is 84. - Wesley Ivan Hurt, Nov 23 2017
Sum of the middle side lengths of all integer-sided triangles with perimeter 3n whose side lengths are in arithmetic progression (For example, when n=5 there are two triangles with perimeter 3(5) = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(5) = 5+5 = 10). - Wesley Ivan Hurt, Nov 01 2020

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A050186, A159797, A180662.

[n*Floor((n-1)/2): n in [0..50]]; // Wesley Ivan Hurt, May 24 2014

A050187:=n->n*floor((n-1)/2); seq(A050187(n), n=0..100); # Wesley Ivan Hurt, Nov 23 2013

Table[n*Floor[(n-1)/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 23 2013 *)

a(n) = n * floor((n-1)/2).
From R. J. Mathar, Aug 08 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x^3*(3+x) / ((1+x)^2*(1-x)^3). (End)
a(n) = binomial(n,2) - (n/2) * ((n+1) mod 2). - Wesley Ivan Hurt, Nov 23 2013
E.g.f.: x*(x*cosh(x) + sinh(x)*(x - 1))/2. - Stefano Spezia, Nov 02 2020

Name change by Wesley Ivan Hurt, Nov 23 2013

A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 112, 210, 387, 728, 1360, 2570, 4845, 9198, 17459, 33288, 63519, 121574, 232960, 447392, 860265, 1657009, 3195465, 6170930, 11930100, 23091222, 44738560, 86767016, 168428805, 327235602, 636289024, 1238188770, 2411205111, 4698767640, 9162588158, 17878237850

Offset: 1

Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009

Arndt's PARI code computes a(n) as the sum, divided by n, of every 3rd term in row n of L = A050186 = Möbius transform of binomials, starting with k = (1-n) mod 3 (nonnegative remainder), where k = 0 and k = n give L(n, k) = 0 and can be omitted. Cf. A053727, EXAMPLE and second PROGRAM. - M. F. Hasler, Sep 27 2018

			Illustrating computation via L = A050186, cf. COMMENTS: a(1) = [L(1,0)] = 0. a(2) = [L(2,2)] = 0. a(3) = L(3,1)/3 = 3/3 = 1. a(4) = ([L(4,0)] + L(4,3))/4 = 4/4 = 1. a(5) = (L(5,2) + [L(5,5)])/5 = 10/5 = 2. In [...] are terms L(n,0) = L(n,n) = 0.

Robert Israel, Table of n, a(n) for n = 1..3331
J. E. Iglesias, Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.

This sequence is the half of A165912 (the number of alternate polynomials). A001037 is the enumeration by degree of the polynomials of I. A000048 is the number of 3-elements orbits of S3 action on I.

f:= proc(n) local D,d;
  D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
  add(numtheory:-mobius(n/d)*(2^d - (-1)^d),d=D)/(3*n)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 14 2019

a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
vector(55,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

A165920(n,k=(1-n)%3)=sum(i=0,(n-k)\3,A050186(n,k+3*i))\n \\ For illustration. - M. F. Hasler, Sep 30 2018

a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).

A011956 Number of close-packings with layer-number 3n and space group R3m.

Original entry on oeis.org

1, 2, 4, 10, 21, 42, 84, 164, 322, 620, 1200, 2300, 4429, 8482, 16303, 31259, 60105, 115472, 222332, 428106, 825774, 1593669, 3080004, 5956902, 11534689, 22352962, 43361663, 84181720, 163574114, 318079104, 619007004, 1205471654, 2349209058, 4581032192

Offset: 7

N. J. A. Sloane

Last column of Table 4 in McLarnan (1981), p. 277. See there for more information. - M. F. Hasler, May 26 2025

Juan E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245. See Table 4.
T. J. McLarnan, The numbers of polytypes in close-packings and related structures, Zeits. Krist. 155, 269-291 (1981).
Eric W. Weisstein, Barlow Packing, on MathWorld-A Wolfram Web Resource.
Wikipedia, Close-packing of equal spheres.

Cf. A011768, A011954, A011955, A371992.

fa[p_, q_] := fa[p, q] = (p+q-1)!/(p!q!) - Sum[fa[p/d, q/d]/d, {d, Rest[Intersection@@(Divisors/@{p, q})]}];
fb[p_, q_] := fb[p, q] = (Quotient[p, 2]+Quotient[q, 2])!/(Quotient[p, 2]!Quotient[q, 2]!) - Sum[fb[p/d, q/d], {d, Rest[Intersection@@(Divisors/@{p, q})]}];
rh[n_] := Sum[fa[n-q, q]+fb[n-q, q], {q, Select[Range[n/2], !Divisible[n-2#, 3]&]}] / 2; (* A371992 *)
fSO[n_] := Sum[fb[2n+1-q,q], {q, Select[Range[n+1,2n], !Divisible[2n+1-2#,3]&]}];(*A011954*)
fb2[p_, q_] := fb2[p, q] = (p+q)!/(p!q!) - Sum[fb2[p/d, q/d], {d, Rest[Intersection@@(Divisors/@{p, q})]}]; (*A050186(p+q, p)*)
fO[n_] := Sum[fb[2n-q, q] - If[EvenQ@q, fb2[n-q/2, q/2] - If[OddQ@n, fb[n-q/2, q/2], 0], 0] / 2, {q, Select[Range[n+1, 2n-1], !Divisible[n-#, 3]&]}]; (*A011955*)
a[n_] := rh[n] - If[OddQ@n, fSO[(n-1)/2], fO[n/2]+fO[n/2-1]];
Table[a[n],{n,7,50}] (* Andrei Zabolotskii, May 30 2025 *)

apply( {A011956(n) = A371992(n) - if(n%2,A011954(n\2), A011955(n/2)+A011955(n/2-1))}, [7..20]) \\ M. F. Hasler, May 27 2025

def A011956(n): return A371992(n) - (A011954(n//2) if n&1 else A011955(n//2)+A011955(n//2-1))
# M. F. Hasler, May 27 2025

a(n) = A371992(n) - A011954((n-1)/2) - A011955(n/2) - A011955(n/2-1), where the terms with non-integer indices are set to 0. - Andrei Zabolotskii and M. F. Hasler, May 27 2025

Name and offset corrected by Andrei Zabolotskii, Feb 14 2024
Name changed by M. F. Hasler, May 26 2025
Terms a(17) onwards from Andrei Zabolotskii, May 30 2025

A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 2, 0, 3, 3, 0, 4, 4, 4, 0, 5, 10, 10, 5, 0, 6, 12, 18, 12, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 24, 56, 64, 56, 24, 8, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 0, 12, 60

Offset: 1

N. J. A. Sloane, Mar 24 2000

Triangle of number of primitive words over {0,1} of length n that contain k 1's, for n,k >= 1. - Benoit Cloitre, Jun 08 2004

			Triangle begins
  1;
  2,  0;
  3,  3,  0;
  4,  4,  4,  0;
  5, 10, 10,  5,  0;
  6, 12, 18, 12,  6,  0;
  ...

J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 29.

Index entries for triangles and arrays related to Pascal's triangle

Cf. A042979, A042980. T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

Same triangle as A050186 except this one does not include column 0.

T[n_, k_] := DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#] &]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)

T(n,k)=sumdiv(gcd(k,n),d,moebius(d)*binomial(n/d,k/d)) \\ Benoit Cloitre, Jun 08 2004

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A051199 T(2n+6,n), array T as in A050186; a count of aperiodic binary words.

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A050187 a(n) = n * floor((n-1)/2).

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A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

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A011956 Number of close-packings with layer-number 3n and space group R3m.

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A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).

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