A000046 -id:A000046 - cp's OEIS frontend

A054199 Binomial transform of A000046 (1, 1, 1, 1, 2, 3, 5, 8, 14, 21, 39, 62, 112, ...).

1, 2, 4, 8, 17, 39, 95, 240, 619, 1615, 4241, 11184, 29601, 78645, 209830, 562454, 1515325, 4104547, 11180365, 30627574, 84375837, 233729358, 650888094, 1821718080, 5122765105, 14468782826, 41031251163, 116789504201, 333544999893, 955496210873, 2744724190487

Offset: 0

N. J. A. Sloane, Apr 29 2000

nonn

Corrected Mar 05 2004. Thanks to Ralf Stephan for noticing this sequence was wrong.

A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 13, 11, 3, 1, 0, 8, 31, 33, 16, 3, 1, 0, 14, 80, 136, 85, 27, 4, 1, 0, 21, 201, 478, 434, 171, 37, 4, 1, 0, 39, 533, 1849, 2270, 1249, 338, 54, 5, 1, 0, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1

Offset: 1

Andrew Howroyd, Apr 09 2017

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

			Triangle starts:
  1
  0  1
  0  1   1
  0  2   2    1
  0  3   5    2    1
  0  5  13   11    3    1
  0  8  31   33   16    3   1
  0 14  80  136   85   27   4  1
  0 21 201  478  434  171  37  4 1
  0 39 533 1849 2270 1249 338 54 5 1
  ...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-6 are A063524, A056366, A056367, A056368, A056369, A056370.
Partial row sums include A000046, A056362, A056363, A056364, A056365.
Row sums are A276548.
Cf. A276550, A152175, A152176, A107424, A137651, A276544, A304972.

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=(R(n)+Ach(n))/2); Mat(vectorv(n,n,sumdiv(n, d, moebius(d)*M[n/d,])))}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n, k) = Sum_{d|n} mu(n/d) * A152176(d, k).

A179781 a(n) = AP(n) is the total number of aperiodic k-palindromes of n, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 12, 14, 27, 31, 54, 63, 119, 123, 240, 255, 490, 511, 990, 1015, 2015, 2047, 4020, 4092, 8127, 8176, 16254, 16383, 32607, 32767, 65280, 65503, 130815, 131061, 261576, 262143, 523775, 524223, 1047540, 1048575, 2096003, 2097151, 4192254

Offset: 1

John P. McSorley, Jul 26 2010

nonn

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of a smaller composition.
A k-palindrome of n is a k-composition of n which is a palindrome.
This sequence is AP(n), the total number of aperiodic k-palindromes of n, 1 <= k <= n.
For example AP(6)=5 because the number n=6
has 1 aperiodic 1-palindrome, namely 6 itself;
has 1 aperiodic 3-palindrome, namely 141;
has 2 aperiodic 4-palindromes, namely 2112 and 1221;
has 1 aperiodic 5-palindrome, namely 11211.
This gives a total of 1+1+2+1=5 aperiodic palindromes of 6.
Number of achiral set partitions of a primitive cycle of n elements having up to two different elements. - Robert A. Russell, Jun 19 2019

			For a(7)=7, the achiral set partitions are 0000001, 0000011, 0000101, 0000111, 0001001, 0010011, and 0010101. - _Robert A. Russell_, Jun 19 2019

John P. McSorley, Counting k-compositions of n with palindromic and related structures. Preprint, 2010.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Hunki Baek, Sejeong Bang, Dongseok Kim, Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014.

Row sums of A179519.

A000048 (oriented), A000046 (unoriented), A308706 (chiral), A016116 (not primitive). - Robert A. Russell, Jun 19 2019

a[n_] := DivisorSum[n, MoebiusMu[n/#] * 2^Floor[#/2]&];
Array[a, 44] (* Jean-François Alcover, Nov 04 2017 *)

a(n) = sumdiv(n, d, moebius(n/d) * 2^(d\2)); \\ Michel Marcus, Dec 09 2014

a(n) = Sum_{d | n} moebius(n/d)*2^(floor(d/2)) (see Baek et al. page 9). - Michel Marcus, Dec 09 2014
a(n) = 2*A000046(n) - A000048(n) = A000048(n) - 2*A308706(n) = A000046(n) - A308706(n). - Robert A. Russell, Jun 19 2019
A016116(n) =  Sum_{d|n} a(d). - Robert A. Russell, Jun 19 2019
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + 2*x^k)/(1 - x^(2*k)). - Andrew Howroyd, Sep 27 2019

More terms from Michel Marcus, Dec 09 2014

A193140 Number of isonemal satins of exact period n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 3, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 0, 3, 0, 1, 0, 3, 3, 0, 1, 3, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 0, 3, 3, 1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 0, 1, 7

Offset: 2

N. J. A. Sloane, Jul 16 2011

nonn

On page 153 of Grünbaum and Shephard (1980) is Table 3 which is a list of all the (n,s)-satins with n<=100. - Michael Somos, Dec 05 2014

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

B. Grünbaum and G. C. Shephard, Satins and twills: an introduction to the geometry of fabrics, Math. Mag., 53 (1980), 139-161. See Theorem 5, page 152.

Cf. A193138, A193139, A086669, A000046, A005441, A262589.

#A193138
U:=proc(n) local j,p3,i,t1,t2,al,even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
p3:=0;
for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:
if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(j-1)); fi;
end;
#A193139:
V:=proc(n) local j,i,t1,t2,al,even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
if (al <= 1) then RETURN(2^(j-1)-1); fi;
if (al = 2) then RETURN(2^j-1); fi;
if (al >= 3) then RETURN(2^(j+1)-1); fi;
end;
#A193140:
[seq(U(n)+V(n), n=3..120)];

a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[
  f[[1, 1]] == 2 && f[[1, 2]] > 1,
  Boole[f[[1, 2]] == 2],
  Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]
]] - 1;
Table[a[n], {n, 2, 100}]
(* Andrey Zabolotskiy, Mar 21 2021 *)

a(n) = A086669(n) - 1. - Andrey Zabolotskiy, Dec 25 2018

a(2) = 0 prepended and name edited by Andrey Zabolotskiy, Mar 21 2021

A371992 Number of different closest packings of equal spheres for rhombohedral crystals having repeat period n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 8, 15, 23, 41, 70, 126, 223, 406, 740, 1370, 2545, 4769, 8977, 16985, 32261, 61469, 117488, 225060, 432159, 831235, 1601796, 3090926, 5973198, 11556533, 22385600, 43405353, 84247085, 163661488, 318209920, 619181766, 1205733457, 2349558582, 4581555964, 8939468450, 17453081143, 34094082857

Offset: 1

R. J. Mathar, Apr 15 2024

nonn

J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 1.

Cf. A371991, A000046, A051168, A180424, A000740, A000048.

fa[p_,q_] := fa[p,q] = (p+q-1)!/(p!q!) - Sum[fa[p/d,q/d]/d, {d, Rest[Intersection@@(Divisors/@{p,q})]}]; (*A051168(p+q,p); Iglesias Eq. (1)*)
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})]}]; (*A180424(p+q,p); Eq. (2)*)
am[p_] := am[p] = 2^(p-1) - Sum[am[p/d], {d, Rest@Divisors@p}]; (*A000740; Eq. (6)*)
atf[p_] := atf[p] = 2^(p-1)/p - Sum[atf[p/d]/d, {d, Select[Rest@Divisors@p, OddQ]}];(*A000048; Eq. (9)*)
a[n_] := Sum[With[{p=n-q}, fa[p,q]+fb[p,q] + If[p==q, am[p]+atf[p]-fa[p,q]-fb[p,q], 0] / 2], {q, Select[Range[n/2], !Divisible[n-2#,3]& (*the opposite condition would give A371991*)]}] / 2; (* Eq. (5) *)
Table[a[n], {n, 2, 40}] (* Andrei Zabolotskii, May 30 2025 *)

apply( {A371992(n)=sum(q=1, n\2, if((n-2*q)%3, A051168(n,q)+A180424(n,q)))/2}, [1..40]) \\ M. F. Hasler, Jun 05 2025

a(n) + A371991(n) = A000046(n).

a(n+1)/a(n) = 2 - 2/n + o(1/n). - M. F. Hasler, Jun 09 2025

Offset changed to 1 and a(1) = 0 prefixed by M. F. Hasler, Jun 05 2025

A005441 Number of isonemal fabrics of period exactly n.

Original entry on oeis.org

1, 1, 4, 4, 9, 8, 55, 21, 104, 62, 429, 196, 800, 698, 4674, 2070, 7721, 7154

Offset: 2

Simon Plouffe

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-97 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. [gives different a(10) which is probably erroneous]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Branko Grünbaum and G. C. Shephard, A catalogue of isonemal fabrics, Discrete Geometry and Convexity, 440 (1985), 279-298.
B. Grünbaum and G. C. Shephard, An extension to the catalogue of isonemal fabrics, Discrete Mathematics, 60, (1986), 155-192.

Cf. A000046, A193140, A262589.

a(n) = A000046(n) + A193140(n) + A262589(n).

a(14)-a(19) from second Grünbaum and Shephard paper added by Sean A. Irvine, Jun 26 2016

a(10) corrected by Andrey Zabolotskiy, Apr 14 2023

A056367 Number of primitive (period n) bracelet structures using exactly three different colored beads.

Original entry on oeis.org

0, 0, 1, 2, 5, 13, 31, 80, 201, 533, 1401, 3822, 10395, 28859, 80201, 225286, 634265, 1796433, 5100325, 14534758, 41513402, 118879249, 341094843, 980661980, 2824223490, 8146897815, 23535345170

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 3 of A276543.

Cf. A056304.

A056362(n)-A000046(n).

A371991 Number of different closest packings of equal spheres for hexagonal crystals having repeat period n.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 6, 6, 16, 21, 42, 63, 129, 201, 404, 685, 1340, 2385, 4625, 8487, 16409, 30735, 59282, 112530, 217182, 415605, 803076, 1545463, 2990945, 5778267, 11201472, 21702645, 42140890, 81830744, 159139428, 309590883, 602935713, 1174779207, 2290915478, 4469734225, 8726815041, 17047041429, 33319598126

Offset: 1

R. J. Mathar, Apr 15 2024

nonn

J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 1.

Cf. A000046, A371992.

apply( {A371991(n)=A000046(n)-A371992(n)}, [1..40]) \\ M. F. Hasler, Jun 09 2025

a(n) + A371992(n) = A000046(n).

a(n+1)/a(n) = 2 - 2/n + O(1/n^2). - M. F. Hasler, Jun 09 2025

Offset changed to 1 and a(1) = 1 prefixed by M. F. Hasler, Jun 09 2025

A262589 Number of other isonemal fabrics of exact period n.

Original entry on oeis.org

0, 0, 2, 0, 4, 0, 40, 0, 64, 0, 316, 6, 448, 90, 3529, 14, 3836, 0

Offset: 2

N. J. A. Sloane, Oct 21 2015

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-97 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. [gives different a(10) which is probably erroneous]

Branko Grünbaum and G. C. Shephard, A catalogue of isonemal fabrics, Discrete Geometry and Convexity, 440 (1985), 279-298.
B. Grünbaum and G. C. Shephard, An extension to the catalogue of isonemal fabrics, Discrete Mathematics, 60, (1986), 155-192.

Cf. A000046, A005441, A193140.

A005441(n) = A000046(n) + A193140(n) + a(n).

a(14)-a(19) using formula added, a(10) corrected by Andrey Zabolotskiy, Apr 14 2023

A308706 Number of chiral pairs of set partitions of a primitive cycle of n elements having exactly two different elements.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 233, 484, 904, 1800, 3395, 6643, 12612, 24457, 46655, 90157, 172750, 333498, 641214, 1238664, 2388618, 4620006, 8931536, 17302033, 33521792, 65042495, 126257160, 245361171, 477087772, 928510506, 1808145395, 3523813566

Offset: 0

Robert A. Russell, Jun 18 2019

nonn

			For a(7)=1, the chiral pair is 0001011-0001101.  For a(8)=2, the chiral pairs are 00001011-00001101 and 00010011-00011001.

Cf. A000048 (oriented), A000046 (unoriented), A179781 (achiral), A059053 (not primitive).

Join[{0}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) - DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]]

a(n) = if (n, (sumdiv(n, d, if (d%2, moebius(d)*2^(n/d)))/(2*n) - sumdiv(n, d, moebius(n/d)*2^(d\2)))/2, 0); \\ Michel Marcus, Jun 27 2019; corrected Jun 12 2022

a(n) = ((1/(2n)) * Sum_{odd d|n} mu(d)*2^(n/d) - Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2, where mu is the Möbius function at A008683.
a(n) = A000048(n) - A000046(n) = (A000048(n) - A179781(n))/2 = A000046(n) - A179781(n).
A059053(n) = Sum_{d|n} a(d).

cp's OEIS Frontend

A054199 Binomial transform of A000046 (1, 1, 1, 1, 2, 3, 5, 8, 14, 21, 39, 62, 112, ...).

Views

Author

Keywords

Extensions

A276543 Triangle read by rows: T(n,k) = number of primitive (period n) n-bead bracelet structures using exactly k different colored beads.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

A179781 a(n) = AP(n) is the total number of aperiodic k-palindromes of n, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A193140 Number of isonemal satins of exact period n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A371992 Number of different closest packings of equal spheres for rhombohedral crystals having repeat period n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A005441 Number of isonemal fabrics of period exactly n.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A056367 Number of primitive (period n) bracelet structures using exactly three different colored beads.

Views

Author

Keywords

Comments

References

Crossrefs

Formula

A371991 Number of different closest packings of equal spheres for hexagonal crystals having repeat period n.

Views

Author

Keywords

Links

Crossrefs

Programs