cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A331140 Number of different coincidence site lattices of index n in lattice Z^4.

Original entry on oeis.org

1, 1, 16, 0, 36, 16, 64, 0, 152, 36, 144, 0, 196, 64, 576, 0, 324, 152, 400, 0, 1024, 144, 576, 0, 912, 196, 1392, 0, 900, 576, 1024, 0, 2304, 324, 2304, 0, 1444, 400, 3136, 0, 1764, 1024, 1936, 0, 5472, 576, 2304, 0, 3152, 912, 5184, 0, 2916, 1392, 5184, 0
Offset: 1

Views

Author

N. J. A. Sloane, Jan 12 2020

Keywords

Links

Crossrefs

Cf. A031361, A331139, A331141, A331142.

Formula

a(2k-1) = a(4k-2) = A331139(k), a(4k) = 0. - Andrey Zabolotskiy, Jan 29 2020

Extensions

Name corrected, a(2) corrected, and terms a(18) and beyond added by Andrey Zabolotskiy, Jan 29 2020

A031360 Number of symmetrically inequivalent coincidence rotations of index 2n-1 in lattice D_4.

Original entry on oeis.org

1, 16, 36, 64, 168, 144, 196, 576, 324, 400, 1024, 576, 960, 1584, 900, 1024, 2304, 2304, 1444, 3136, 1764, 1936, 6048, 2304, 3248, 5184, 2916, 5184, 6400, 3600, 3844, 10752, 7056, 4624, 9216, 5184, 5476, 15360, 9216, 6400, 14472, 7056, 11664, 14400
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The aerated sequence 1, 0, 16, 0, 36, 0, 64, 0, 168,.. is multiplicative. - R. J. Mathar, Sep 30 2011
Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331139(n). - Andrey Zabolotskiy, Jan 29 2020

Links

Crossrefs

Cf. A031361, A331139, A331141.

Programs

  • Maple
    read("transforms") : maxOrd := 120 :
ZetaNum := proc(p,nmax,f) local n ; L := [1,seq(0,n=2..p-1),f,seq(0,n=p+1..nmax)] ; end proc:
Zeta := proc(p,nmax,f) local L,e; L := [1,seq(0,n=2..nmax)] ; for e from 1 do if p^e > nmax then break; else L := subsop(p^e=f^e,L) ; end if; end do: L ; end proc:
Zetap := [1,seq(0,n=2..maxOrd)] : for e from 3 to maxOrd do if isprime(e) then ZetaNum(e,maxOrd,1) ; Zetap := DIRICHLET(Zetap,%) ; ZetaNum(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e^2) ; Zetap := DIRICHLET(Zetap,%) ; end if; end do:
seq( Zetap[2*e+1],e=0..nops(Zetap)/2-1) ; # R. J. Mathar, Jul 16 2010
  • Mathematica
    a[1]=1; a[n_ /; n >= 2 && IntegerQ[Log[2, n]]] = 0; a[p_?PrimeQ] := (p+1)^2; a[n_] := a[n] = If[Length[f = FactorInteger[n]] == 1, {p, r} = First[f]; (p+1)/(p-1)*p^(r-1)*(p^(r+1)+p^(r-1)-2), Times @@ (a /@ Power @@@ f)]; Table[a[n], {n, 1, 87, 2}] (* Jean-François Alcover, Apr 17 2013 *)
  • PARI
    a(n,f=factor(2*n-1))=prod(i=1,#f~, my(p=f[i,1],e=f[i,2]); (p+1)/(p-1)*p^(e-1)*(p^(e+1)+p^(e-1)-2)) \\ Charles R Greathouse IV, Aug 26 2017

Formula

Dirichlet series for the aerated 1, 0, 16, 0, 36, 0, 64 ..: Product_{odd primes p} (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))).
Dirichlet g.f. for the aerated sequence is Zeta(s) *Zeta(s-1)^2 *Zeta(s-2) / (Zeta(2*s) * Zeta(2*s-2)) *(1-2^(1-s)) *(1-2^(2-s))/ ( (1+2^(-s))*(1+2^(1-s)) ). - R. J. Mathar, Sep 30 2011
Sum_{k=1..n} a(k) ~ 1680 * Zeta(3) * n^3 / Pi^6. - Vaclav Kotesovec, Feb 07 2019

Extensions

More terms from R. J. Mathar, Jul 16 2010
Name corrected by Andrey Zabolotskiy, Jan 29 2020

A031361 Number of symmetrically inequivalent coincidence rotations of index n in lattice Z^4.

Original entry on oeis.org

1, 2, 16, 0, 36, 32, 64, 0, 168, 72, 144, 0, 196, 128, 576, 0, 324, 336, 400, 0, 1024, 288, 576, 0, 960, 392, 1584, 0, 900, 1152, 1024, 0, 2304, 648, 2304, 0, 1444, 800, 3136, 0, 1764, 2048, 1936, 0, 6048, 1152, 2304, 0, 3248, 1920, 5184, 0, 2916, 3168, 5184, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Dirichlet product of 1 + 2/2^s with Sum_{n>=1} A031360(n)/(2n-1)^s. - R. J. Mathar, Jul 16 2010
Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331140(n). - Andrey Zabolotskiy, Jan 29 2020

Links

Crossrefs

Cf. A031360, A331140, A331141, A350872.

Programs

  • Maple
    read("transforms") : maxOrd := 120 :
ZetaNum := proc(p,nmax,f) local n ; L := [1,seq(0,n=2..p-1),f,seq(0,n=p+1..nmax)] ; end proc:
Zeta := proc(p,nmax,f) local L,e; L := [1,seq(0,n=2..nmax)] ; for e from 1 do if p^e > nmax then break; else L := subsop(p^e=f^e,L) ; end if; end do: L ; end proc:
Zetap := ZetaNum(2,maxOrd,2): for e from 3 to maxOrd do if isprime(e) then ZetaNum(e,maxOrd,1) ; Zetap := DIRICHLET(Zetap,%) ; ZetaNum(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e^2) ; Zetap := DIRICHLET(Zetap,%) ; end if; end do: Zetap ;
# R. J. Mathar, Jul 16 2010
  • Mathematica
    maxOrd = 120;
did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
DIRICHLET[a_List, b_List] := Module[{c = {}, i, s, d}, For[i = 1, i <= Min[Length[a], Length[b]], i++, s = 0; For[d = 1, d <= i, d++, If[did[i, d] == 1, s = s + a[[d]] b[[i/d]]]]; c = Append[c, s]]; c];
zetaNum[p_, nmax_, f_] := Module[{n}, L = Join[{1}, Table[0, {n, 2, p-1}], {f}, Table[0, {n, p+1, nmax}]]];
zeta[p_, nmax_, f_] := Module[{L, e}, L = Join[{1}, Table[0, {n, 2, nmax}] ]; For[e = 1, True, e++, If[p^e > nmax, Break[], L = ReplacePart[L, p^e -> f^e]]]; L];
zetap = zetaNum[2, maxOrd, 2];
For[e = 3, e <= maxOrd, e++, If[PrimeQ[e], ze = zetaNum[e, maxOrd, 1];
  zetap = DIRICHLET[zetap, ze]; ze = zetaNum[e, maxOrd, e];
  zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e];
  zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e^2];
  zetap = DIRICHLET[zetap, ze]]];
zetap (* Jean-François Alcover, Apr 20 2020, after R. J. Mathar *)

Formula

Dirichlet series: (1+2^(1-s))* Product (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))); p != 2.
From Vaclav Kotesovec, Jul 18 2025: (Start)
Dirichlet g.f.: (2^s-4) * (2^s-2) * zeta(s-2) * zeta(s-1)^2 * zeta(s) / (2^s * (2^s+1) * zeta(2*s) * zeta(2*s-2)).
Sum_{k=1..n} a(k) ~ 525 * zeta(3) * n^3 / (2*Pi^6). (End)

Extensions

More terms from R. J. Mathar, Jul 16 2010
Typo in formula (exclamation mark for 1) corrected by R. J. Mathar, Jul 23 2010
Name corrected by Andrey Zabolotskiy, Jan 29 2020

A331139 Number of different coincidence site lattices of index 2n-1 in lattice D_4.

Original entry on oeis.org

1, 16, 36, 64, 152, 144, 196, 576, 324, 400, 1024, 576, 912, 1392, 900, 1024, 2304, 2304, 1444, 3136, 1764, 1936, 5472, 2304, 3152, 5184, 2916, 5184, 6400, 3600, 3844, 9728, 7056, 4624, 9216, 5184, 5476, 14592, 9216, 6400, 12552, 7056, 11664, 14400, 8100
Offset: 1

Views

Author

N. J. A. Sloane, Jan 12 2020

Keywords

Comments

Differs from A031360 if n is divisible by the square of an odd prime.

Links

Crossrefs

Cf. A031360, A331140, A331141, A331142.

Extensions

Terms a(10) and beyond added and name corrected by Andrey Zabolotskiy, Jan 29 2020

A331142 Number of different coincidence site lattices of index n in lattice A_4.

Original entry on oeis.org

1, 5, 10, 20, 6, 50, 50, 80, 90, 30, 144, 200, 170, 250, 60, 320, 290, 450, 400, 120, 500, 720, 530, 800, 150, 850, 810, 1000, 900, 300, 1024, 1280, 1440, 1450, 300, 1800, 1370, 2000, 1700, 480, 1764, 2500, 1850, 2880, 540, 2650, 2210, 3200, 2450, 750, 2900
Offset: 1

Views

Author

N. J. A. Sloane, Jan 12 2020

Keywords

Links

Crossrefs

Cf. A031360, A031361, A331139, A331140, A331141.

Extensions

Name corrected and terms a(12) and beyond added by Andrey Zabolotskiy, Jan 29 2020
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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