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 1-7 of 7 results.

A226782 If n == 0 (mod 2) then a(n) = 0, otherwise a(n) = 4^(-1) in Z/nZ*.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 2, 0, 7, 0, 3, 0, 10, 0, 4, 0, 13, 0, 5, 0, 16, 0, 6, 0, 19, 0, 7, 0, 22, 0, 8, 0, 25, 0, 9, 0, 28, 0, 10, 0, 31, 0, 11, 0, 34, 0, 12, 0, 37, 0, 13, 0, 40, 0, 14, 0, 43, 0, 15, 0, 46, 0, 16, 0, 49, 0, 17
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226783, A226784, A226785, A226786, A226787.

Programs

  • Maple
    A226782 := proc(n)
    local x ,a,m;
    a := 4 ;
    m := 2 ;
    if n mod m = 0 or n = 1 then
        0;
    else
        msolve(x*a=1,n) ;
        op(%) ;
        op(2,%) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013
  • Mathematica
    Inv[a_, mod_] := Which[mod == 1, 0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]]; Table[Inv[4, n], {n, 1, 122}]
(* Second program: *)
Table[If[EvenQ[n], 0, ModularInverse[4, n], 0], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(n%2,lift(Mod(1, n)/4),0) \\ Charles R Greathouse IV, Jun 18 2013

Formula

From Colin Barker, Jun 20 2013: (Start)
G.f.: -x^3*(x^6 - 4*x^2 - 1) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ).
a(2n+1) = A225126(n+1). (End)

A226787 If n=0 (mod 3) then a(n)=0, otherwise a(n)=9^(-1) in Z/nZ*.

Original entry on oeis.org

0, 1, 0, 1, 4, 0, 4, 1, 0, 9, 5, 0, 3, 11, 0, 9, 2, 0, 17, 9, 0, 5, 18, 0, 14, 3, 0, 25, 13, 0, 7, 25, 0, 19, 4, 0, 33, 17, 0, 9, 32, 0, 24, 5, 0, 41, 21, 0, 11, 39, 0, 29, 6, 0, 49, 25, 0, 13, 46, 0, 34, 7, 0, 57, 29
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226782-A226786.

Programs

  • Mathematica
    Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[9, n], {n, 1, 122}]
(* Second program: *)
Table[If[Mod[n, 3] == 0, 0, ModularInverse[9, n], 0], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(n%3,lift(Mod(1,n)/9),0) \\ Charles R Greathouse IV, Jun 18 2013

Formula

Empirical g.f.: -x^2*(x^17-x^14-3*x^12-x^11-3*x^9-9*x^8-x^6-4*x^5-4*x^3-x^2-1) / (x^18 -2*x^9 +1). - Colin Barker, Jun 20 2013

A226786 If n=0 (mod 2) then a(n)=0, otherwise a(n)=8^(-1) in Z/nZ*.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 1, 0, 8, 0, 7, 0, 5, 0, 2, 0, 15, 0, 12, 0, 8, 0, 3, 0, 22, 0, 17, 0, 11, 0, 4, 0, 29, 0, 22, 0, 14, 0, 5, 0, 36, 0, 27, 0, 17, 0, 6, 0, 43, 0, 32, 0, 20, 0, 7, 0, 50, 0, 37, 0, 23, 0, 8, 0, 57, 0, 42
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226782-A226787.

Programs

  • Mathematica
    Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[8, n], {n, 1, 122}]
(* Second program: *)
Table[If[EvenQ[n], 0, ModularInverse[8, n], 0], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(n%2,lift(Mod(1,n)/8),0) \\ Charles R Greathouse IV, Jun 18 2013

Formula

G.f.: -x^3*(x^14-x^10-3*x^8-8*x^6-x^4-2*x^2-2)/(x^16-2*x^8+1). - Colin Barker, Jun 20 2013

A226783 If n=0 (mod 5) then a(n)=0, otherwise a(n)=5^(-1) in Z/nZ*.

Original entry on oeis.org

0, 1, 2, 1, 0, 5, 3, 5, 2, 0, 9, 5, 8, 3, 0, 13, 7, 11, 4, 0, 17, 9, 14, 5, 0, 21, 11, 17, 6, 0, 25, 13, 20, 7, 0, 29, 15, 23, 8, 0, 33, 17, 26, 9, 0, 37, 19, 29, 10, 0, 41, 21, 32, 11, 0, 45, 23, 35, 12, 0, 49, 25, 38
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226782-A226787.

Programs

  • Maple
    A226783 := proc(n)
    local x ;
    a := 5 ;
    m := 5 ;
    if n mod m = 0 or n = 1 then
        0;
    else
        msolve(x*a=1,n) ;
        op(%) ;
        op(2,%) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013
  • Mathematica
    Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[5, n], {n, 1, 122}]
CoefficientList[Series[-x^2(x^9-x^6-x^5-5x^4-x^2-2x-1)/((x-1)^2 (x^4+ x^3+ x^2+ x+ 1)^2),{x,0,120}],x] (* Harvey P. Dale, Oct 08 2016 *)
Table[If[Mod[n, 5]==0, 0, ModularInverse[5, n]], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(n%5,lift(Mod(1, n)/5),0) \\ Charles R Greathouse IV, Jun 18 2013

Formula

G.f.: -x^2*(x^9-x^6-x^5-5*x^4-x^2-2*x-1) / ( (x-1)^2*(x^4+x^3+x^2+x+1)^2 ). - Colin Barker, Jun 20 2013
a(5n+1) = A016813(n), n>0. a(5n+2)= A005408(n), n>0. a(5n+3) = A016789(n). a(5n+4)=n+1. - R. J. Mathar, Jun 28 2013
a(n) = Sum_{k=1..n} k*(floor((5k-1)/n)-floor((5k-2)/n)), n>1. - Anthony Browne, Jun 19 2016

A226785 If n=0 (mod 7) then a(n)=0, otherwise a(n)=7^(-1) in Z/nZ*.

Original entry on oeis.org

0, 1, 1, 3, 3, 1, 0, 7, 4, 3, 8, 7, 2, 0, 13, 7, 5, 13, 11, 3, 0, 19, 10, 7, 18, 15, 4, 0, 25, 13, 9, 23, 19, 5, 0, 31, 16, 11, 28, 23, 6, 0, 37, 19, 13, 33, 27, 7, 0, 43, 22, 15, 38, 31, 8, 0, 49, 25, 17, 43, 35, 9, 0
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226782-A226787.

Programs

  • Maple
    A226785 := proc(n)
    local x,a,m ;
    a := 7 ;
    m := 7 ;
    if igcd(n,m) > 1 or n =1 then
        0;
    else
        msolve(x*a=1,n) ;
        op(%) ;
        op(2,%) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013
  • Mathematica
    Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[7, n],{n, 1, 122}]
Join[{0},LinearRecurrence[{0,0,0,0,0,0,2,0,0,0,0,0,0,-1},{1,1,3,3,1,0,7,4,3,8,7,2,0,13},70]] (* Harvey P. Dale, Nov 15 2014 *)
Table[If[Mod[n, 7]==0, 0, ModularInverse[7, n]], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(n%7,lift(Mod(1,n)/7),0) \\ Charles R Greathouse IV, Jun 18 2013

Formula

G.f.: -x^2*(x^13 -x^10 -2*x^9 -x^8 -2*x^7 -7*x^6 -x^4 -3*x^3 -3*x^2 -x -1) / (x^14 -2*x^7 +1). a(n) = 2*a(n-7)-a(n-14). - Colin Barker, Jun 20 2013
a(7n+1) = 6*n+1, n>0. a(7n+2)=A016777(n). a(7n+3) = A005408(n). a(7n+4) = A016885(n). a(7n+5)= A004767(n). a(7n+6)= n+1. - R. J. Mathar, Jun 28 2013

A226784 If gcd(n,6) != 1 then a(n)=0, otherwise a(n)=6^(-1) in Z/nZ*.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 6, 0, 0, 0, 2, 0, 11, 0, 0, 0, 3, 0, 16, 0, 0, 0, 4, 0, 21, 0, 0, 0, 5, 0, 26, 0, 0, 0, 6, 0, 31, 0, 0, 0, 7, 0, 36, 0, 0, 0, 8, 0, 41, 0, 0, 0, 9, 0, 46, 0, 0, 0, 10, 0, 51, 0, 0, 0, 11, 0, 56, 0, 0
Offset: 1

Views

Author

José María Grau Ribas, Jun 18 2013

Keywords

Links

Crossrefs

Cf. A092092, A226782, A226783, A226785, A226786, A226787.

Programs

  • Maple
    A226784 := proc(n)
    local x,a,m ;
    a := 6 ;
    m := 6 ;
    if igcd(n,m) > 1 or n =1 then
        0;
    else
        msolve(x*a=1,n) ;
        op(%) ;
        op(2,%) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013
  • Mathematica
    Inv[a_, mod_] := Which[mod == 1,0, GCD[a, mod] > 1, 0, True, Last@Reduce[a*x == 1, x, Modulus -> mod]];Table[Inv[6, n], {n, 1, 122}]
(* Second program: *)
Table[If[GCD[n, 6] != 1, 0, ModularInverse[6, n], 0], {n, 1, 100}] (* Jean-François Alcover, Mar 14 2023 *)
  • PARI
    a(n)=if(gcd(n,6)>1,0,lift(Mod(1,n)/6)) \\ Charles R Greathouse IV, Jun 18 2013

Formula

G.f.: -x^5*(x^8-6*x^2-1) / (x^12-2*x^6+1). a(n) = 2*a(n-6)-a(n-12). - Colin Barker, Jun 20 2013
a(6n+1) = A016861(n), n>0. a(6n+2) = a(6n+3) = a(6n+4) = 0. a(6n+5)=n+1. - R. J. Mathar, Jun 28 2013
a(n) = Sum_{k=1..n} k*(floor((6k-1)/n)-floor((6k-2)/n)), n>1. - Anthony Browne, Jun 19 2016

Extensions

Name corrected by David A. Corneth, Jun 20 2016

A092093 Back and Forth Summant S(n, 5): a(n) = sum{i = 0..floor(2n/5)} n-5i.

Original entry on oeis.org

1, 2, 1, 3, 0, 3, 6, 2, 6, 0, 5, 10, 3, 9, 0, 7, 14, 4, 12, 0, 9, 18, 5, 15, 0, 11, 22, 6, 18, 0, 13, 26, 7, 21, 0, 15, 30, 8, 24, 0, 17, 34, 9, 27, 0, 19, 38, 10, 30, 0, 21, 42, 11, 33, 0, 23, 46, 12, 36, 0, 25, 50, 13, 39, 0, 27, 54, 14, 42, 0, 29, 58, 15, 45, 0, 31, 62, 16, 48, 0, 33
Offset: 1

Views

Author

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

Keywords

References

  • J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004.
  • F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
  • F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.

Links

Crossrefs

Cf. A092096, A092399.
Other values of k: A000004 (k = 1, 2), A092092 (k = 3), A027656 (k = 4).

Programs

  • PARI
    S(n, k=5) = local(s, x); s = n; x = n - k; while (x >= -n, s = s + x; x = x - k); s;

Formula

a(5n) = 0; a(5n+1) = 2n+1; a(5n+2) = 4n+2; a(5n+3) = n+1; a(5n+4) = 3n+3.
G.f.: x*(2*x^6+x^5+3*x^3+x^2+2*x+1) / ((x-1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, Jul 28 2013

Extensions

Edited and extended by David Wasserman, Dec 19 2005
Showing 1-7 of 7 results.

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