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 1-2 of 2 results.

A096023 Numbers congruent to {63, 123, 183, 243, 303, 363} mod 420.

Original entry on oeis.org

63, 123, 183, 243, 303, 363, 483, 543, 603, 663, 723, 783, 903, 963, 1023, 1083, 1143, 1203, 1323, 1383, 1443, 1503, 1563, 1623, 1743, 1803, 1863, 1923, 1983, 2043, 2163, 2223, 2283, 2343, 2403, 2463, 2583, 2643, 2703, 2763, 2823, 2883, 3003, 3063, 3123
Offset: 1

Views

Author

Klaus Brockhaus, Jun 15 2004

Keywords

Comments

Numbers n such that (n+j) mod (2+j) = 1 for j from 0 to 4 and (n+5) mod 7 <> 1.
Numbers n such that n mod 60 = 3 and n mod 420 <> 3.

Examples

			63 mod 2 = 64 mod 3 = 65 mod 4 = 66 mod 5 = 67 mod 6 = 1 and 68 mod 7 = 5, hence 63 is in the sequence.

Links

Crossrefs

Cf. A007310, A017629, A096022, A096024, A096025, A096026, A096027.
Cf. A047391 (see MAGMA code). - Bruno Berselli, Mar 25 2011

Programs

  • Magma
    [ n : n in [1..3500] | n mod 420 in [63, 123, 183, 243, 303, 363] ] // Vincenzo Librandi, Mar 24 2011
  • Magma
    /* Alternatively:*/ &cat[ [ 60*n+3, 60*n+63 ]: n in [1..52] | n mod 7 in [1,3,5] ]; // Bruno Berselli, Mar 25 2011
  • Maple
    A096023:=n->420*floor(n/6)+[63, 123, 183, 243, 303, 363][(n mod 6)+1]: seq(A096023(n), n=0..80); # Wesley Ivan Hurt, Jul 22 2016
  • Mathematica
    Select[Range[0, 5*10^3], MemberQ[{63, 123, 183, 243, 303, 363}, Mod[#, 420]] &] (* Wesley Ivan Hurt, Jul 22 2016 *)
  • PARI
    {k=5;m=3150;for(n=1,m,j=0;b=1;while(b&&j

Formula

G.f.: 3*x*(21+20*x+20*x^2+20*x^3+20*x^4+20*x^5+19*x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jul 22 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 420 for n>6.
a(n) = (210*n - 96 - 30*cos(n*Pi/3) - 30*cos(2*n*Pi/3) - 15*cos(n*Pi) + 30*sqrt(3)*sin(n*Pi/3) + 10*sqrt(3)*sin(2*n*Pi/3))/3.
a(6k) = 420k-57, a(6k-1) = 420k-117, a(6k-2) = 420k-177, a(6k-3) = 420k-237, a(6k-4) = 420k-297, a(6k-5) = 420k-357. (End)

Extensions

New definition from Ralf Stephan, Dec 01 2004

A047390 Numbers that are congruent to {0, 3, 5} mod 7.

Original entry on oeis.org

0, 3, 5, 7, 10, 12, 14, 17, 19, 21, 24, 26, 28, 31, 33, 35, 38, 40, 42, 45, 47, 49, 52, 54, 56, 59, 61, 63, 66, 68, 70, 73, 75, 77, 80, 82, 84, 87, 89, 91, 94, 96, 98, 101, 103, 105, 108, 110, 112, 115, 117, 119, 122, 124, 126, 129, 131, 133, 136, 138, 140, 143
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also numbers k such that k*(k+2)*(k+4) is divisible by 7. - Bruno Berselli, Dec 28 2017

Links

Crossrefs

Cf. A011655, A047391, A049347, A079978.

Programs

  • Magma
    [n: n in [0..122] | n mod 7 in [0, 3, 5]];  // Bruno Berselli, Mar 29 2011
  • Maple
    seq(2*n+floor(n/3)+(n^2 mod 3), n=0..52); # Gary Detlefs, Mar 19 2010
  • Mathematica
    Select[Range[0,150], MemberQ[{0,3,5}, Mod[#,7]]&] (* Harvey P. Dale, Dec 07 2011 *)
CoefficientList[Series[x (3 + 2 x + 2 x^2)/((1 - x)^2 (1 + x + x^2)), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 02 2014 *)
  • PARI
    is(n)=!!setsearch([0,3,5],n%7) \\ Charles R Greathouse IV, Nov 09 2014
  • PARI
    a(n)=(7*n-5)\3 \\ Charles R Greathouse IV, Nov 09 2014
  • Python
    import math
a = lambda n: 2*(n-1)+math.ceil((n-1)/3.0)
for n in range(1,101): print(a(n), end = ", ") # Karl V. Keller, Jr., Nov 01 2014

Formula

a(n) = 2*n + floor(n/3) + (n^2 mod 3), with offset 0, a(0)=0. - Gary Detlefs, Mar 19 2010
From Bruno Berselli, Mar 29 2011: (Start)
G.f.: x^2*(3 + 2*x + 2*x^2)/((1 - x)^2*(1 + x + x^2)).
a(n) = (1/3)*(7*n - 6 - A049347(n-1)) = A047391(n) - A079978(n-1). (End)
a(n) = n + ceiling(4*(n-1)/3) - 1. - Arkadiusz Wesolowski, Sep 18 2012
a(n) = 2*(n-1) + ceiling((n-1)/3). - Karl V. Keller, Jr., Nov 01 2014
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 7*n/3 - 2 - 2*sin(2*n*Pi/3)/(3*sqrt(3)).
a(3*k) = 7*k-2, a(3*k-1) = 7*k-4, a(3*k-2) = 7*k-7. (End)
Showing 1-2 of 2 results.

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

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