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 1-4 of 4 results.

A047336 Numbers that are congruent to {1, 6} mod 7.

Original entry on oeis.org

1, 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 62, 64, 69, 71, 76, 78, 83, 85, 90, 92, 97, 99, 104, 106, 111, 113, 118, 120, 125, 127, 132, 134, 139, 141, 146, 148, 153, 155, 160, 162, 167, 169, 174, 176, 181, 183, 188, 190, 195, 197, 202, 204, 209
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 7). - Bruno Berselli, Nov 17 2010

Links

Crossrefs

Cf. A007310, A019674, A047522, A045472 (primes), A195041 (partial sums), A005408, A047209, A056020, A090771, A091998, A113801, A160389, A175885, A175886, A175887, A178818, A371858.

Programs

  • Haskell
    a047336 n = a047336_list !! (n-1)
a047336_list = 1 : 6 : map (+ 7) a047336_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [n: n in [1..210]| n mod 7 in {1,6}]; // Bruno Berselli, Feb 22 2011
  • Mathematica
    Rest[Flatten[Table[{7i-1,7i+1},{i,0,40}]]] (* Harvey P. Dale, Nov 20 2010 *)
  • PARI
    a(n)=n\2*7-(-1)^n \\ Charles R Greathouse IV, May 02 2016

Formula

a(1) = 1; a(n) = 7(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 (corrected by Jon E. Schoenfield, Dec 22 2008)
a(n) = (7/2)*(n-(1-(-1)^n)/2) - (-1)^n. - Rolf Pleisch, Nov 02 2010
From Bruno Berselli, Nov 17 2010: (Start)
G.f.: x*(1+5*x+x^2)/((1+x)*(1-x)^2).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = a(n-2)+7.
a(n) = 7*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 7*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/7)*cot(Pi/7) = A019674 * A178818. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 01 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/7) (A160389).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/7) * cosec(Pi/7) (A371858). (End)

Extensions

More terms from Jon E. Schoenfield, Jan 18 2009

A113801 Numbers that are congruent to {1, 13} mod 14.

Original entry on oeis.org

1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379
Offset: 1

Views

Author

Giovanni Teofilatto, Jan 22 2006

Keywords

Comments

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010
More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

Links

Crossrefs

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

Programs

  • Haskell
    a113801 n = a113801_list !! (n-1)
a113801_list = 1 : 13 : map (+ 14) a113801_list
-- Reinhard Zumkeller, Jan 07 2012
  • Mathematica
    LinearRecurrence[{1,1,-1},{1,13,15},60] (* or *) Select[Range[500], MemberQ[{1,13},Mod[#,14]]&] (* Harvey P. Dale, May 11 2011 *)
  • PARI
    a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

Formula

a(n) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010
From Bruno Berselli, Oct 26 2010: (Start)
a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.
G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2)+14 for n>2.
a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)
a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 5*exp(-x))/2. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/14).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/14)*cosec(Pi/14). (End)

Extensions

Corrected and extended by Giovanni Teofilatto, Nov 14 2008
Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

A045463 Primes congruent to {0, 1, 6} mod 7.

Original entry on oeis.org

7, 13, 29, 41, 43, 71, 83, 97, 113, 127, 139, 167, 181, 197, 211, 223, 239, 251, 281, 293, 307, 337, 349, 379, 419, 421, 433, 449, 461, 463, 491, 503, 547, 587, 601, 617, 631, 643, 659, 673, 701, 727, 743, 757
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The prime factors of x^3 - x^2 - 2x + 1 come exclusively from this sequence. - Charles R Greathouse IV, Mar 18 2022

Links

Crossrefs

Cf. A000040.

Programs

  • Magma
    [ p: p in PrimesUpTo(900) | p mod 7 in {0, 1, 6} ]; // Vincenzo Librandi, Aug 13 2012
  • Mathematica
    Select[Prime[Range[200]],MemberQ[{0,1,6},Mod[#,7]]&] (* Vincenzo Librandi, Aug 13 2012 *)
  • PARI
    select(p->abs(centerlift(Mod(p,7)))<2, primes(100)) \\ Charles R Greathouse IV, Mar 17 2022

Formula

A045472 UNION {7}. - R. J. Mathar, Oct 18 2008
a(n) ~ 3n log n. - Charles R Greathouse IV, Mar 17 2022

A284786 Pisano period of sequence A006054 modulo n.

Original entry on oeis.org

1, 7, 26, 14, 62, 182, 42, 28, 78, 434, 266, 182, 12, 42, 806, 56, 614, 546, 254, 434, 546, 266, 1106, 364, 310, 84, 234, 42, 28, 5642, 1986, 112, 3458, 4298, 1302, 546, 2814, 1778, 156, 868, 40, 546, 42, 266, 2418, 1106, 4514, 728, 294, 2170, 7982, 84, 5726, 1638, 8246, 84, 3302, 28, 7082, 5642, 582
Offset: 1

Views

Author

Patrick D McLean, Apr 02 2017

Keywords

Links

Crossrefs

Cf. A001175 Pisano periods of Fibonacci numbers mod n.
Cf. A045472.

Programs

  • Maple
    f:= proc(n) option remember; local F, t, k, a;
F:= ifactors(n)[2];
if nops(F) > 1 then
  return(ilcm(seq(procname(t[1]^t[2]),t=F)))
fi;
a:= [0,0,1];
for k from 1 do
  a:= [a[2],a[3],2*a[3]+a[2]-a[1] mod n];
  if  a = [0,0,1] then return k fi;
od:
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jun 14 2017
  • Mathematica
    Table[FindTransientRepeat[
    Mod[LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 100000], n], 2] //
   Last // Length, {n, 1, 20}]

Formula

From Robert Israel, Jun 14 2017: (Start)
If m and n are coprime, a(m*n) = lcm(a(m),a(n)).
If p is a prime such that the polynomial x^3-x^2-2*x+1 splits into distinct factors mod p, then a(p) divides p-1. These primes are A045472. (End)

Extensions

More terms from Robert Israel, Jun 14 2017
Showing 1-4 of 4 results.

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

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