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.

A169609 Period 3: repeat [1, 3, 3].

Original entry on oeis.org

1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3
Offset: 0

Views

Author

Klaus Brockhaus, Dec 03 2009

Keywords

Comments

Interleaving of A000012, A010701 and A010701.
Also continued fraction expansion of (5+sqrt(65))/10 = 1.3062257748....
Also decimal expansion of 133/999.
a(n) = A144437(n) for n > 0.
Unsigned version of A154595.
Binomial transform of A168615.
Inverse binomial transform of A168673.
Essentially first differences of A047347.

Links

Crossrefs

Cf. A000012 (all 1's sequence), A010701 (all 3's sequence), A144437 (repeat 3, 3, 1), A154595 (repeat 1, 3, 3, -1, -3, -3), A168615, A168673, A047347 (congruent to {0, 1, 4} mod 7), A010684 (repeat 1, 3).
Cf. A171419 (decimal expansion of (5+sqrt(65))/10).
Cf. A146094.

Programs

  • Magma
    [ n mod 3 eq 0 select 1 else 3: n in [0..104] ];
  • Magma
    &cat [[1, 3, 3]^^30]; // Wesley Ivan Hurt, Jul 02 2016
  • Maple
    seq(op([1, 3, 3]), n=0..50); # Wesley Ivan Hurt, Jul 02 2016
  • Mathematica
    PadRight[{},120,{1,3,3}] (* or *) LinearRecurrence[{0,0,1},{1,3,3},120] (* Harvey P. Dale, Apr 29 2015 *)

Formula

a(n) = a(n-3) for n > 2, with a(0) = 1, a(1) = 3, a(2) = 3.
G.f.: (1+3*x+3*x^2)/(1-x^3).
a(n) = (7/3)+(2/3)*cos((2*Pi/3)*(n+1))-(2*sqrt(3)/3)*sin((2*Pi/3)*(n+1)). [Richard Choulet, Mar 15 2010]
a(n) = a(n-a(n-2)) for n>=2. Example: a(5) = a(5-a(3)) = a(5-a(3-a(1))) = a(5-a(3-3)) = a(5-a(0)) = a(5-1) = a(4) = a(4-a(2)) = a(4-3) = a(1) = 3. [Richard Choulet, Mar 15 2010; edited by Klaus Brockhaus, Nov 21 2010]
a(n) = 1 + 2*sgn(n mod 3). - Wesley Ivan Hurt, Jul 02 2016
a(n) = 3/gcd(n,3). - Wesley Ivan Hurt, Jul 11 2016

Extensions

Keywords cofr, cons added by Klaus Brockhaus, Apr 20 2010
Minor edits, crossref added by Klaus Brockhaus, May 03 2010

A168673 Binomial transform of A169609.

Original entry on oeis.org

1, 4, 10, 20, 38, 74, 148, 298, 598, 1196, 2390, 4778, 9556, 19114, 38230, 76460, 152918, 305834, 611668, 1223338, 2446678, 4893356, 9786710, 19573418, 39146836, 78293674, 156587350, 313174700, 626349398, 1252698794, 2505397588, 5010795178, 10021590358
Offset: 0

Views

Author

Paul Curtz, Dec 02 2009

Keywords

Comments

Sequence and successive differences are identical to their third differences. See principal sequence A024495. Main diagonal of the array of successive differences is A083595 (1,6,8,20,36,...).

Links

Crossrefs

Cf. A169609, A144437, A168615, A024495, A083595, A130772, A062092, A130151.

Programs

  • Magma
    I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)- 3*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 30 2016
  • Mathematica
    LinearRecurrence[{3,-3,2},{1,4,10},25] (* G. C. Greubel, Jul 29 2016 *)
RecurrenceTable[{a[0] == 1, a[1] == 4, a[2] == 10, a[n] == 3 a[n-1] - 3 a[n-2] + 2 a[n-3]}, a, {n, 40}] (* Vincenzo Librandi, Jul 30 2016 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; 2,-3,3]^n*[1;4;10])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

Formula

a(n+1) - 2a(n) = A130772(n).
a(n) = A062092(n) - A130151(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) for n > 2; a(0) = 1, a(1) = 4, a(2) = 10.
G.f.: (1 + x + x^2)/(1 -3*x +3*x^2 -2*x^3). - Philippe Deléham, Dec 03 2009

Extensions

Edited and extended by Klaus Brockhaus, Dec 03 2009
Showing 1-2 of 2 results.

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