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.

A131039 Expansion of (1-x)*(2*x^2-4*x+1)/(1-2*x+5*x^2-4*x^3+x^4).

Original entry on oeis.org

1, -3, -5, 7, 26, 0, -97, -97, 265, 627, -362, -2702, -1351, 8733, 13775, -18817, -70226, 0, 262087, 262087, -716035, -1694157, 978122, 7300802, 3650401, -23596563, -37220045, 50843527, 189750626, 0, -708158977, -708158977, 1934726305, 4577611587, -2642885282, -19726764302, -9863382151
Offset: 0

Views

Author

Creighton Dement, Jun 11 2007

Keywords

Comments

Unsigned bisection gives match to A002316 (Related to Bernoulli numbers). Note that all numbers in A002316 appear to be in A002531 (Numerators of continued fraction convergents to sqrt(3)) as well. a(12*n+5) = (0,0,0,0,...)
Floretion Algebra Multiplication Program, FAMP Code: 2tesseq['i + .5i' + .5j' + .5k' + .5e]

Links

Crossrefs

Cf. A131040, A131041, A131042, A002316, A002531.

Programs

  • Maple
    f:= gfun:-rectoproc({a(0)=1, a(1)=-3, a(2)=-5, a(3)=7, a(n)=2*a(n-1)-5*a(n-2)+4*a(n-3)-a(n-4)},a(n),remember):
map(f, [$0..100]); # Robert Israel, Dec 25 2016
  • Mathematica
    CoefficientList[Series[(1-x)(2x^2-4x+1)/(1-2x+5x^2-4x^3+x^4),{x, 0, 50}], x] (* or *) LinearRecurrence[{2,-5,4,-1},{1,-3,-5,7},50] (* Harvey P. Dale, Aug 31 2011 *)

Formula

a(0)=1, a(1)=-3, a(2)=-5, a(3)=7, a(n)=2*a(n-1)-5*a(n-2)+4*a(n-3)-a(n-4) [Harvey P. Dale, Aug 31 2011]

A131040 a(n) = (1/2+1/2*i*sqrt(11))^n + (1/2-1/2*i*sqrt(11))^n, where i=sqrt(-1).

Original entry on oeis.org

1, -5, -8, 7, 31, 10, -83, -113, 136, 475, 67, -1358, -1559, 2515, 7192, -353, -21929, -20870, 44917, 107527, -27224, -349805, -268133, 781282, 1585681, -758165, -5515208, -3240713, 13304911, 23027050, -16887683, -85968833, -35305784
Offset: 0

Views

Author

Creighton Dement, Jun 11 2007

Keywords

Comments

Generating floretion is 1.5i' + .5j' + .5k' + .5e whereas in A131039 it is 'i + .5i' + .5j' + .5k' + .5e
Essentially the Lucas sequence V(1,3). - Peter Bala, Jun 23 2015

Links

Crossrefs

Cf. A131039, A131041, A131042, A002316, A002531.

Programs

  • Maple
    Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[ 1.5i' + .5j' + .5k' + .5e]
  • Sage
    [lucas_number2(n,1,3) for n in range(1, 34)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = a(n-1) - 3*a(n-2); G.f. (1 - 6*x)/(1 - x + 3*x^2).
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x - 11*x^2))/2 )^n. - Peter Bala, Jun 23 2015

A131041 a(n) = 2*a(n-1) - a(n-2) - a(n-4).

Original entry on oeis.org

1, 1, 1, -1, -4, -8, -13, -17, -17, -9, 12, 50, 105, 169, 221, 223, 120, -152, -645, -1361, -2197, -2881, -2920, -1598, 1921, 8321, 17641, 28559, 37556, 38232, 21267, -24257, -107337, -228649, -371228, -489550, -500535, -282871, 306021
Offset: 0

Views

Author

Creighton Dement, Jun 11 2007

Keywords

Comments

Generating floretion is .5i' + .5j' + .5k' + .5e + 'ii' (for A131039 it is 'i + .5i' + .5j' + .5k' + .5e and for A131040 it is 1.5i' + .5j' + .5k' + .5e)

Links

Crossrefs

Cf. A131039, A131040, A131042, A002316, A002531.

Programs

  • Maple
    Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[.5i' + .5j' + .5k' + .5e + 'ii']
  • Mathematica
    LinearRecurrence[{2,-1,0,-1},{1,1,1,-1},40] (* Harvey P. Dale, Oct 14 2012 *)

Formula

G.f. (1-x-2*x^3)/(1-2*x+x^2+x^4)

A131717 Natural numbers A000027 with 6n+4 and 6n+5 terms swapped.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 23, 22, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38, 39, 41, 40, 42, 43, 44, 45, 47, 46, 48, 49, 50, 51, 53, 52, 54, 55, 56, 57, 59, 58, 60, 61, 62, 63, 65, 64, 66, 67, 68, 69, 71, 70, 72
Offset: 1

Views

Author

Paul Curtz, Sep 15 2007

Keywords

Comments

Hexaperiodic differences: 1, 1, 2, -1, 2, 1; 0, 1, -3, 3, -1, 0 (even palindromic signed); 1,-4, 6, -4, 1, 0.

Links

Crossrefs

Cf. A131042.

Programs

  • Maple
    seq(seq(6*i+s,s=[1,2,3,5,4,6]),i=0..100); # Robert Israel, Nov 11 2014
  • Mathematica
    Drop[CoefficientList[Series[x (2x^5 - x^4 + 2x^3 + x^2 + x + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x], 1] (* Indranil Ghosh, Apr 18 2017 *)
Table[Sum[(7 #1 - 13 #2 + 17 #3 - 3 #4 + 2 #5 + 2 #6)/30 & @@ Mod[k + Range[0, 5], 6], {k, 0, n}], {n, 0, 71}] (* Michael De Vlieger, Apr 22 2017 *)
  • PARI
    Vec(x*(2*x^5-x^4+2*x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 11 2014

Formula

a(n) = A008585(n/3) if n is congruent to 0 mod 3. - Frieder Mittmann, Nov 11 2014
a(n) = A007310((n-1)/3) if n is congruent to 1 mod 3. - Frieder Mittmann, Nov 11 2014
a(n) = A047235((n-2)/3) if n is congruent to 2 mod 3. - Frieder Mittmann, Nov 11 2014
G.f.: x*(2*x^5-x^4+2*x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Nov 11 2014
a(n) = (24*floor(n/6)-3*(n^2-3*n-2)-9*floor(n/3)*(3*floor(n/3)-2*n+3)+(-1)^floor(n/3)*(3*n^2-5*n-6+3*floor(n/3)*(9*floor(n/3)-6*n+5)))/4. - Luce ETIENNE, Apr 18 2017

A274631 First column of array in A274630.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 7, 8, 10, 9, 11, 12, 13, 14, 16, 15, 17, 18, 19, 20, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 32, 31, 34, 33, 36, 35, 38, 37, 39, 40, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 56, 55, 58, 57, 59, 60, 62, 61, 64, 63, 66, 65, 68, 67, 70, 69, 72, 71, 74, 73, 76, 75, 78, 77, 79
Offset: 1

Views

Author

N. J. A. Sloane, Jul 08 2016

Keywords

Crossrefs

Cf. A269526, A274630.
First differs from A131042 at a(23). - Omar E. Pol, Jul 08 2016
Showing 1-5 of 5 results.

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