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.

A110048 Expansion of 1/((1+2*x)*(1-4*x-4*x^2)).

Original entry on oeis.org

1, 2, 16, 64, 336, 1568, 7680, 36864, 178432, 860672, 4157440, 20070400, 96915456, 467935232, 2259419136, 10909384704, 52675280896, 254338531328, 1228055511040, 5929575645184, 28630525673472, 138240403177472
Offset: 0

Views

Author

Creighton Dement, Jul 10 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code:
-kbasejseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
See also comment for A110047.

Links

Crossrefs

Cf. A079291, A084159, A086346, A086348, A110046, A110047, A110049, A110050.

Programs

  • Magma
    [2^(n-2)*(Evaluate(DicksonFirst(n+1,-1), 2) +2*(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 18 2022
  • Maple
    seriestolist(series(1/((1+2*x)*(1-4*x-4*x^2)), x=0,40));
  • Mathematica
    CoefficientList[Series[1/((1+2x)(1-4x-4x^2)), {x,0,40}], x] (* or *) LinearRecurrence[{2,12,8}, {1,2,16}, 41] (* Harvey P. Dale, Nov 02 2011 *)
  • SageMath
    [2^(n-2)*(lucas_number2(n+1,2,-1) +2*(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 18 2022

Formula

Superseeker finds: a(n+1) = 2*A086348(n+1) (A086348's offset is 1: On a 3 X 3 board, number of n-move routes of chess king ending at central cell); binomial transform matches A084159 (Pell oblongs); j-th coefficient of g.f.*(1+x)^j matches A079291 (Squares of Pell numbers); a(n) + a(n+1) = A086346(n+2) (A086346's offset is 1: On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner cell.)
From Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3), where a(1)=1, a(2)=2, a(3)=16.
a(n) = 2^(n-3)*( 4*(-1)^(1-n) + (sqrt(2)-1)^(-n) + (-sqrt(2)-1)^(-n)) . (End)
a(n) = 2^n*A097076(n+1). - R. J. Mathar, Mar 08 2021

A110046 Expansion of (1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

Original entry on oeis.org

1, 8, 28, 144, 656, 3200, 15296, 73984, 356608, 1722368, 8313856, 40144896, 193826816, 935886848, 4518821888, 21818834944, 105350496256, 508677324800, 2456110759936, 11859152338944, 57261050298368, 276480810549248
Offset: 0

Views

Author

Creighton Dement, Jul 10 2005

Keywords

Links

Crossrefs

Cf. A110047, A110048, A110049, A110050.

Programs

  • Maple
    seriestolist(series((1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
  • Mathematica
    CoefficientList[Series[(1+4x-12x^2-16x^3)/((2x+1)(2x-1)(4x^2+4x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{4,8,-16,-16},{1,8,28,144},40] (* Harvey P. Dale, Jun 12 2016 *)
  • PARI
    Vec((1 + 4*x - 12*x^2 - 16*x^3) / ((1 - 2*x)*(1 + 2*x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, May 01 2019

Formula

From Colin Barker, May 01 2019: (Start)
a(n) = ((2 - 2*sqrt(2))^(1+n) + 2*(-(-2)^n + 2^n + 2^n*(1+sqrt(2))^(1+n))) / 4.
a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) for n>3.
(End)

A110051 Expansion of (1-x+2*x^3+x^2)/((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)).

Original entry on oeis.org

1, 4, 25, 119, 599, 2887, 14039, 67767, 327607, 1581751, 7638967, 36883895, 178097591, 859930039, 4152135095, 20048276919, 96801746359, 467400158647, 2256808013239, 10896832949687, 52614565424567, 254045594545591
Offset: 0

Views

Author

Creighton Dement, Jul 10 2005

Keywords

Links

Crossrefs

Cf. A110050, A110052.

Programs

  • Maple
    seriestolist(series((1-x+2*x^3+x^2)/((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -basejrokseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; RokType: Y[15] = Y[15] + 1/2
  • Mathematica
    CoefficientList[Series[(1-x+2x^3+x^2)/((1-x)(2x+1) (2x-1) (4x^2+4x-1)), {x,0,30}],x] (* or *) LinearRecurrence[{5,4,-24,0,16},{1,4,25,119,599},30] (* Harvey P. Dale, Sep 07 2012 *)

Formula

a(0)=1, a(1)=4, a(2)=25, a(3)=119, a(4)=599, a(n)=5*a(n-1)+4*a(n-2)- 24*a(n-3)+ 16*a(n-5). - Harvey P. Dale, Sep 07 2012
a(n)=1/112*(-2*(7*(-2)^n+7*2^(n+1)-8)+(13-9*Sqrt[2])*(2-2*Sqrt[2])^n+ 9* 2^(n+1/2)*(1+Sqrt[2])^n+13*(2*(1+Sqrt[2]))^n). - Harvey P. Dale, Sep 07 2012

A110052 Expansion of x*(-1+4*x)/((x-1)*(2*x-1)*(4*x^2+4*x-1)).

Original entry on oeis.org

0, 1, 3, 11, 43, 187, 859, 4059, 19419, 93403, 450267, 2172635, 10487515, 50632411, 244463323, 1180350171, 5699188443, 27518023387, 132868585179, 641545909979, 3097656932059, 14956809271003, 72217860617947, 348698671167195
Offset: 0

Views

Author

Creighton Dement, Jul 10 2005

Keywords

Links

Crossrefs

Cf. A110050, A110051, A110053, A110054.

Programs

  • Maple
    seriestolist(series(x*(-1+4*x)/((x-1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2basekrokseq:[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; RokType: Y[15] = Y[15] + 1/2
  • PARI
    concat(0, Vec(x*(1 - 4*x) / ((1 - x)*(1 - 2*x)*(1 - 4*x - 4*x^2)) + O(x^30))) \\ Colin Barker, May 01 2019

Formula

From Colin Barker, May 01 2019: (Start)
a(n) = (-3/7 + 2^(-1+n) - ((-4+sqrt(2))*(2*(1+sqrt(2)))^n + (2-2*sqrt(2))^n*(4+sqrt(2))) / (28*sqrt(2))).
a(n) = 7*a(n-1) - 10*a(n-2) - 4*a(n-3) + 8*a(n-4) for n>3.
(End)
Showing 1-4 of 4 results.

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