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.

A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)).

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 12, 16, 25, 42, 58, 92, 141, 206, 324, 488, 737, 1138, 1714, 2612, 3989, 6038, 9212, 14016, 21289, 32442, 49322, 75020, 114205, 173662, 264244, 402072, 611569, 930562, 1415714, 2153700, 3276837, 4985126, 7584236, 11538800
Offset: 0

Views

Author

Creighton Dement, May 25 2005

Keywords

Links

Crossrefs

Cf. A107850, A107851, A107852.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)),{x,0,39}],x] (* James C. McMahon, Feb 19 2024 *)
  • PARI
    Vec((1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)) + O(x^45)) \\ Colin Barker, Apr 30 2019

Formula

a(n) = A052947(n+2) + A014017(n+6). - Ralf Stephan, Nov 30 2010
a(n) = a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) + 2*a(n-7) for n>6. - Colin Barker, Apr 30 2019

A107850 Expansion of g.f. (x^2+x+1)*(2*x^2+2*x+1)*(x-1)^2/((1-x^2-2*x^3)*(x^4+1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 3, 6, 7, 13, 17, 24, 41, 57, 91, 142, 207, 325, 489, 736, 1137, 1713, 2611, 3990, 6039, 9213, 14017, 21288, 32441, 49321, 75019, 114206, 173663, 264245, 402073, 611568, 930561, 1415713, 2153699, 3276838, 4985127, 7584237, 11538801
Offset: 0

Views

Author

Creighton Dement, May 25 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code: 1lesforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Links

Crossrefs

Cf. A107849, A107851, A107852.

Programs

  • Mathematica
    CoefficientList[Series[(x^2+x+1)(2x^2+2x+1)(x-1)^2/((1-x^2-2x^3)(x^4+1)),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{1,1,1,0,1,1,3},50] (* Harvey P. Dale, Dec 26 2015 *)

Formula

a(n) = A052947(n-1)+A118831(n+6). - R. J. Mathar, Apr 18 2008
a(0)=1, a(1)=1, a(2)=1, a(3)=0, a(4)=1, a(5)=1, a(6)=3, a(n)=a(n-2)+ 2*a(n-3)- a(n-4)+a(n-6)+2*a(n-7). - Harvey P. Dale, Dec 26 2015

A107851 Expansion of g.f. x*(-1-x-3*x^2-x^3+2*x^5)/((2*x^3+x^2-1)*(x^4+1)).

Original entry on oeis.org

0, 1, 1, 4, 4, 5, 9, 10, 18, 29, 41, 68, 100, 149, 233, 346, 530, 813, 1225, 1876, 2852, 4325, 6601, 10026, 15250, 23229, 35305, 53732, 81764, 124341, 189225, 287866, 437906, 666317, 1013641, 1542132, 2346276, 3569413, 5430537, 8261962, 12569362
Offset: 0

Views

Author

Creighton Dement, May 25 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code: 1jesforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Links

Crossrefs

Cf. A107849, A107850, A107852.

Programs

  • Mathematica
    CoefficientList[Series[x(-1-x-3x^2-x^3+2x^5)/((2x^3+x^2-1)(x^4+1)), {x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,4,4,5,9},51] (* Harvey P. Dale, Jul 19 2011 *)
  • PARI
    a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;4;4;5;9])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = A159284(n+1) + A132380(n+7).
a(0)=0, a(1)=1, a(2)=1, a(3)=4, a(4)=4, a(5)=5, a(6)=9, a(n)= a(n-2)+ 2*a(n-3)-a(n-4)+a(n-6)+2*a(n-7). - Harvey P. Dale, Jul 19 2011

A107853 Expansion of g.f. x*(x-1)*(x+1)^3/((2*x^3+x^2-1)*(x^4+1)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 7, 10, 13, 26, 33, 50, 83, 114, 183, 282, 413, 650, 977, 1474, 2275, 3426, 5223, 7978, 12077, 18426, 28033, 42578, 64883, 98642, 150039, 228410, 347325, 528490, 804145, 1223138, 1861123, 2831426, 4307399, 6553674, 9970253
Offset: 0

Views

Author

Creighton Dement, May 25 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code: - 2basejforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Links

Crossrefs

Cf. A107849, A107850, A107851, A107852, A107854.

Programs

  • Mathematica
    LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,2,1,2,3,2},50] (* Harvey P. Dale, Jan 24 2018 *)

Formula

a(n) = 2*A159287(n) + A091337(n+6).

A107854 G.f. x*(x^2+1)*(x^3-x-1)/((2*x^3+x^2-1)*(x^4+1)).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 8, 11, 19, 29, 42, 67, 99, 149, 232, 347, 531, 813, 1226, 1875, 2851, 4325, 6600, 10027, 15251, 23229, 35306, 53731, 81763, 124341, 189224, 287867, 437907, 666317, 1013642, 1542131, 2346275, 3569413, 5430536, 8261963, 12569363
Offset: 0

Views

Author

Creighton Dement, May 25 2005

Keywords

Comments

The sequence A078028 is given by 1em[I* ]forzapseq and is from the same "batch" (i.e., corresponding to the same floretion and symmetry settings) as A107849, A107850, A107851, A107852, A107853 and (a(n)).
Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]forzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Links

Crossrefs

Cf. A107849, A107850, A107851, A107852, A107853, A078028.

Programs

  • Mathematica
    CoefficientList[Series[x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,2,3,3,5},50] (* Harvey P. Dale, Jun 21 2022 *)
  • PARI
    a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;2;3;3;5])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = A159284(n) + A014017(n+5).
Showing 1-5 of 5 results.

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