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.

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A175773 Expansion of 1/(1 - x - x^6 - x^11 + x^12).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 22, 28, 37, 48, 62, 80, 103, 133, 172, 223, 289, 374, 483, 625, 808, 1045, 1352, 1749, 2262, 2926, 3785, 4896, 6333, 8191, 10595, 13704, 17726, 22929, 29659, 38363, 49622, 64185, 83022, 107388, 138905, 179672
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

The ratio a(n+1)/a(n) is 1.2934859531254534... for n->infinity.

Links

Crossrefs

Cf. A175739.
Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^6-x^11+x^12))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^6 - x^11 + x^12), {x, 0, 50}], x]
  • PARI
    x='x+O('x^50); Vec(1/(1-x-x^6-x^11+x^12)) \\ G. C. Greubel, Nov 03 2018

Formula

G.f.: 1/((1 - x + x^2)*(1 - x^2 - x^3 + x^5 - x^7 - x^8 + x^10)).
a(n) = a(n-1) + a(n-6) + a(n-11) - a(n-12), n >= 12. - Franck Maminirina Ramaharo, Oct 31 2018

A175782 Expansion of 1/(1 - x - x^20 - x^39 + x^40).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 27, 31, 36, 42, 49, 57, 66, 76, 87, 99, 112, 126, 141, 157, 174, 192, 211, 231, 254, 279, 307, 339, 376, 419, 469, 527, 594
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

Limiting ratio of a(n)/a(n-1) = 1.119189829034646... .
A quasi - Salem polynomial based on the symmetrical polynomial defined by p(x,0) = 1, p(x,n) = x^(2*n) - x^(2*n - 1) - x^n - x + 1 for n>=1.
The polynomial has one real and two complex roots outside the unit circle.

Links

Crossrefs

Cf. A175739.
Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^20-x^39+x^40))); // G. C. Greubel, Nov 03 2018
  • Maple
    gf:= 1/(1-x-x^20-x^39+x^40):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 27 2012
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^20 - x^39 + x^40), {x, 0, 50}], x]
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22},70] (* Harvey P. Dale, Jun 30 2023 *)
  • PARI
    Vec(O(x^99)+1/(1 - x - x^20 - x^39 + x^40)) \\ N.B.: This yields a vector whose first component v[1] equals a(0), i.e., the offset is shifted by one. - M. F. Hasler, Dec 11 2010

Formula

a(n) = a(n-1) + a(n-20) + a(n-39) - a(n-40). - Franck Maminirina Ramaharo, Oct 31 2018

A181600 Expansion of 1/(1 - x - x^2 + x^8 - x^10).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 85, 136, 218, 349, 559, 895, 1434, 2297, 3679, 5893, 9439, 15119, 24217, 38790, 62132, 99520, 159407, 255331, 408978, 655083, 1049283, 1680695, 2692063, 4312028, 6906816, 11063033, 17720278, 28383559, 45463532, 72821479
Offset: 0

Views

Author

Roger L. Bagula, May 06 2013

Keywords

Comments

Limiting ratio is 1.60176..., the largest real root of -1 + x^2 - x^8 - x^9 + x^10. Compare this constant to Lehmer's Salem constant A073011 and the golden mean.

Links

Crossrefs

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x-x^2+x^8-x^10))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^2 + x^8 - x^10), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 0, 0, 0, 0, 0, -1, 0, 1}, {1, 1, 2, 3, 5, 8, 13, 21, 33, 53}, 50] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    Vec(1/(1 -x -x^2 +x^8 -x^10) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

Formula

a(n) = a(n-1) + a(n-2) - a(n-8) + a(n-10). - Franck Maminirina Ramaharo, Oct 31 2018

A185357 Expansion of 1/(1 - x - x^2 + x^18 - x^20).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4180, 6763, 10942, 17703, 28642, 46340, 74974, 121301, 196254, 317521, 513720, 831152, 1344728, 2175647, 3519998, 5695035, 9214046, 14907484, 24118947, 39022252, 63134437, 102145749
Offset: 0

Views

Author

Roger L. Bagula, Jan 21 2012

Keywords

Comments

Limiting ratio is 1.61791..., the real root of -1 + x^2 - x^18 - x^19 + x^20. Signature in Mathematica is:
-CoefficientList[1 - x - x^2 + x^18 - x^20, x]
{-1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1}.
The sequence agrees with the Fibonacci numbers (A000045) for the first 18 terms.

Links

Crossrefs

Cf. A117791, A107293, A204631.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x - x^2 + x^18 - x^20), {x, 0, 50}], x]
  • PARI
    Vec(1/(1-x-x^2+x^18-x^20) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

A202907 Expand 1/(1 - (3/2)*x + (2/3)*x^4 - x^5) in powers of x, then multiply coefficient of x^n by 3^floor(n/4)*2^n to get integers.

Original entry on oeis.org

1, 3, 9, 27, 211, 633, 1899, 5697, 52297, 156891, 470673, 1412019, 12675403, 38026209, 114078627, 342235881, 3081171505, 9243514515, 27730543545, 83191630635, 748691121283, 2246073363849, 6738220091547
Offset: 0

Views

Author

Roger L. Bagula, Jan 27 2012

Keywords

Links

Crossrefs

Cf. A167602, A167602, A117791, A107293, A204631, A185357.

Programs

  • Mathematica
    Table[3^Floor[k/4]*2^k*SeriesCoefficient[ Series[1/(1 - (3/2)* x + (2/3) x^4 - x^5), {x, 0, 30}], k], {k, 0, 30}] (* Bagula *)
a[ n_] := 2^n 3^Quotient[ n, 4] SeriesCoefficient[ 1 / (1 - 3/2 x + 2/3 x^4 - x^5), {x, 0, n}] (* Michael Somos, Jan 27 2012 *)

Formula

From Chai Wah Wu, Aug 01 2020: (Start)
a(n) = 211*a(n-4) + 7776*a(n-8) for n > 7.
G.f.: -(3*x + 1)*(9*x^2 + 1)/(7776*x^8 + 211*x^4 - 1). (End)

A225484 Expansion of 1/(1 - x^3 - x^4 - x^5 - x^6 + x^9).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 8, 10, 14, 20, 26, 36, 49, 66, 90, 123, 167, 227, 308, 420, 571, 776, 1056, 1436, 1952, 2656, 3612, 4912, 6680, 9085, 12356, 16804, 22853, 31081, 42269, 57486, 78182, 106327, 144604
Offset: 0

Views

Author

Roger L. Bagula, May 08 2013

Keywords

Comments

Limiting ratio is 1.3599997117115008..., the largest real root of 1 - x^3 - x^4 - x^5 - x^6 + x^9 = 0.

Links

Crossrefs

Cf. A117791, A107293, A204631, A225391, A225482.

Programs

  • Mathematica
    SeriesCoefficient[Series[1/(1 - x^3 - x^4 - x^5 - x^6 + x^9), {x, 0, 50}], n]
  • PARI
    Vec(1/(1-x^3-x^4-x^5-x^6+x^9)+O(x^99)) \\ Charles R Greathouse IV, May 08 2013

A205961 Expansion of 1/(-32*x^5 + 8*x^3 - 4*x^2 - x + 1).

Original entry on oeis.org

1, 1, 5, 1, 13, 9, 85, 177, 477, 921, 1701, 4289, 9389, 28201, 60917, 153041, 308349, 733625, 1645125, 4062177, 9670989, 22625865, 52288405, 118067953, 276204317, 639640537, 1523941861
Offset: 0

Views

Author

Roger L. Bagula, Feb 02 2012

Keywords

Comments

Previous name was: Expand 1/(1 - x/2 - x^2 + x^3 - x^5) in powers of x, then multiply coefficient of x^n by 2^n to get integers.
The sequence is from -1 + x^2 - x^3 - x^4/2 + x^5 with real root 1.1647612555333289.
The limiting ratio of successive terms is 2*1.1647612555333289.
Recurrence: -32 *a (n) + 8 *a (n + 2) - 4 *a (n + 4) + a (n + 5) == 0; with a (1) == 1; a (2) == 1; a (3) == 5; a (4) == 1; a (5) == 13 (from FindSequenceFunction[]).

Links

Crossrefs

Cf. A202907, A167602, A167602, A117791, A107293, A204631, A185357.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x/2 - x^2 + x^3 - x^5), {x, 0, 50}], x] * 2^Range[0, 50]
LinearRecurrence[{1,4,-8,0,32}, {1,1,5,1,13}, 100] (* G. C. Greubel, Nov 16 2016 *)
  • PARI
    for(n=0,30, print1(2^n*polcoeff(1/(1-x/2 - x^2 + x^3 - x^5) + O(x^32), n), ", ")) \\ G. C. Greubel, Nov 16 2016

Extensions

New name from Joerg Arndt, Nov 19 2016

A206568 Expand 1/(8 - 8 x + 3 x^3 - 2 x^4) in powers of x, then multiply coefficient of x^n by 8^(1 + floor(n/3)) to get integers.

Original entry on oeis.org

1, 1, 1, 5, 4, 3, 25, 23, 22, 149, 130, 110, 785, 693, 623, 4389, 3880, 3397, 23977, 21115, 18684, 131893, 116502, 102680, 724705, 638985, 563949, 3980357, 3512812, 3098935, 21873593, 19295871, 17024690
Offset: 0

Views

Author

Roger L. Bagula, Feb 09 2012

Keywords

Comments

Bob Hanlon (hanlonr(AT)cox.net) helped convert the expansion to a recursion.

Links

Crossrefs

Cf. A202907, A167602, A167602, A117791, A107293, A204631, A185357, A205961.

Programs

  • Maple
    a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <64|69|21|-1>>^ iquo(n, 3, 'r'). `if`(r=0, <<1, 5, 25, 149>>, `if`(r=1, <<1, 4, 23, 130>>, <<1, 3, 22, 110>>)))[1, 1]: seq (a(n), n=0..40); # Alois P. Heinz, Feb 11 2012
  • Mathematica
    (* expansion*)
Table[8^(1 + Floor[n/3])*SeriesCoefficient[Series[1/(8 - 8 x + 3 x^3 - 2 x^4), {x, 0, 50}], n], {n, 0,50}]
(*recursion*)
a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 5; a[5] = 4; a[6] = 3;
a[7] = 25; a[8] = 23; a[9] = 22; a[10] = 149; a[11] = 130;
a[12] = 110;
a[n_Integer?Positive] := a[n] = 64*a[-12 + n] + 69*a[-9 + n] + 21*a[-6 +n] - a[-3 + n]
Table[a[n], {n, 1, 50}]

Formula

G.f.: (-4*x^8-6*x^7-9*x^6-4*x^5-5*x^4-6*x^3-x^2-x-1) / (64*x^12 +69*x^9 +21*x^6 -x^3-1).

A225490 Expansion of 1/(1 - x - x^2 + x^5 + x^6 - x^7).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 30, 42, 60, 84, 118, 166, 233, 327, 458, 643, 901, 1263, 1770, 2481, 3477, 4872, 6828, 9568, 13408, 18788, 26328, 36893, 51697, 72442, 101511, 142245, 199323, 279306, 391383, 548433
Offset: 0

Views

Author

Roger L. Bagula, May 08 2013

Keywords

Comments

Limiting ratio is 1.401268367939855..., the largest real root of -1 + x + x^2 - x^5 - x^6 + x^7.

Links

Crossrefs

Cf. A117791, A107293, A204631, A225391, A225482, A225484.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x - x^2 + x^5 + x^6 - x^7), {x, 0, 50}], x]

A225500 Expansion of 1/(1 - x - x^5 + x^6 - x^7 - x^11 + x^12).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 33, 43, 56, 73, 95, 124, 161, 210, 273, 355, 463, 603, 786, 1023, 1332, 1735, 2259, 2942, 3831, 4989, 6497, 8461, 11019, 14350, 18687, 24335, 31691, 41270, 53745
Offset: 0

Views

Author

Roger L. Bagula, May 09 2013

Keywords

Links

Crossrefs

Cf. A117791, A107293, A204631, A225391, A225482, A225484, A225490, A225490.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x - x^5 + x^6 - x^7 - x^11 + x^12), {x, 0, 50}], x]
LinearRecurrence[{1,0,0,0,1,-1,1,0,0,0,1,-1},{1,1,1,1,1,2,2,3,4,5,7,9},100] (* G. C. Greubel, Nov 16 2016 *)
  • PARI
    Vec(1/(1-x -x^5 +x^6 -x^7 -x^11 +x^12) + O(x^50)) \\ G. C. Greubel, Nov 16 2016
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