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.

Previous Showing 21-30 of 33 results. Next

A173925 Expansion of 1/(1 - x - x^8 - x^15 + x^16).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 19, 24, 30, 37, 45, 56, 69, 85, 105, 130, 161, 199, 246, 304, 376, 465, 575, 711, 879, 1086, 1343, 1660, 2052, 2537, 3137, 3879, 4796, 5929, 7330, 9062, 11203, 13850, 17123, 21170, 26173, 32359, 40006
Offset: 0

Views

Author

Roger L. Bagula, Nov 26 2010

Keywords

Comments

Limiting ratio is 1.2303914344072246.
The polynomial is the 10th Salem on Mossinghoff's list.

Links

Crossrefs

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 60); Coefficients(R!(1/(1-x-x^8-x^15+x^16))); // G. C. Greubel, Nov 03 2018
  • Maple
    seq(coeff(series(1/(1-x-x^8-x^15+x^16), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 15 2019
  • Mathematica
    CoefficientList[Series[1/(1-x-x^8-x^15+x^16), {x, 0, 60}] ,x] (* Harvey P. Dale, Apr 02 2012 *)
  • PARI
    my(x='x+O('x^60)); Vec(1/(1-x-x^8-x^15+x^16)) \\ G. C. Greubel, Nov 03 2018
  • Sage
    def A173925_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^8-x^15+x^16) ).list()
A173925_list(60) # G. C. Greubel, Dec 15 2019

Formula

a(n) = a(n-1) + a(n-8) + a(n-15) - a(n-16). - Harvey P. Dale, Apr 02 2012

A174522 Expansion of 1/(1 - x - x^4 + x^6).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 7, 8, 11, 14, 15, 16, 20, 26, 30, 32, 37, 47, 57, 63, 70, 85, 105, 121, 134, 156, 191, 227, 256, 291, 348, 419, 484, 548, 640, 768, 904, 1033, 1189, 1409, 1673, 1938, 2223, 2599, 3083, 3612, 4162, 4823, 5683, 6696, 7775, 8986
Offset: 0

Views

Author

Roger L. Bagula, Nov 28 2010

Keywords

Comments

Low limiting ratio in 100th iteration near 1.16663.
The polynomial is interesting for the puzzling low ratio and the Salem like root structure with two complex roots outside the unit circle.

Links

Crossrefs

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 - x - x^4 + x^6))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^4 + x^6), {x, 0, 60}], x]
  • PARI
    x='x+O('x^50); Vec(1/(1 - x - x^4 + x^6)) \\ G. C. Greubel, Nov 03 2018

Formula

a(n) = a(n-1) + a(n-4) + a(n-6). - Franck Maminirina Ramaharo, Oct 31 2018

A175740 Expansion of 1/(1 - x - x^10 - x^19 + x^20).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 21, 26, 32, 39, 47, 56, 66, 79, 94, 112, 134, 161, 194, 234, 282, 339, 407, 488, 585, 701, 840, 1007, 1208, 1450, 1741, 2090, 2510, 3013, 3616, 4339, 5206, 6246, 7494, 8992, 10790, 12948
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

Limiting ratio is 1.2000265239873915.

Links

Crossrefs

Cf. A175739.
Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 60); Coefficients(R!(1/(1 - x - x^10 - x^19 + x^20))); // G. C. Greubel, Nov 03 2018
  • Maple
    seq(coeff(series(1/(1 -x -x^10 -x^19 +x^20), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 05 2019
  • Mathematica
    CoefficientList[Series[1/(1 -x -x^10 -x^19 +x^20), {x, 0, 60}], x]
  • PARI
    my(x='x+O('x^60)); Vec(1/(1 -x -x^10 -x^19 +x^20)) \\ G. C. Greubel, Nov 03 2018
  • Sage
    def A175740_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1 -x -x^10 -x^19 +x^20) ).list()
A175740_list(60) # G. C. Greubel, Dec 05 2019

Formula

G.f.: 1/((1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14)).
a(n) = a(n-1) + a(n-10) + a(n-19) + a(n-20). - Franck Maminirina Ramaharo, Oct 31 2018

A175772 Expansion of 1/(1 - x - x^9 - x^17 + x^18).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 20, 25, 31, 38, 46, 55, 67, 81, 98, 119, 145, 177, 216, 263, 320, 389, 473, 575, 699, 850, 1034, 1258, 1530, 1862, 2265, 2755, 3351, 4076, 4958, 6031, 7336, 8923, 10854, 13203, 16060, 19535, 23762
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

The ratio a(n+1)/a(n) is 1.216391661138265... as n->infinity.

Links

Crossrefs

Cf. A175739.
Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^9-x^17+x^18))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^9 - x^17 + x^18), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} ,60] (* Harvey P. Dale, Jul 13 2014 *)
  • PARI
    x='x+O('x^50); Vec(1/(1-x-x^9-x^17+x^18)) \\ G. C. Greubel, Nov 03 2018

Formula

G.f.: 1/((1 - x^2 + x^4)*(1 - x^4 - x^5 - x^6 + x^10)*(1 - x + x^2 - x^3 + x^4)).
a(n) = a(n-1) + a(n-9) + a(n-17) - a(n-18). - Harvey P. Dale, Jul 13 2014

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

A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 22, 25, 30, 35, 41, 49, 57, 67, 79, 93, 109, 129, 151, 178, 209, 246, 290, 340, 401, 471, 554, 652, 767, 902, 1061, 1248, 1468, 1727, 2031, 2390, 2810, 3306, 3889, 4574, 5381, 6329
Offset: 0

Views

Author

Luis A Restrepo (luisiii(AT)mac.com), Feb 04 2007

Keywords

Comments

a(n) = O(n^c), where c is the larger real root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 1.176280818..., the smallest known Salem constant.

References

  • Wolfram, S., A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002.

Links

Crossrefs

Cf. A070178, A029826, A107480, A127193, A127194, A127624.

Programs

  • Mathematica
    LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1},{1,1,1,1,1,1,1,1,1,1},70] (* Harvey P. Dale, May 31 2013 *)

Formula

G.f.: ( 1+2*x+2*x^2+x^3-x^5-2*x^6-3*x^7-3*x^8-2*x^9 ) / ( 1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10 ). [R. J. Mathar, Jun 30 2010]

Extensions

Edited by Don Reble, Mar 09 2007

A143335 Expansion of (1 - 2*x^3 - x^4 - 2*x^5 - x^6 - x^7 - x^8 + 2*x^9)/(1 + x - x^3 - x^4 - x^5 - x^6 - x^7 + x^9 + x^10).

Original entry on oeis.org

1, -1, 1, -2, 1, -2, 0, -1, -3, 2, -6, 1, -4, -3, -3, -5, -4, -7, -6, -9, -8, -14, -10, -18, -18, -20, -28, -27, -38, -39, -50, -57, -67, -79, -94, -109, -128, -154, -175, -213, -244, -292, -341, -400, -475, -553, -655, -768, -905, -1062, -1253, -1470, -1732
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 22 2008

Keywords

Comments

Shares the same 10th-order "Salem" linear recurrence with A029826, A173243 and A125950.

Links

Crossrefs

Cf. A029826, A070178, A087612, A107479, A107480, A109538.
Cf. A109543, A109544, A114749, A125950, A130844, A147851.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1-2*x^3-x^4 -2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7+x^9 +x^10) )); // G. C. Greubel, Nov 03 2018
  • Maple
    seq(coeff(series((1-2*x^3-x^4 -2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7 +x^9+x^10), x, n+1), x, n), n = 0..65); # G. C. Greubel, Mar 13 2020
  • Mathematica
    LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1}, {1,-1,1,-2,1,-2,0,-1,-3,2}, 65] (* Franck Maminirina Ramaharo, Nov 02 2018 *)
  • PARI
    my(x='x+O('x^65)); Vec((1-2*x^3-x^4-2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10)) \\ G. C. Greubel, Nov 03 2018

Formula

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10). - Franck Maminirina Ramaharo, Nov 02 2018

Extensions

Edited by Assoc. Eds. of the OEIS - Jun 30 2010

A173243 G.f. (x + 1)^10/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1).

Original entry on oeis.org

1, 9, 36, 85, 135, 163, 178, 208, 265, 341, 419, 485, 549, 642, 778, 940, 1110, 1290, 1498, 1761, 2095, 2487, 2921, 3413, 4000, 4712, 5565, 6563, 7711, 9044, 10627, 12516, 14745, 17352, 20397, 23969, 28187, 33176, 39048, 45931, 54007, 63509
Offset: 0

Views

Author

Roger L. Bagula, Feb 13 2010

Keywords

Comments

Limiting ratio is a(n+1)/a(n)->1.1762808182599176..

Links

Crossrefs

Cf. A029826

Programs

  • Mathematica
    p[x_] = (x + 1)^10/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1);
a = Table[SeriesCoefficient[ Series[p[x], {x, 0, 50}], n], {n, 0, 50}]
LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1},{1,9,36,85,135,163,178,208,265,341,419},50] (* Harvey P. Dale, Sep 04 2021 *)

Extensions

Removed unused variables - The Assoc. Editors of the OEIS, Feb 24 2010
Previous Showing 21-30 of 33 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.