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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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

A143420 Row sums of triangle A373101.

Original entry on oeis.org

1, 8, 55, 370, 2520, 17472, 123151, 880070, 6360706, 46402312, 341153384, 2524722928, 18789734496, 140521154048, 1055383259791, 7956220907758, 60179579570382, 456545145078408, 3472804505717170
Offset: 2

Views

Author

Gary W. Adamson, Aug 14 2008

Keywords

Comments

Each term in the sequence is a sum of tetrahedral numbers.
The underlying triangle mentioned as A143419 was lost and is now restored in A373101. - Georg Fischer, May 23 2024

Examples

			a(5) = 370 = (20 + 165 + 165 + 20) = C(6,3) + C(11,3) + C(11,8) + C(6,3).

Crossrefs

Cf. A108958 (row sums of A143418), A373101.

Programs

  • Maple
    seq(add((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1),n=2..20); # Georg Fischer, May 23 2024

Extensions

Definition changed, a(7) corrected and more terms from Georg Fischer, May 23 2024

A373101 Triangle read by rows, T(n,k) = (binomial(n,k)^3 - binomial(n,k))/6 for k=1..n-1 and n >= 2.

Original entry on oeis.org

1, 4, 4, 10, 35, 10, 20, 165, 165, 20, 35, 560, 1330, 560, 35, 56, 1540, 7140, 7140, 1540, 56, 84, 3654, 29260, 57155, 29260, 3654, 84, 120, 7770, 98770, 333375, 333375, 98770, 7770, 120, 165, 15180, 287980, 1543465, 2667126, 1543465, 287980, 15180, 165
Offset: 2

Views

Author

Georg Fischer, May 23 2024

Keywords

Comments

This triangle was mentioned in A143420 with the wrong A-number A143419.

Examples

			T(n,k) for n=2..7:
   1;
   4,    4;
  10,   35,   10;
  20,  165,  165,   20;
  35,  560, 1330,  560,   35;
  56, 1540, 7140, 7140, 1540, 56;

Crossrefs

Cf. A143418, A143420.

Programs

  • Maple
    seq(print(n,seq((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1)),n=2..10);
Previous Showing 21-24 of 24 results.

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