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 1-2 of 2 results.

A159674 Expansion of (1 - x)/(1 - 32*x + x^2).

Original entry on oeis.org

1, 31, 991, 31681, 1012801, 32377951, 1035081631, 33090234241, 1057852414081, 33818187016351, 1081124132109151, 34562154040476481, 1104907805163138241, 35322487611179947231, 1129214695752595173151, 36099547776471865593601, 1154056314151347103822081
Offset: 0

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integers.
Positive values of x (or y) satisfying x^2 - 32*x*y + y^2 + 30 = 0. - Colin Barker, Feb 24 2014

Links

Crossrefs

Cf. A029548, A157456, A159675, A159677.
Cf. similar sequences listed in A238379.

Programs

  • Magma
    A029548:= func< n | Evaluate(ChebyshevSecond(n),16) >;
[A029548(n+1) -A029548(n): n in [0..30]]; // G. C. Greubel, Sep 25 2022
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
# Second program
seq(simplify(ChebyshevU(n, 16) -ChebyshevU(n-1, 16)), n=0..30); # G. C. Greubel, Sep 25 2022
  • Mathematica
    CoefficientList[Series[(1-x)/(1-32*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{32,-1},{1,31},30] (* Harvey P. Dale, Mar 21 2017 *)
  • PARI
    concat([0], Vec((-x+1)/(x^2-32*x+1) + O(x^100))) \\ Colin Barker, Feb 24 2014
  • SageMath
    def A159674(n): return chebyshev_U(n, 16) - chebyshev_U(n-1, 16)
[A159674(n) for n in range(31)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is: a(0)=1, a(1)=31, a(t+2) = 32*a(t+1) - a(t) resulting in terms 1, 31, 991, 31681, ... (this sequence).
The b(j) recurrence is: b(0)=1, b(1)=33, b(t+2) = 32*b(t+1) - b(t) resulting in terms 1, 33, 1055, 33727, ... (A159675).
The n(j) recurrence is: n(-1) = n(0) = 0, n(1) = 64, n(t+3) = 1023*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 64, 65472, 66912384, ... (A159677).
a(n) = (1/34)*(17-sqrt(255))*(1+(16+sqrt(255))^(2*n+1))/(16+sqrt(255))^n. - Bruno Berselli, Feb 25 2014
a(n) = ChebyshevU(n, 16) - ChebyshevU(n-1, 16) = A029548(n) - A029548(n-1). - G. C. Greubel, Sep 25 2022

Extensions

More terms and new name from Colin Barker, Feb 24 2014
Set offset to 0 by Joerg Arndt, Feb 25 2014

A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2).

Original entry on oeis.org

1, 33, 1055, 33727, 1078209, 34468961, 1101928543, 35227244415, 1126169892737, 36002209323169, 1150944528448671, 36794222701034303, 1176264181904649025, 37603659598247734497, 1202140842962022854879, 38430903315186483621631, 1228586765243005453037313
Offset: 1

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Links

Crossrefs

Cf. A157456, A159674, A159677.

Programs

  • Magma
    [n le 2 select (33)^(n-1) else 32*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 25 2022
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
  • Mathematica
    LinearRecurrence[{32,-1},{1,33},20] (* or *)
CoefficientList[Series[(1+x)/(1-32 x+x^2),{x,0,20}], x] (* Harvey P. Dale, Apr 22 2011 *)
  • PARI
    Vec(x*(1+x)/(1-32*x+x^2) + O(x^20)) \\ Colin Barker, Feb 24 2014
  • PARI
    a(n) = round((16+sqrt(255))^(-n)*(-15-sqrt(255)+(-15+sqrt(255))*(16+sqrt(255))^(2*n))/30) \\ Colin Barker, Jul 25 2016
  • SageMath
    def A159675(n): return chebyshev_U(n-1, 16) + chebyshev_U(n-2, 16)
[A159675(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is a(1)=1; a(2)=31; a(t+2)=32*a(t+1)-a(t) resulting in terms 1, 31, 991, 31681... (A159674).
The b(j) recurrence is b(1)=1; b(2)=33; b(t+2)=32*b(t+1)-b(t) resulting in terms 1, 33, 1055, 33727... (this sequence).
The n(j) recurrence is n(0)=n(1)=0; n(2)=64; n(t+3)=1023*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 64, 65472, 66912384... (A159677).
G.f.: x*(1 + x)/(1 - 32*x + x^2). - Harvey P. Dale, Apr 22 2011
a(n) = (16+sqrt(255))^(-n)*(-15 - sqrt(255) + (-15 + sqrt(255))*(16 + sqrt(255))^(2*n))/30. - Colin Barker, Jul 25 2016
a(n) = ChebyshevU(n-1, 16) + ChebyshevU(n-2, 16). - G. C. Greubel, Sep 25 2022

Extensions

More terms from Harvey P. Dale, Apr 22 2011
New name from Colin Barker, Feb 24 2014
Showing 1-2 of 2 results.

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

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