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

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

Offset: 0

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (32,-1).

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