OFFSET
1,2
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
Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (32,-1).
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
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Apr 19 2009
EXTENSIONS
More terms from Harvey P. Dale, Apr 22 2011
New name from Colin Barker, Feb 24 2014
STATUS
approved