A159677 Expansion of 64*x^2/(1 - 1023*x + 1023*x^2 - x^3).

0, 0, 64, 65472, 66912384, 68384391040, 69888780730560, 71426265522241344, 72997573474949923072, 74603448665133299138304, 76244651538192756769423680, 77921959268584332285051862720, 79636166127841649402566234276224, 81388083860694897105090406378438272

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 integer numbers.

Links

• Colin Barker, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (1023,-1023,1).

Crossrefs

Cf. A157456, A159674, A159675.

Programs

Magma

I:=[0,0,64]; [n le 3 select I[n] else 1023*Self(n-1) - 1023*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 03 2018

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

RecurrenceTable[{a[0]==a[1]==0,a[2]==64,a[n]==1023(a[n-1]-a[n-2])+ a[n-3]}, a,{n,20}] (* Harvey P. Dale, Jan 01 2014 *)
LinearRecurrence[{1023,-1023,1},{0,0,64},20] (* Harvey P. Dale, Jan 01 2014 *)

PARI

concat([0, 0], Vec(64/(-x^3+1023*x^2-1023*x+1) + O(x^20))) \\ Colin Barker, Mar 04 2014

PARI

a(n) = round(-((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510) \\ Colin Barker, Jul 25 2016

SageMath

def A159677(n): return (16/255)*(-1 +chebyshev_U(n, 511) -1021*chebyshev_U(n-1, 511))
[A159677(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, ... (A159674).

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, ... (this sequence).

a(n) = -((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510. - Colin Barker, Jul 25 2016

a(n) = (16/255)*(-1 + ChebyshevU(n, 511) - 1021*ChebyshevU(n-1, 511)). - G. C. Greubel, Sep 25 2022

Extensions

More terms from Harvey P. Dale, Jan 01 2014

New name from Colin Barker, Feb 24 2014

Offset changed to 0 by Colin Barker, Mar 04 2014