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

0,3

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).

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A157456, A159674, A159675.

Sequence in context: A123394 A069445 A227604 * A013832 A320862 A034989

Adjacent sequences: A159674 A159675 A159676 * A159678 A159679 A159680

Keyword

nonn,easy

Author

Paul Weisenhorn, Apr 19 2009

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

Status

approved