A159683 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 3*n(j) + 1 = a(j)*a(j) and 5*n(j) + 1 = b(j)*b(j) with positive integer numbers.

0, 16, 1008, 62496, 3873760, 240110640, 14882985936, 922505017408, 57180428093376, 3544264036771920, 219687189851765680, 13617061506772700256, 844038126230055650208, 52316746764756677612656, 3242794261288683956334480, 201000927453133648615125120

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..559

Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Formula

The a(j) recurrence is a(1)=1, a(2)=7, a(t+2) = 8*a(t+1) - a(t) resulting in terms 1, 7, 55. 433, 3409, ... (A070997).

The b(j) recurrence is b(1)=1, b(2)=9, b(t+2) = 8*b(t+1) - b(t) resulting in terms 1, 9, 71, 559, 4401, ... (A057080).

The n(j) recurrence is n(0) = n(1) = 0, n(2)=16, n(t+3) = 63*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 16, 1008, 62496, ... (this sequence).

From Colin Barker, Sep 25 2015: (Start)

a(n) = 63*a(n-1) - 63*a(n-2) + a(n-3) for n>3.

G.f.: 16*x^2 / ((1-x)*(1-62*x+x^2)). (End)

a(n) = (-8+(4+sqrt(15))*(31+8*sqrt(15))^(-n) -(-4+sqrt(15))*(31+8*sqrt(15))^n)/30. - Colin Barker, Mar 03 2016

a(n) = (4/15)*(-1 + ChebyshevU(n, 31) - 61*ChebyshevU(n-1, 31)). - G. C. Greubel, Sep 27 2022

Maple

for a from 1 by 2 to 100000 do b:=sqrt((5*a*a-2)/3): if (trunc(b)=b) then
n:=(a*a-1)/3: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: end if: end do:
# Second program
seq((4/15)*(simplify(ChebyshevU(n, 31) - 61*ChebyshevU(n-1, 31)) -1), n=1..30); # G. C. Greubel, Sep 27 2022

Mathematica

CoefficientList[Series[16*x/((1-x)*(1-62*x+x^2)), {x, 0, 30}], x] (* G. C. Greubel, Jun 02 2018 *)
LinearRecurrence[{63, -63, 1}, {0, 16, 1008}, 30] (* Harvey P. Dale, May 07 2022 *)

Prog

(PARI) concat(0, Vec(16*x^2/((1-x)*(1-62*x+x^2)) + O(x^30))) \\ Colin Barker, Sep 25 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!(16*x^2/((1-x)*(1-62*x+x^2)))); // G. C. Greubel, Jun 02 2018
(SageMath) [(4/15)*(-1 + chebyshev_U(n, 31) - 61*chebyshev_U(n-1, 31)) for n in range(1, 30)] # G. C. Greubel, Sep 27 2022

Crossrefs

Cf. A057080, A070997, A157456, A245031.

Sequence in context: A211090 A160012 A070307 * A197104 A358051 A067490

Adjacent sequences: A159680 A159681 A159682 * A159684 A159685 A159686

Keyword

nonn,easy

Author

Paul Weisenhorn, Apr 19 2009

Status

approved