A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2).

1, 33, 1055, 33727, 1078209, 34468961, 1101928543, 35227244415, 1126169892737, 36002209323169, 1150944528448671, 36794222701034303, 1176264181904649025, 37603659598247734497, 1202140842962022854879, 38430903315186483621631, 1228586765243005453037313

Offset: 1

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 = 1..650
• Index entries for linear recurrences with constant coefficients, signature (32,-1).

Crossrefs

Cf. A157456, A159674, A159677.

Programs

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

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 *)

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

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

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

Extensions

More terms from Harvey P. Dale, Apr 22 2011

New name from Colin Barker, Feb 24 2014