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A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2). 3
1, 33, 1055, 33727, 1078209, 34468961, 1101928543, 35227244415, 1126169892737, 36002209323169, 1150944528448671, 36794222701034303, 1176264181904649025, 37603659598247734497, 1202140842962022854879, 38430903315186483621631, 1228586765243005453037313 (list; graph; refs; listen; history; text; internal format)
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

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

CROSSREFS

Cf. A157456, A159674, A159677.

Sequence in context: A197358 A299074 A101632 * A162837 A163216 A163567

Adjacent sequences:  A159672 A159673 A159674 * A159676 A159677 A159678

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

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Last modified July 22 16:39 EDT 2019. Contains 325224 sequences. (Running on oeis4.)