A218735 -id:A218735

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A237132

Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.

+10
6

3, 4, 5, 9, 12, 23, 31, 60, 81, 157, 212, 411, 555, 1076, 1453, 2817, 3804, 7375, 9959, 19308, 26073, 50549, 68260, 132339, 178707, 346468, 467861, 907065, 1224876, 2374727, 3206767, 6217116, 8395425, 16276621, 21979508, 42612747, 57543099, 111561620

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 704 = 0.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

FORMULA

a(n) = 3*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(3*x^2+7*x+3) / ((x^2-x-1)*(x^2+x-1)).

a(n) = F(n+2) + (-1)^n*F(n-3), n>1, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014

Let h(n) = hypergeom([(1 - n)/2, n mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 3. - Peter Luschny, Sep 03 2019

EXAMPLE

9 is in the sequence because (x, y) = (9, 23) is a solution to x^2 - 3xy + y^2 + 11 = 0.

PROG

(PARI) Vec(-x*(x-1)*(3*x^2+7*x+3)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

CROSSREFS

Cf. A001519, A005248, A055819, A237133, A218735.

KEYWORD

nonn,easy

AUTHOR

Colin Barker, Feb 04 2014

STATUS

approved

A237133

Values of x in the solutions to x^2 - 3xy + y^2 + 19 = 0, where 0 < x < y.

+10
4

4, 5, 7, 11, 17, 28, 44, 73, 115, 191, 301, 500, 788, 1309, 2063, 3427, 5401, 8972, 14140, 23489, 37019, 61495, 96917, 160996, 253732, 421493, 664279, 1103483, 1739105, 2888956, 4553036, 7563385, 11920003, 19801199, 31206973, 51840212, 81700916, 135719437

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1216 = 0.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

FORMULA

a(n) = 3*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(4*x^2+9*x+4) / ((x^2-x-1)*(x^2+x-1)).

a(n) = (1/2) * (F(n+4) + (-1)^n*F(n-5)), n>4, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014

EXAMPLE

11 is in the sequence because (x, y) = (11, 28) is a solution to x^2 - 3xy + y^2 + 19 = 0.

MATHEMATICA

LinearRecurrence[{0, 3, 0, -1}, {4, 5, 7, 11}, 40] (* Harvey P. Dale, Dec 15 2014 *)