OFFSET
1,1
COMMENTS
Given explicitly as the numerators of the convergents to the continued fractions
[2,(1,1,1,4)^i,5,(1,1,1,4)^{i-1},1,2] (for n odd and i = (n-1)/2)
and
[2,(1,1,1,4)^i,1,1,2,(1,4,1,1)^i,1] (for n even and i = n/2 - 1).
a(n) == 10 + 8*(-1)^n (mod 21). - Robert Israel, Dec 13 2017
REFERENCES
E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 02 2022
LINKS
Colin Barker, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).
FORMULA
a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6).
From Colin Barker, Dec 13 2017: (Start)
G.f.: x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)).
a(n) = a(n-1) + 254*a(n-2) - 254*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)
4*a(n) = 7*( -A077412(n) +17*A077412(n-1) ) -3*( (-1)^n*A077412(n) -15*(-1)^n*A077412(n-1) ) - 2 . - R. J. Mathar, Mar 07 2022
EXAMPLE
For n = 3 the pair is (x,y) = (653,247).
MAPLE
f:= gfun:-rectoproc({a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6), a(1)=2, a(2)=18, a(3)=653, a(4)=4701, a(5)= 165986, a(6)=1194162}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Dec 12 2017
PROG
(PARI) Vec(x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)) + O(x^25)) \\ Colin Barker, Dec 13 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jeffrey Shallit, Dec 11 2017
EXTENSIONS
More terms from Robert Israel, Dec 12 2017
STATUS
approved