OFFSET
0,2
COMMENTS
The radius of convergence of g.f. equals 1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)) = 0.216845335...
The following remarks assume an offset of 1. This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. It is the case P1 = 4, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014
LINKS
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (4,2,4,-1).
FORMULA
G.f.: (1 - x^2)/(1 - 4*x - 2*x^2 - 4*x^3 + x^4).
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
a(n) = 1/(2*sqrt(2))*(T(n,1 + sqrt(2)) - T(n,1 - sqrt(2))), where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 2]. Note, the bottom left element of the matrix M^n gives the Lucas sequence A000129.
a(n) = U(n-1,exp(2*i*Pi/8))*U(n-1,exp(-2*i*Pi/8)) = U(n-1,(1 + i)/sqrt(2))*U(n-1,(1 - i)/sqrt(2)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
The o.g.f. is the Chebyshev transform of the rational function x/(1 - 4*x - 4*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
EXAMPLE
G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
The terms equal the norm of the complex coefficients in the expansion:
1/(1 - 2*x - i*x^2) = 1 + 2*x + (4 + i)*x^2 + (8 + 4*i)*x^3 + (15 + 12*i)*x^4 + (26 + 32*i)*x^5 + (40 + 79*i)*x^6 + (48 + 184*i)*x^7 +...
so that
a(1) = 2^2, a(2) = 4^2 + 1, a(3) = 8^2 + 4^2, a(4) = 15^2 + 12^2, a(5) = 26^2 + 32^2, ...
MATHEMATICA
LinearRecurrence[{4, 2, 4, -1}, {1, 4, 17, 80}, 25] (* Jean-François Alcover, Nov 02 2019 *)
PROG
(PARI) {a(n)=norm(polcoeff(1/(1-2*x-I*x^2+x*O(x^n)), n))}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 21 2012
STATUS
approved