OFFSET
0,3
COMMENTS
A001047(n) = p(0) where p(x) is the unique degree-(n-1) polynomial such that p(k) = a(k) for k = 1, 2, ..., n.
a(n) = p(n) where p(x) is the unique degree-(n-1) polynomial such that p(k) = (-1)^k for k = 0, 1, ..., n-1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (-3, -2).
FORMULA
G.f.: x / ((1 + x) * (1 + 2*x)) = 1 / (1+x) - 1 / (1 + 2*x).
E.g.f.: exp(-x) - exp(-2*x). a(n) = -2 * a(n-1) - (-1)^n if n>0.
a(n) = -(-1)^n * A000225(n). a(n) = -3 * a(n-1) - 2 * a(n-2) if n>1.
REVERT transform is A001003 omitting a(0)=0.
INVERT transform is A108520.
2^n = a(n+1)^2 - a(n) * a(n+2).
EXAMPLE
G.f. = x - 3*x^2 + 7*x^3 - 15*x^4 + 31*x^5 - 63*x^6 + 127*x^7 - 255*x^8 + 511*x^9 + ...
MATHEMATICA
a[ n_] := If[ n<0, 0, (-1)^n (1 - 2^n)];
LinearRecurrence[{-3, -2}, {0, 1}, 50] (* G. C. Greubel, Aug 09 2018 *)
PROG
(PARI) {a(n) = if( n<0, 0, (-1)^n * (1 - 2^n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( x / ((1 + x) * (1 + 2*x)) + x * O(x^n), n))};
(Magma) [(-1)^n*(1 - 2^n): n in [0..50]]; // G. C. Greubel, Aug 09 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Michael Somos, May 19 2013
STATUS
approved