OFFSET
0,3
COMMENTS
Partial sums of A097073.
This is the sequence A(1,1;1,2;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [Wolfdieter Lang, Oct 18 2010]
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [Wolfdieter Lang, Oct 18 2010]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).
FORMULA
a(n) = 2*A001045(n+1) - 1.
a(n) = (2^(n+2) + 2*(-1)^n - 3)/3.
From Wolfdieter Lang, Oct 18 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=1, a(1)=1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=1=a(1), a(2)=5. Observed by G. Detlefs. See the W. Lang link. (End)
a(n) = 3*a(n-1) - 2*a(n-2) + 4*(-1)^n. - Gary Detlefs, Dec 19 2010
E.g.f.: (1/3)*(2*exp(-x) - 3*exp(x) + 4*exp(2*x)). - G. C. Greubel, Aug 18 2022
MATHEMATICA
CoefficientList[Series[(1-x+2x^2)/((1-x)(1-x-2x^2)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, -2}, {1, 1, 5}, 40] (* Harvey P. Dale, Apr 09 2018 *)
PROG
(Magma) [(2^(n+2) +2*(-1)^n -3)/3: n in [0..40]]; // G. C. Greubel, Aug 18 2022
(SageMath) [(2^(n+2) +2*(-1)^n -3)/3 for n in (0..40)] # G. C. Greubel, Aug 18 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 22 2004
EXTENSIONS
Correction of the homogeneous recurrence and index link added by Wolfdieter Lang, Nov 16 2013
STATUS
approved