OFFSET
1,2
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).
FORMULA
a(n) = a(n-1) + 2^((n-1)/2) = 2*a(n-2) + 3 = a(n-1) + A016116(n-1) = A027383(n-1) - 1 = 2*A027383(n-3) + 1 = 4*A052955(n-4) + 1. a(2n) = 2^(n+1) - 3; a(2n+1) = 3*2^n - 3.
From Colin Barker, Jan 12 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n > 5.
G.f.: x*(2*x^4-x^2+x+1) / ((x-1)*(2*x^2-1)). (End)
E.g.f.: 1 + x + x^2/2 - 3*cosh(x) + 2*cosh(sqrt(2)*x) - 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jul 25 2024
MATHEMATICA
LinearRecurrence[{1, 2, -2}, {1, 2, 3, 5, 9}, 50] (* Harvey P. Dale, Sep 11 2016 *)
PROG
(PARI) { for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009
CROSSREFS
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Mar 19 2001
STATUS
approved