|
|
A060816
|
|
a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.
|
|
20
|
|
|
1, 2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602, 3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667, 8716961002, 26150883007, 78452649022, 235357947067, 706073841202
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1), A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010
If n > 0 and H = hex number (A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021
|
|
LINKS
|
|
|
FORMULA
|
The following is a summary of formulas added over the past 18 years.
a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001
a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019
a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
|
|
MATHEMATICA
|
LinearRecurrence[{4, -3}, {1, 2, 7}, 30] (* Harvey P. Dale, Nov 15 2022 *)
|
|
PROG
|
(PARI) { for (n=0, 200, if (n>1, a1=a=3*a1 + 1, if (n==0, a=1, a1=a=2)); write("b060816.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 13 2009
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|