|
Search: id:A075118
|
|
|
| A075118 |
|
Variant on Lucas numbers: a(n)=a(n-1)+3*a(n-2) with a(0)=2 and a(1)=1. |
|
+0
2
|
|
| 2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, 22258, 51169, 117943, 271450, 625279, 1439629, 3315466, 7634353, 17580751, 40483810, 93226063, 214677493, 494355682, 1138388161, 2621455207, 6036619690, 13900985311, 32010844381
(list; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
The sequence 4,1,7,.. = 2*0^n+A075118(n) is given by trace(A^n) where A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry (pbarry(AT)wit.ie), Oct 01 2004
|
|
REFERENCES
|
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
|
|
FORMULA
|
a(n) = ((1+sqrt(13))/2)^n+((1-sqrt(13))/2)^n = 2*A006130(n)-A006130(n-1) = A075117(3, n).
G.f.: (2-x)/(1-x-3x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2008]
|
|
EXAMPLE
|
a(4) = a(3)+3*a(2) = 10+3*7 = 31.
|
|
MAPLE
|
a := n -> (Matrix([[1, 2]]). Matrix([[1, 1], [3, 0]])^n)[1, 2]; seq (a(n), n=0..29); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 15 2008]
|
|
MATHEMATICA
|
a[0] = 2; a[1] = 1; a[n_] := a[n] = a[n - 1] + 3a[n - 2]; Table[ a[n], {n, 0, 30}]
|
|
PROGRAM
|
(Other) sage: [lucas_number2(n, 1, -3) for n in xrange(0, 30)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
|
|
CROSSREFS
|
Cf. A000032, A006130, A014551, A072265, A075117.
Sequence in context: A032210 A032135 A032039 this_sequence A100245 A095137 A141488
Adjacent sequences: A075115 A075116 A075117 this_sequence A075119 A075120 A075121
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Henry Bottomley (se16(AT)btinternet.com), Sep 02 2002
|
|
|
Search completed in 0.002 seconds
|
