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A015449 Expansion of (1-4*x)/(1-5*x-x^2). 13
1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686, 11913527111, 61861971241, 321223383316, 1667978887821, 8661117822421, 44973567999926, 233528957822051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Binomial transform of A152187. - Johannes W. Meijer, Aug 01 2010

For n>=1, row sums of triangle

m/k.|..0.....1.....2.....3.....4.....5.....6.....7

==================================================

.0..|..1

.1..|..1.....5

.2..|..1.....5....25

.3..|..1....10....25.....125

.4..|..1....10....75.....125....625

.5..|..1....15....75.....500....625....3125

.6..|..1....15...150.....500...3125....3125...15625

.7..|..1....20...150....1250...3125...18750...15625...78125

which is triangle for numbers 5^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012

a(n+1) is (for n>=0) the number of length-n strings of 6 letters {0,1,2,3,4,5} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). [Joerg Arndt, Oct 11 2012]

With offset 1, the sequence is the INVERT transform (1, 5, 5*4, 5*4^2, 5*4^3,...); i.e. of (1, 5, 20, 80, 320, 1280,...). The sequence can also be obtained by taking powers of the matrix [(1,5); (1,4)] and extracting the upper left terms. - Gary W. Adamson, Jul 31 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook)

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (5,1).

FORMULA

a(n) = 5*a(n-1) + a(n-2).

a(n) = Sum_{k, 0<=k<=n} 4^k*A055830(n,k) . - Philippe Deléham, Oct 18 2006

G.f.: (1-4*x)/(1-5*x-x^2). [Philippe Deléham, Nov 20 2008]

a(n) = (1/2)*[(5/2)+(1/2)*sqrt(29)]^n-(3/58)*[(5/2)+(1/2)*sqrt(29)]^n*sqrt(29)+(1/2)*[(5/2)-(1/2) *sqrt(29)]^n+(3/58)*sqrt(29)*[(5/2)-(1/2)*sqrt(29)]^n, with n>=0. [Paolo P. Lava, Nov 21 2008]

For n>=2, a(n)=F_n(5)+F_(n+1)(5), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x)=sum{i=0,...,floor((n-1)/2)}C(n-i-1,i)x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012

MAPLE

a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # Zerinvary Lajos, Jul 26 2006

MATHEMATICA

Transpose[NestList[Flatten[{Rest[#], ListCorrelate[{1, 5}, #]}]&, {1, 1}, 40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)

LinearRecurrence[{5, 1}, {1, 1}, 50] (* Vincenzo Librandi, Nov 06 2012 *)

PROG

(MAGMA) [n le 2 select 1 else 5*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012

CROSSREFS

Row m=5 of A135597.

Cf. A084057, A108306, A164549.

Sequence in context: A047665 A003128 A058146 * A162475 A036729 A275403

Adjacent sequences:  A015446 A015447 A015448 * A015450 A015451 A015452

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified September 26 06:24 EDT 2017. Contains 292502 sequences.