OFFSET
0,3
COMMENTS
Row m=5 of A135597.
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 A003947. 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)
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
Milan 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..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
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
a(n) = Sum_{k=0..n} A046854(n-1,k)*5^k. - R. J. Mathar, Feb 10 2024
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}, 30] (* Vincenzo Librandi, Nov 06 2012 *)
CoefficientList[Series[(1-4*x)/(1-5*x-x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 19 2017 *)
Sum[Fibonacci[Range[30] +k-2, 5], {k, 0, 1}] (* G. C. Greubel, Oct 23 2019 *)
PROG
(Magma) [n le 2 select 1 else 5*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 06 2012
(PARI) Vec((1-4*x)/(1-5*x-x^2) +O('x^30)) \\ _G. C. Greubel, Dec 19 2017
(Sage)
def A015449_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-4*x)/(1-5*x-x^2)).list()
A015449_list(30) # G. C. Greubel, Oct 23 2019
(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=5*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Oct 23 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved