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A015449
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Expansion of (1-4*x)/(1-5*x-x^2).
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13
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1, 1, 6, 31, 161, 836, 4341, 22541, 117046, 607771, 3155901, 16387276, 85092281, 441848681, 2294335686, 11913527111, 61861971241, 321223383316, 1667978887821, 8661117822421, 44973567999926, 233528957822051
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = 5*a(n-1) + a(n-2).
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
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MAPLE
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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
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MATHEMATICA
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Transpose[NestList[Flatten[{Rest[#], ListCorrelate[{1, 5}, #]}]&, {1, 1}, 40]][[1]] (* Harvey P. Dale, Mar 23 2011 *)
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 *)
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PROG
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(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)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-4*x)/(1-5*x-x^2)).list()
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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