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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 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 (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 *) CoefficientList[Series[(1-4*x)/(1-5*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *) 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) x='x+O('x^30); Vec((1-4*x)/(1-5*x-x^2)) \\ _G. C. Greubel, Dec 19 2017 CROSSREFS 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 STATUS approved

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Last modified October 19 09:57 EDT 2018. Contains 316349 sequences. (Running on oeis4.)