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A015521
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a(n) = 3*a(n-1) + 4*a(n-2).
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41
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0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861, 3355443, 13421773, 53687091, 214748365, 858993459, 3435973837, 13743895347, 54975581389, 219902325555, 879609302221, 3518437208883, 14073748835533
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OFFSET
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0,3
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COMMENTS
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Inverse binomial transform of powers of 5 (A000351) preceded by 0. - Paul Barry, Apr 02 2003
Number of walks of length n between any two distinct vertices of the complete graph K_5. Example: a(2)=3 because the walks of length 2 between the vertices A and B of the complete graph ABCDE are: ACB, ADB, AEB. - Emeric Deutsch, Apr 01 2004
The terms of the sequence are the number of segments (sides) per iteration of the space-filling Peano-Hilbert curve. - Giorgio Balzarotti, Mar 16 2006
General form: k=4^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878 [From Vladimir Orlovsky, Dec 11 2008]
A further inverse binomial transform generates A015441. - Paul Curtz, Nov 01 2009
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 3's along the central diagonal, and 2's along the subdiagonal and the superdiagonal. [From John M. Campbell, Jul 19 2011]
Pisano period lengths: 1, 1, 2, 2, 10, 2, 6, 2, 6, 10, 10, 2, 6, 6, 10, 2, 4, 6, 18, 10,... - R. J. Mathar, Aug 10 2012
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,4).
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FORMULA
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a(n)=4^n/5-(-1)^n/5. E.g.f.: (exp(4*x)-exp(-x))/5. - Paul Barry, Apr 02 2003
a(n)=sum{k=1..n, binomial(n, k)(-1)^(n+k)*5^(k-1) }. - Paul Barry, May 13 2003
a(2*n) = 4*a(2*n-1) -1, a(2*n+1) = 4*a(2*n) +1. In general this is true for all sequences of the type a(n) +a(n+1) = q^(n): i.e. a(2*n) = q*a(2n-1) -1 and a(2*n+1) = q*a(2*n) +1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 15 2003
a(n)=4^(n-1) - a(n-1). G.f.: x/(1-3*x-4*x^2). - Emeric Deutsch, Apr 01 2004
a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)*3^(n-2k)*4^k} - Paul Barry, Jul 29 2004
a(n)=4*a(n-1)-(-1)^n, n>0, a(0)=0. - Paul Barry, Aug 25 2004
a(n)=Sum_{k, 0<=k<=n} A155161(n,k)*2^(n-k), n>=1 . [Philippe DELEHAM, Jan 27 2009]
a(n)=round(4^n/5) [From Mircea Merca, Dec 28 2010]
The logarithmic generating function 1/5*log((1+x)/(1-4*x)) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + ... has compositional inverse 5/(4+exp(-5*x)) - 1, the e.g.f. for a signed version of A213127. - Peter Bala, Jun 24 2012
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MAPLE
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seq(round(4^n/5), n=0..25) [From Mircea Merca, Dec 28 2010]
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MATHEMATICA
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k=0; lst={k}; Do[k=4^n-k; AppendTo[lst, k], {n, 0, 5!}]; lst [From Vladimir Orlovsky, Dec 11 2008]
LinearRecurrence[{3, 4}, {0, 1}, 30] (* Harvey P. Dale, Jun 26 2012 *)
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PROG
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(Sage) [lucas_number1(n, 3, -4) for n in xrange(0, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(MAGMA) [Round(4^n/5): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011
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CROSSREFS
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Cf. A015518, A001045.
Cf. A001045, A078008, A097073, A115341, A015518, A054878, A213127.
Cf. A201455.
Sequence in context: A163774 A197074 A014985 * A146279 A098619 A086608
Adjacent sequences: A015518 A015519 A015520 * A015522 A015523 A015524
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KEYWORD
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nonn,easy
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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