A159612 INVERT transform of (1, 3, 1, 3, 1, ...).

1, 4, 8, 24, 56, 152, 376, 984, 2488, 6424, 16376, 42072, 107576, 275864, 706168, 1809624, 4634296, 11872792, 30409976, 77901144, 199541048, 511145624, 1309309816, 3353892312, 8591131576, 22006700824, 56371227128, 144398030424, 369882938936, 947475060632, 2427006816376

Offset: 1

Author

Gary W. Adamson, Apr 17 2009

Keywords

• nonn
• easy

Comments

The sequence 1,1,4,8,24,... is an eigensequence of the sequence triangle of 1,3,1,3,1,3,1,..., which is the Riordan array ((1+3x)/(1-x^2),x). - Paul Barry, Feb 10 2011

From Sean A. Irvine, Jun 07 2025: (Start)

Also, the number of walks of length n-1 starting at vertex 1 in the following graph:

0 2

|\ /|

| 1 |

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4 3. (End)

Examples

			a(4) = 24 = (1, 3, 1, 3) dot (8, 4, 1, 1) = (8 + 12, + 1 + 3).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Sean A. Irvine, Walks on Graphs.
• Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 33.
• Index entries for linear recurrences with constant coefficients, signature (1,4).

Crossrefs

Cf. A006131, A119473.

Cf. A026597 (vertices 0, 2, 3, 4), A384604 (missing edge {0,4}).

Programs

Mathematica

LinearRecurrence[{1, 4}, {1, 4}, 50] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)

PARI

Vec(x*(1+3*x)/(1-x-4*x^2) + O(x^40)) \\ Colin Barker, Dec 22 2016

Formula

G.f.: x*(1+3*x)/(1-x-4*x^2). - Philippe Deléham, Mar 01 2012

a(n) = a(n-1) + 4*a(n-2), a(1)=1, a(2)=4. - Vincenzo Librandi, Mar 11 2011

a(n+1) = Sum_{k=0..n} A119473(n,k)*3^k. - Philippe Deléham, Oct 05 2012

a(n) = 2^(-3-n)*((1-sqrt(17))^n*(-5+3*sqrt(17)) + (1+sqrt(17))^n*(5+3*sqrt(17))) / sqrt(17) for n > 0. - Colin Barker, Dec 22 2016

a(n) = A006131(n)+3*A006131(n-1). - R. J. Mathar, Jun 07 2025

E.g.f.: (exp(x/2)*(51*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2)) - 51)/68. - Stefano Spezia, Jun 07 2025