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

1,2

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

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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).

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A119473.

Sequence in context: A153334 A332871 A116719 * A099176 A190156 A291024

Adjacent sequences: A159609 A159610 A159611 * A159613 A159614 A159615

Keyword

nonn,easy

Author

Gary W. Adamson, Apr 17 2009

Status

approved