A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.

1, 2, 3, 7, 14, 27, 49, 86, 147, 247, 410, 675, 1105, 1802, 2931, 4759, 7718, 12507, 20257, 32798, 53091, 85927, 139058, 225027, 364129, 589202, 953379, 1542631, 2496062, 4038747, 6534865, 10573670, 17108595, 27682327, 44790986, 72473379, 117264433, 189737882

Offset

0,2

Comments

Row sums of A131299 and A131301.

a(n)/a(n-1) tends to phi.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Formula

From Bruno Berselli, May 03 2012: (Start)

G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2).

a(n) = (3*A131269(n)-n-1)/2.

a(n) = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). (End)

a(n) = 3*A000045(n+2)-2*(n+1). - R. J. Mathar, Mar 24 2018

Example

a(4) = 14 = (1 + 1 + 7 + 4 + 1).

Maple

seq(add(3*binomial(floor((n+k)/2), k)-2, k=0..n), n=0..50); # Nathaniel Johnston, Jun 29 2011

Mathematica

LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* Bruno Berselli, May 03 2012 *)

Prog

(PARI) Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38)) \\ Bruno Berselli, May 03 2012
(Magma) /* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]]; // Bruno Berselli, May 03 2012
(Maxima) makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); /* Bruno Berselli, May 03 2012 */

Crossrefs

Cf. A131269, A131299, A131301.

Sequence in context: A300569 A054194 A138651 * A333342 A078043 A294627

Adjacent sequences: A131297 A131298 A131299 * A131301 A131302 A131303

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 27 2007

Extensions

Terms after a(9) from Nathaniel Johnston, Jun 29 2011

New definition from Bruno Berselli, May 03 2012

Status

approved