A048745 Partial sums of A048654.

1, 5, 14, 36, 89, 217, 526, 1272, 3073, 7421, 17918, 43260, 104441, 252145, 608734, 1469616, 3547969, 8565557, 20679086, 49923732, 120526553, 290976841, 702480238, 1695937320, 4094354881, 9884647085, 23863649054, 57611945196, 139087539449, 335787024097

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = 2*a(n-1) + a(n-2) + 3, a(0)=1, a(1)=5.

a(n) = ( ((4+(5/2)*sqrt(2))*(1+sqrt(2))^n - (4-(5/2)*sqrt(2))*(1-sqrt(2))^n)/ 2*sqrt(2) ) - 3/2.

G.f.: (1+2*x)/((1-x)*(1-2*x-x^2)). - Paul D. Hanna, Feb 22 2005

a(n) = 3*a(n-1) - a(n-2) - a(n-3), n>2, a(0)=1, a(1)=5, a(2)=14. - Philippe Deléham, Dec 16 2008

2*a(n) = A135532(n+2) - 3. - R. J. Mathar, Mar 06 2013

a(n) = (1/2)*( 5*P(n+1) + 3*P(n) - 3), where P(n) = A000129(n). - G. C. Greubel, May 23 2021

Mathematica

t={1, 5}; Do[AppendTo[t, t[[-2]] + 2*t[[-1]] + 3], {n, 40}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
Accumulate[LinearRecurrence[{2, 1}, {1, 4}, 30]] (* or *) LinearRecurrence[{3, -1, -1}, {1, 5, 14}, 30] (* Harvey P. Dale, Aug 03 2020 *)

Prog

(PARI) a(n)=polcoeff((1+2*x)/(1-3*x+x^2+x^3)+x*O(x^n), n) \\ Paul D. Hanna
(Magma) I:=[1, 5, 14]; [n le 3 select I[n] else 3*Self(n-1) -Self(n-2) -Self(n-3): n in [1..31]]; // G. C. Greubel, May 23 2021
(Sage) [(5*lucas_number1(n+1, 2, -1) + 3*lucas_number1(n, 2, -1) -3)/2 for n in (0..30)] # G. C. Greubel, May 23 2021

Crossrefs

Cf. A000129, A005409, A098790.

Sequence in context: A187198 A097507 A052951 * A307462 A292170 A224716

Adjacent sequences: A048742 A048743 A048744 * A048746 A048747 A048748

Keyword

easy,nonn

Author

Barry E. Williams

Status

approved