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

Author

Barry E. Williams

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000129, A005409, A098790.

Programs

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

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 *)

PARI

a(n)=polcoeff((1+2*x)/(1-3*x+x^2+x^3)+x*O(x^n),n) \\ Paul D. Hanna

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

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