A117585 a(n) = 2*a(n-1) + a(n-2) + n.

1, 3, 9, 24, 61, 151, 369, 896, 2169, 5243, 12665, 30584, 73845, 178287, 430433, 1039168, 2508785, 6056755, 14622313, 35301400, 85225133, 205751687, 496728529, 1199208768, 2895146089, 6989500971, 16874148057, 40737797112, 98349742309

Offset: 0

Author

Gary W. Adamson, Mar 29 2006

Keywords

• nonn

Comments

A modified Pellian sequence.

Examples

			a(4) = 61 = 2*(a(3)) + a(2) + 4 = 2*24 + 9 + 4.
a(4) = 61 = sum of terms in row 5 of A117584: 1 + 5 + 9 + 17 + 29.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Crossrefs

Row sums of triangle A117584.

Programs

Magma

P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >;
[(1/2)*(P(n+2) + 2*P(n+1) - (n+2)): n in [0..30]]; // G. C. Greubel, Jul 05 2021

Mathematica

RecurrenceTable[{a[0]==1,a[1]==3, a[n]==2a[n-1]+a[n-2]+n}, a, {n,30}] (* or *) LinearRecurrence[{4,-4,0,1}, {1,3,9,24}, 30] (* Harvey P. Dale, Mar 11 2015 *)

Sage

def a(n): return (1/2)*(lucas_number1(n+2,2,-1) + 2*lucas_number1(n+1,2,-1) -n-2)
[a(n) for n in (0..30)] # G. C. Greubel, Jul 05 2021

Formula

a(n)/a(n-1) tends to 1 + sqrt(2) = 2.414213562...(a(14)/a(13) = 430433/178287 = 2.4142702...).

a(n) = (1/2)*(Pell(n+2) + 2*Pell(n+1) - n - 2), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007

From R. J. Mathar, Aug 05 2009: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

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

Extensions

Terms from a(20) on corrected by R. J. Mathar, Aug 05 2009