A131269 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)=6.

1, 2, 3, 6, 11, 20, 35, 60, 101, 168, 277, 454, 741, 1206, 1959, 3178, 5151, 8344, 13511, 21872, 35401, 57292, 92713, 150026, 242761, 392810, 635595, 1028430, 1664051, 2692508, 4356587, 7049124, 11405741, 18454896, 29860669, 48315598, 78176301, 126491934

Offset: 0

Author

Gary W. Adamson, Jun 23 2007

Keywords

• nonn
• easy

Comments

Row sums of triangles A131268 and A131270.

a(n)/a(n-1) tends to phi (A001622).

Examples

			a(4) = 11 = sum of row 4 terms of triangle A131268: (1 + 1 + 5 + 3 + 1), or the reversed terms of triangle A131270, row 4.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A000045, A001622, A065941, A131268, A131270.

Cf. A001595 (first differences).

Programs

GAP

List([0..40], n-> 2*Fibonacci(n+2)-n-1); # G. C. Greubel, Jul 09 2019

Magma

/* By the first comment: */ [&+[2*Binomial(n-Floor((k+1)/2), Floor(k/2))-1: k in [0..n]]: n in [0..40]]; /* Bruno Berselli, May 03 2012 */

Magma

[2*Fibonacci(n+2)-n-1: n in [0..40]]; // G. C. Greubel, Jul 09 2019

Mathematica

LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 6}, 41] (* Bruno Berselli, May 03 2012 *)
Table[2*Fibonacci[n+2]-n-1, {n,0,40}] (* G. C. Greubel, Jul 09 2019 *)

Maxima

makelist(expand(((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2) )/(2^(n+1)*sqrt(5))-n-1), n, 0, 40); /* Bruno Berselli, May 03 2012 */

PARI

Vec((1-x-x^2+2*x^3)/((1-x-x^2)*(1-x)^2)+O(x^40)) \\ Bruno Berselli, May 03 2012

PARI

vector(40, n, n--; 2*fibonacci(n+2)-n-1) \\ G. C. Greubel, Jul 09 2019

Python

prpr = 1
prev = 2
for n in range(2,99):
    current = prpr + prev + n - 2
    print(prpr, end=',')
    prpr = prev
    prev = current  # Alex Ratushnyak, May 02 2012

Sage

[2*fibonacci(n+2)-n-1 for n in (0..40)] # G. C. Greubel, Jul 09 2019

Formula

a(n) = a(n-2) + a(n-1) + n - 2 with n>1, a(0)=1, a(1)=2. - Alex Ratushnyak, May 02 2012

From Bruno Berselli, May 03 2012: (Start)

G.f.: (1-x-x^2+2*x^3)/((1-x-x^2)*(1-x)^2). - Bruno Berselli, May 03 2012

a(n) = A001595(n+1) - n = A006355(n+3) - n - 1 = ((1+sqrt(5))^(n+2) - (1-sqrt(5))^(n+2))/(2^(n+1)*sqrt(5))-n-1. (End)

Extensions

Better definition and more terms from Bruno Berselli, May 03 2012