A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, 17, 14, 20, 15, 22, 17, 24, 19, 27, 20, 29, 22, 31, 24, 34, 25, 36, 27, 38, 29, 41, 30, 43, 32, 45, 34, 48, 35, 50, 37, 52, 39, 55, 40, 57, 42, 59, 44, 62, 45, 64, 47, 66, 49, 69, 50, 71, 52, 73, 54, 76, 55, 78

Offset: 0

Author

Paul Curtz, Nov 22 2015

Keywords

• nonn
• easy

Comments

A260708 difference table rows have the same nine-step recurrence:

0, 1, 3, 6, 10, 16, 21, 29, 36, 46, 55, 65, 78, 93, ...

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, ... = a(n)

1, 1, 1, 2, -1, 3, -1, 3, -1, 4, -3, 5, -3, 5, ... = b(n)

0, 0, 1, -3, 4, -4, 4, -4, 5, -7, 8, -8, 8, -8, ... (see A042965(n)).

(b(2n) + b(2n+1) = A052901(n+2).)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1500
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Crossrefs

Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.

Programs

Magma

I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

Mathematica

RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* G. C. Greubel, Nov 23 2015 *)

PARI

Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 22 2015

PARI

vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015

Formula

a(2n) = A047282(n). a(2n+1) = A047212(n+1).

a(n) = A260708(n+1) - A260708(n).

a(n+6) = a(n) + period of length 2: repeat 7, 5.

a(2n) + a(2n+1) = 3 + 4*n.

a(n) = n + 1 + (-1)^n*A152467(n+2).

From Colin Barker, Nov 22 2015: (Start)

a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.

G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).

(End)