A213586 Antidiagonal sums of the convolution array A213584.

1, 6, 20, 51, 112, 224, 421, 758, 1324, 2263, 3808, 6336, 10457, 17158, 28036, 45675, 74256, 120544, 195485, 316790, 513116, 830831, 1344960, 2176896, 3523057, 5701254, 9225716, 14928483, 24155824, 39086048, 63243733, 102331766

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• easy

Comments

a(n) is the number of bit strings of length n+5 with the pattern 01 at least thrice, and without the pattern 110, see example. - John M. Campbell, Jan 25 2013

Examples

			From _John M. Campbell_, Jan 25 2013: (Start)
There are a(3) = 20 bit strings of length 3+5 with the pattern 01 at least thrice, and without the pattern 110:
00010101, 00100101, 00101001, 00101010, 00101011,
01000101, 01001001, 01001010, 01001011, 01010001,
01010010, 01010011, 01010100, 01010101, 01010111,
10010101, 10100101, 10101001, 10101010, 10101011.
(End)
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Crossrefs

Cf. A213500, A213584, A213590.

Programs

GAP

List([1..40], n-> Fibonacci(n+8) -(21+10*n+2*n^2)) # G. C. Greubel, Jul 06 2019

Magma

[Fibonacci(n+8) -(21+10*n+2*n^2): n in [1..40]]; // G. C. Greubel, Jul 06 2019

Mathematica

(See A213584.)
With[{F = Fibonacci}, Table[F[n+8] -(21+10*n+2*n^2), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)

PARI

vector(40, n, fibonacci(n+8) -(21+10*n+2*n^2)) \\ G. C. Greubel, Jul 06 2019

Sage

[fibonacci(n+8) -(21+10*n+2*n^2) for n in (1..40)] # G. C. Greubel, Jul 06 2019

Formula

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

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

a(n) = Fibonacci(n+8) - (21 + 10*n + 2*n^2). - G. C. Greubel, Jul 06 2019