A110679 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 1, a(1) = 2, a(2) = 11.

1, 2, 11, 44, 189, 798, 3383, 14328, 60697, 257114, 1089155, 4613732, 19544085, 82790070, 350704367, 1485607536, 6293134513, 26658145586, 112925716859, 478361013020, 2026369768941, 8583840088782, 36361730124071, 154030760585064, 652484772464329

Offset: 0

Author

Creighton Dement, Aug 02 2005

Keywords

• easy
• nonn

Comments

2tesseq[A*B*cyc(A)] (see program code) gives an alternative formula for A110528.

a(n) is the number of tilings of a 2 X n rectangle by using 1 X 1 squares, dominoes and right trominoes. - Roberto Tauraso, Mar 21 2017

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Robert Munafo, Sequences Related to Floretions
• Hermann Stamm-Wilbrandt, 6 interlaced bisections
• Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Crossrefs

Cf. A110528, A110680, A001076, A110526, A110526, A033887.

Programs

Magma

[(Fibonacci(3*n+2) +(-1)^n)/2: n in [0..30]]; // G. C. Greubel, Apr 19 2019

Maple

seriestolist(series((-1+x)/((x+1)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -1jesseq[A*B*cyc(A)] with A = - 'j + 'k - 'ii' - 'ij' - 'ik' and B = - .5'i - .5i' - .5'ii' + .5'jj' - .5'kk' + .5'jk' + .5'kj' - .5e

Mathematica

a[n_] := (Fibonacci[3*n+2] + (-1)^n)/2; a /@ Range[0, 22] (* Giovanni Resta, Mar 21 2017 *)

PARI

Vec((1 - x) / ((1 + x)*(1 - 4*x - x^2)) + O(x^30)) \\ Colin Barker, Mar 21 2017

PARI

{a(n) = -(-1)^n * (fibonacci(-2 - 3*n)\2)}; /* Michael Somos, Mar 26 2017 */

Sage

[(fibonacci(3*n+2) +(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

Formula

Program "FAMP" finds: 2*(-1^(n+1)) = A110528(n) - A001076(n+1) - 2*a(n). Program "Superseeker" finds: a(n) = A110526(n+1) - A110526(n); a(n) + a(n+1) = A033887(n+1).

a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*Fibonacci(3*k+1). - Gary Detlefs, Jan 22 2013

a(n) = (Fibonacci(3*n+2)+(-1)^n)/2. - Roberto Tauraso, Mar 21 2017

From Colin Barker, Mar 21 2017: (Start)

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

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

(End)

a(n) = -(-1)^n * A049651(-1 - n) for all n in Z. - Michael Somos, Mar 26 2017

a(2*n) = A254627(2*n+1); a(2*n+1) = A077259(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019

2*a(n) = A015448(n+1)+(-1)^n. - R. J. Mathar, Oct 03 2021

Extensions

Typo in program code fixed by Creighton Dement, Dec 11 2009