A110679 - OEIS

%I #43 Sep 08 2022 08:45:20

%S 1,2,11,44,189,798,3383,14328,60697,257114,1089155,4613732,19544085,

%T 82790070,350704367,1485607536,6293134513,26658145586,112925716859,

%U 478361013020,2026369768941,8583840088782,36361730124071,154030760585064,652484772464329

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

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

%C 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

%H Colin Barker, <a href="/A110679/b110679.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert Munafo, <a href="http://www.mrob.com/pub/math/seq-floretion.html">Sequences Related to Floretions</a>

%H Hermann Stamm-Wilbrandt, <a href="/A110679/a110679_2.svg">6 interlaced bisections</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,1).

%F 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).

%F a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*Fibonacci(3*k+1). - _Gary Detlefs_, Jan 22 2013

%F a(n) = (Fibonacci(3*n+2)+(-1)^n)/2. - _Roberto Tauraso_, Mar 21 2017

%F From _Colin Barker_, Mar 21 2017: (Start)

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

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

%F (End)

%F a(n) = -(-1)^n * A049651(-1 - n) for all n in Z. - _Michael Somos_, Mar 26 2017

%F 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

%F 2*a(n) = A015448(n+1)+(-1)^n. - _R. J. Mathar_, Oct 03 2021

%p 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

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

%o (PARI) Vec((1 - x) / ((1 + x)*(1 - 4*x - x^2)) + O(x^30)) \\ _Colin Barker_, Mar 21 2017

%o (PARI) {a(n) = -(-1)^n * (fibonacci(-2 - 3*n)\2)}; /* _Michael Somos_, Mar 26 2017 */

%o (Magma) [(Fibonacci(3*n+2) +(-1)^n)/2: n in [0..30]]; // _G. C. Greubel_, Apr 19 2019

%o (Sage) [(fibonacci(3*n+2) +(-1)^n)/2 for n in (0..30)] # _G. C. Greubel_, Apr 19 2019

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

%K easy,nonn

%O 0,2

%A _Creighton Dement_, Aug 02 2005

%E Typo in program code fixed by _Creighton Dement_, Dec 11 2009