A177704 - OEIS

%I #44 Mar 21 2023 15:41:18

%S 1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,

%T 1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,

%U 1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1

%N Period 4: repeat [1, 1, 1, 2].

%C Continued fraction expansion of (3 + 2*sqrt(6))/5.

%C Decimal expansion of 1112/9999.

%C a(n) = A164115(n + 1) = (-1)^(n + 1) * A164117(n + 1) = A138191(n + 3) = A107453(n + 5).

%H Vincenzo Librandi, <a href="/A177704/b177704.txt">Table of n, a(n) for n = 0..1000</a>

%H David Nacin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Nacin/nacin5.html">Van der Laan Sequences and a Conjecture on Padovan Numbers</a>, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

%F a(n) = (5-(-1)^n + i*i^n-i*(-i)^n)/4 where i = sqrt(-1).

%F a(n) = a(n-4) for n > 3; a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2.

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

%F a(n) = 1 + (1-(-1)^n) * (1+i^(n+1))/4 where i = sqrt(-1). - _Bruno Berselli_, Apr 05 2011

%F a(n) = 5/4 - sin(Pi*n/2)/2 - (-1)^n/4. - _R. J. Mathar_, Oct 08 2011

%F a(n) = floor((n+1)*5/4) - floor(n*5/4). - _Hailey R. Olafson_, Jul 23 2014

%F From _Wesley Ivan Hurt_, Jun 15 2016: (Start)

%F a(n+3) - a(n+2) = A219977(n).

%F Sum_{i=0..n-1} a(i) = A001068(n). (End)

%F E.g.f.: (-sin(x) + 3*sinh(x) + 2*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 15 2016

%p A177704:=n->floor((n+1)*5/4) - floor(n*5/4): seq(A177704(n), n=0..100); # _Wesley Ivan Hurt_, Jun 15 2016

%t Table[Floor[(n + 1)*5/4] - Floor[n*5/4], {n, 0, 100}] (* _Wesley Ivan Hurt_, Jun 15 2016 *)

%t LinearRecurrence[{0, 0, 0, 1}, {1, 1, 1, 2}, 100] (* _Vincenzo Librandi_, Jun 16 2016 *)

%o (Magma) &cat[ [1, 1, 1, 2]: k in [1..27] ];

%o (PARI) a(n) = if(n%4==3, 2, 1) \\ _Felix Fröhlich_, Jun 15 2016

%Y Cf. A001068, A107453, A138191, A164115, A164117, A177705, A219977, A236398.

%K nonn,cofr,easy

%O 0,4

%A _Klaus Brockhaus_, May 11 2010