This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A077846 #74 May 06 2025 11:51:02
%S A077846 1,3,9,25,69,189,517,1413,3861,10549,28821,78741,215125,587733,
%T A077846 1605717,4386901,11985237,32744277,89459029,244406613,667731285,
%U A077846 1824275797,4984014165,13616579925,37201188181,101635536213,277673448789,758617970005,2072582837589,5662401615189
%N A077846 Expansion of g.f. 1/(1 - 3*x + 2*x^3).
%C A077846 Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1..n+2, s(0) = 1, s(n+2) = 3. - _Herbert Kociemba_, Jun 17 2004
%C A077846 A Whitney transform of 2^n (see Benoit Cloitre formula and A004070). The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - _Paul Barry_, Feb 16 2005
%H A077846 Vincenzo Librandi, <a href="/A077846/b077846.txt">Table of n, a(n) for n = 0..1000</a>
%H A077846 Fan Chung and R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194.
%H A077846 William J. Keith, <a href="https://arxiv.org/abs/1911.04755">Partitions into parts simultaneously regular, distinct, and/or flat</a>, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
%H A077846 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
%H A077846 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-2).
%F A077846 a(n) = 3*a(n-1) - 2*a(n-3) = 2*A057960(n) - 1 = round(2*A028859(n)/sqrt(3) - 1/3) = Sum_{i} b(n, i), where b(n, 0) = b(n, 6) = 0, b(0, 3) = 1, b(0, i) = 0 if i <> 3 and b(n+1, i) = b(n, i-1) + b(n, i) + b(n, i+1) if 0 < i < 6 (i.e., the number of three-choice paths along a corridor width 5, starting from the middle). - _Henry Bottomley_, Mar 06 2003
%F A077846 Binomial transform of A068911. a(n) = (1+sqrt(3))^n*(2+sqrt(3))/3 + (1-sqrt(3))^n*(2-sqrt(3))/3 - 1/3. - _Paul Barry_, Feb 17 2004
%F A077846 a(0)=1; for n >= 1, a(n) = ceiling((1+sqrt(3))*a(n-1)). - _Benoit Cloitre_, Jun 19 2004
%F A077846 a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^j*binomial(j, i-j). - _Benoit Cloitre_, Oct 23 2004
%F A077846 a(n) = 2*(a(n-1) + a(n-2)) + 1, n > 1. - _Gary Detlefs_, Jun 20 2010
%F A077846 a(n) = (2*A052945(n+1) - 1)/3. - _R. J. Mathar_, Mar 31 2011
%F A077846 a(n) = floor((1+sqrt(3))^(n+2)/6). - _Bruno Berselli_, Feb 05 2013
%F A077846 a(n) = (-2 + (1-sqrt(3))^(n+2) + (1+sqrt(3))^(n+2))/6. - _Alexander R. Povolotsky_, Feb 13 2016
%F A077846 E.g.f.: exp(x)*(4*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x) - 1)/3. - _Stefano Spezia_, Mar 02 2024
%t A077846 CoefficientList[Series[1 / (1 - 3 x + 2 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)
%t A077846 LinearRecurrence[{3,0,-2},{1,3,9},40] (* _Harvey P. Dale_, Apr 27 2014 *)
%o A077846 (PARI) a(n)=sum(i=0,n,sum(j=0,n,2^j*binomial(j,i-j)))
%o A077846 (PARI) Vec(1/(1-3*x+2*x^3) + O(x^100)) \\ _Altug Alkan_, Mar 24 2016
%Y A077846 First differences are in A002605.
%Y A077846 Cf. A028859, A052945, A057960, A068911,
%K A077846 nonn,easy
%O A077846 0,2
%A A077846 _N. J. A. Sloane_, Nov 17 2002
%E A077846 Name changed by _Arkadiusz Wesolowski_, Dec 06 2011