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%I #94 Jul 17 2023 16:24:22
%S 1,1,2,3,6,9,18,27,54,81,162,243,486,729,1458,2187,4374,6561,13122,
%T 19683,39366,59049,118098,177147,354294,531441,1062882,1594323,
%U 3188646,4782969,9565938,14348907,28697814,43046721,86093442,129140163,258280326,387420489
%N a(0) = 1; thereafter a(2*n + 1) = 3^n, a(2*n + 2) = 2 * 3^n.
%C Row sums of triangle in A123149. - _Philippe Deléham_, May 04 2012
%C This is simply the classic sequence A038754 prefixed by a 1. - _N. J. A. Sloane_, Nov 23 2017
%C Binomial transform is A057960.
%C Range of row n of the circular Pascal array of order 6. - _Shaun V. Ault_, May 30 2014
%C a(n) is also the number of achiral color patterns in a row or cycle of length n using three or fewer colors. Two color patterns are the same if we permute the colors, so ABCAB=BACBA. For a cycle, we can rotate the colors, so ABCAB=CABAB. A row is achiral if it is the same as some color permutation of its reverse. Thus the reversal of ABCAB is BACBA, which is equivalent to ABCAB when we permute A and B. A cycle is achiral if it is the same as some rotation of some color permutation of its reverse. Thus CABAB reversed is BABAC. We can permute A and B to get ABABC and then rotate to get CABAB, so CABAB is achiral. It is interesting that the number of achiral color patterns is the same for rows and cycles. - _Robert A. Russell_, Mar 10 2018
%H G. C. Greubel, <a href="/A182522/b182522.txt">Table of n, a(n) for n = 0..1000</a>
%H Shaun V. Ault and Charles Kicey, <a href="http://arxiv.org/abs/1407.2197">Counting paths in corridors using circular Pascal arrays</a>, arXiv:1407.2197 [math.CO], 2014.
%H Shaun V. Ault and Charles Kicey, <a href="http://dx.doi.org/10.1016/j.disc.2014.05.020">Counting paths in corridors using circular Pascal arrays</a>, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.
%H Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, <a href="https://arxiv.org/abs/2306.03155">Pattern-avoiding stabilized-interval-free permutations</a>, arXiv:2306.03155 [math.CO], 2023.
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H Johann Cigler, <a href="https://arxiv.org/abs/2212.02118">Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials</a>, arXiv:2212.02118 [math.NT], 2022.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F G.f.: (1 + x - x^2) / (1 - 3*x^2).
%F Expansion of 1 / (1 - x / (1 - x / (1 + x / (1 + x)))) in powers of x.
%F a(n+1) = A038754(n).
%F a(n) = Sum_{k=0..n} A123149(n,k). - _Philippe Deléham_, May 04 2012
%F a(n) = (3-(1+(-1)^n)*(3-2*sqrt(3))/2)*sqrt(3)^(n-3) for n>0, a(0)=1. - _Bruno Berselli_, Mar 19 2013
%F a(0) = 1, a(1) = 1, a(n) = a(n-1) + a(n-2) if n is odd, and a(n) = a(n-1) + a(n-2) + a(n-3) if n is even. - _Jon Perry_, Mar 19 2013
%F For odd n = 2m-1, a(2m-1) = T(m,1)+T(m,2)+T(m,3) for triangle T(m,k) of A140735; for even n = 2m, a(2m) = T(m,1)+T(m,2)+T(m,3) for triangle T(m,k) of A293181. - _Robert A. Russell_, Mar 10 2018
%F From _Robert A. Russell_, Oct 21 2018: (Start)
%F a(2m) = S2(m+3,3) - 4*S2(m+2,3) + 5*S2(m+1,3) - 2*S2(m,3).
%F a(2m-1) = S2(m+2,3) - 3*S2(m+1,3) + 2*S2(m,3), where S2(n,k) is the Stirling subset number A008277.
%F a(n) = 2*A001998(n-1) - A124302(n) = A124302(n) - 2*A107767(n-1) = A001998(n-1) - A107767(n-1).
%F a(n) = 2*A056353(n) - A002076(n) = A002076(n) - 2*A320743(n) = A056353(n) - A320743(n).
%F a(n) = A057427(n) + A052551(n-2) + A304973(n). (End)
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 18*x^6 + 27*x^7 + 54*x^8 + ...
%e From _Robert A. Russell_, Mar 10 2018: (Start)
%e For a(4) = 6, the achiral color patterns for rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. Note that for cycles AABB=ABBA and ABBC=ABCA. The achiral patterns for cycles are AAAA, AAAB, AABB, ABAB, ABAC, and ABBC. Note that AAAB and ABAC are not achiral rows.
%e For a(5) = 9, the achiral color patterns (for both rows and cycles) are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, and ABCBA. (End)
%t Join[{1}, RecurrenceTable[{a[1]==1, a[2]==2, a[n]==3 a[n-2]}, a, {n, 40}]] (* _Bruno Berselli_, Mar 19 2013 *)
%t CoefficientList[Series[(1+x-x^2)/(1-3*x^2), {x,0,50}], x] (* _G. C. Greubel_, Apr 14 2017 *)
%t Table[If[EvenQ[n], StirlingS2[(n+6)/2,3] - 4 StirlingS2[(n+4)/2,3] + 5 StirlingS2[(n+2)/2,3] - 2 StirlingS2[n/2,3], StirlingS2[(n+5)/2,3] - 3 StirlingS2[(n+3)/2,3] + 2 StirlingS2[(n+1)/2,3]], {n,0,40}] (* _Robert A. Russell_, Oct 21 2018 *)
%t Join[{1},Table[If[EvenQ[n], 2 3^((n-2)/2), 3^((n-1)/2)],{n,40}]] (* _Robert A. Russell_, Oct 28 2018 *)
%o (PARI) {a(n) = if( n<1, n==0, n--; (n%2 + 1) * 3^(n \ 2))}
%o (Magma) I:=[1,1,2]; [n le 3 select I[n] else 3*Self(n-2): n in [1..40]]; // _Bruno Berselli_, Mar 19 2013
%o (Maxima) makelist(if n=0 then 1 else (1+mod(n-1,2))*3^floor((n-1)/2), n, 0, 40); /* _Bruno Berselli_, Mar 19 2013 */
%o (PARI) my(x='x+O('x^50)); Vec((1+x-x^2)/(1-3*x^2)) \\ _G. C. Greubel_, Apr 14 2017
%o (SageMath)
%o def A182522(n): return (3 -(3-2*sqrt(3))*((n+1)%2))*3^((n-3)/2) + int(n==0)/3
%o [A182522(n) for n in range(41)] # _G. C. Greubel_, Jul 17 2023
%Y Cf. A038754 (essentially the same sequence).
%Y Cf. A008277, A052551, A057427, A057960.
%Y Cf. A123149, A140735, A293181, A304973.
%Y Also row sums of triangle in A169623.
%Y Column 3 of A305749.
%Y Cf. A124302 (oriented), A001998 (unoriented), A107767 (chiral), for rows, varying offsets.
%Y Cf. A002076 (oriented), A056353 (unoriented), A320743 (chiral), for cycles.
%K nonn,easy
%O 0,3
%A _Michael Somos_, May 03 2012
%E Edited by _Bruno Berselli_, Mar 19 2013
%E Definition simplified by _N. J. A. Sloane_, Nov 23 2017
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