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A182522
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a(0) = 1; thereafter a(2*n + 1) = 3^n, a(2*n + 2) = 2 * 3^n.
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9
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1, 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489
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OFFSET
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0,3
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COMMENTS
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Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, May 30 2014
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
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LINKS
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FORMULA
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G.f.: (1 + x - x^2) / (1 - 3*x^2).
Expansion of 1 / (1 - x / (1 - x / (1 + x / (1 + x)))) in powers of x.
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
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
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
a(2m) = S2(m+3,3) - 4*S2(m+2,3) + 5*S2(m+1,3) - 2*S2(m,3).
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.
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EXAMPLE
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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 + ...
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.
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)
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MATHEMATICA
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Join[{1}, RecurrenceTable[{a[1]==1, a[2]==2, a[n]==3 a[n-2]}, a, {n, 40}]] (* Bruno Berselli, Mar 19 2013 *)
CoefficientList[Series[(1+x-x^2)/(1-3*x^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 14 2017 *)
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 *)
Join[{1}, Table[If[EvenQ[n], 2 3^((n-2)/2), 3^((n-1)/2)], {n, 40}]] (* Robert A. Russell, Oct 28 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, n--; (n%2 + 1) * 3^(n \ 2))}
(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
(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 */
(PARI) my(x='x+O('x^50)); Vec((1+x-x^2)/(1-3*x^2)) \\ G. C. Greubel, Apr 14 2017
(SageMath)
def A182522(n): return (3 -(3-2*sqrt(3))*((n+1)%2))*3^((n-3)/2) + int(n==0)/3
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CROSSREFS
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Cf. A038754 (essentially the same sequence).
Also row sums of triangle in A169623.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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