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A015523
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a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.
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23
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0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469, 5477472, 22964761, 96281643, 403668734, 1692414417, 7095586921, 29748832848, 124724433149, 522917463687, 2192374556806, 9191710988853, 38537005750589
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OFFSET
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0,3
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COMMENTS
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Contribution from Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds for n>=1 to 24 red king vectors, i.e. A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).
Inverse binomial transform of A052918 (with extra leading 0).
(End)
First differences in A197189. - Bruno Berselli, Oct 11 2011
Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12,120, 12, 12, 3, 4, 24,288, 12, 72, 12,... - R. J. Mathar, Aug 10 2012
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (3,5).
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FORMULA
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a(n) = 3*a(n-1) + 5*a(n-2).
a(n) = ((3/2+sqrt(29)/2)^n -(3/2-sqrt(29)/2)^n)/sqrt(29); a(n)=sum(k=0..floor((n-1)/2), binomial(n-k-1, k)*5^k*3^(n-2*k-1) ). - Paul Barry, Jul 20 2004
G.f.: x/(1-3*x-5*x^2). - R. J. Mathar, Nov 16 2007
Contribution from Johannes W. Meijer, Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = (A072263(n)+A015523(n)*sqrt(29))/2
Limit(A072263(n)/A015523(n)) = sqrt(29). (End)
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MATHEMATICA
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Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* From Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* From Alonso del Arte, Jan 16 2011 *)
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PROG
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(Sage) [lucas_number1(n, 3, -5) for n in xrange(0, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011
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CROSSREFS
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Cf. A072263, A072264, A152187, A179606, A197189.
Sequence in context: A037793 A037093 A135926 * A127363 A133444 A126875
Adjacent sequences: A015520 A015521 A015522 * A015524 A015525 A015526
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KEYWORD
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nonn,easy
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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