Search: id:A057960
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%I A057960
%S A057960 1,2,5,13,35,95,259,707,1931,5275,14411,39371,107563,293867,802859,
%T A057960 2193451,5992619,16372139,44729515,122203307,333865643,912137899,
%U A057960 2492007083,6808289963,18600594091,50817768107,138836724395
%N A057960 Number of three-choice paths along a corridor width 5, starting from 
               one side.
%F A057960 a(n) = sum(b(n, i)) where b(n, 0) = b(n, 6) = 0, b(0, 1) = 1, b(0, n) 
               = 0 if n<> 1 and b(n+1, i) = b(n, i-1) + b(n, i) + b(n, i+1) if 1<=i<=5.
%F A057960 a(n) = 3*a(n-1)-2*a(n-3) = 2*A052948(n)-A052948(n-2).
%F A057960 a(n) = ceiling((1+sqrt(3))^(n+2)/12). - Mitch Harris (harris.mitchell(AT)mgh.harvard.edu), 
               Apr 26 2006
%F A057960 a(n) = floor(a(n-1)*(a(n-1)+1/2)/a(n-2). - Frank Adams-Watters and Max 
               Alekseyev, Apr 25 2006
%F A057960 a(n) = floor(a(n-1)*(1+3^0, 5)) - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jul 25 2003
%F A057960 G.f.: (1-x-x^2)/((1-x)(1-2x-2x^2)); a(n)=1/3+(2+sqrt(3))(1+sqrt(3))^n/
               6+(2-sqrt(3))(1-sqrt(3))^n/6. Binomial transform of A038754 (with 
               extra leading 1). - Paul Barry (pbarry(AT)wit.ie), Sep 16 2003
%e A057960 a(6) = 259 since a(5) = 21+30+25+14+5 so a(6) = (21+30)+(21+30+25)+(30+25+14)+(25+14+5)+(14+5) 
               = 51+76+69+44+19.
%p A057960 with(combstruct): ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), b): ZL1:=Prod(begin_blockP, 
               Z, end_blockP): ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, 
               Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, 
               Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL2, ZL3), b=ZL3], 
               ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, 
               end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, 
               mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, 
               {Q}, unlabelled], size=n), n=2..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%Y A057960 The "three-choice" comes in the recurrence b(n+1, i) = b(n, i-1) + b(n, 
               i) + b(n, i+1) if 1<=i<=5. Narrower corridors produce A000012, A000079, 
               A000129, A001519. An infinitely wide corridor (i.e. just one wall) 
               would produce A005773. Two-choice corridors are A000124, A000125, 
               A000127.
%Y A057960 Sequence in context: A160438 A054657 A024576 this_sequence A007075 A000107 
               A063028
%Y A057960 Adjacent sequences: A057957 A057958 A057959 this_sequence A057961 A057962 
               A057963
%K A057960 nonn
%O A057960 0,2
%A A057960 Henry Bottomley (se16(AT)btinternet.com), May 18 2001

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