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A057960
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Number of three-choice paths along a corridor of width 5, starting from one side.
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7
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1, 2, 5, 13, 35, 95, 259, 707, 1931, 5275, 14411, 39371, 107563, 293867, 802859, 2193451, 5992619, 16372139, 44729515, 122203307, 333865643, 912137899, 2492007083, 6808289963, 18600594091, 50817768107, 138836724395
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| 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.
a(n) = 3*a(n-1)-2*a(n-3) = 2*A052948(n)-A052948(n-2).
a(n) = ceiling((1+sqrt(3))^(n+2)/12). - Mitch Harris (harris.mitchell(AT)mgh.harvard.edu), Apr 26 2006
a(n) = floor(a(n-1)*(a(n-1)+1/2)/a(n-2). - Frank Adams-Watters and Max Alekseyev, Apr 25 2006
a(n) = floor(a(n-1)*(1+3^0, 5)) - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2003
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
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EXAMPLE
| 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.
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MAPLE
| 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
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MATHEMATICA
| Join[{a=1, b=2}, Table[c=(a+b)*2-1; a=b; b=c, {n, 0, 50}]] (*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), 22 Nov 2010*)
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CROSSREFS
| 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.
Appears to be essentially the same as A126359. [From DELEHAM Philippe, Sep 28 2011]
Sequence in context: A160438 A054657 A024576 * A007075 A000107 A063028
Adjacent sequences: A057957 A057958 A057959 * A057961 A057962 A057963
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KEYWORD
| nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), May 18 2001
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