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A057960
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Number of base-5 (n+1)-digit numbers starting with a zero and with adjacent digits differing by one or less.
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14
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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, 379308985003, 1036291418795, 2831200807595
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OFFSET
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0,2
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COMMENTS
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Or, number of three-choice paths along a corridor of width 5 and length n, starting from one side.
If b(n) is the number of three-choice paths along a corridor of width 5 and length n, starting from any of the five positions at the beginning of the corridor, then b(n) = a(n+2) for n >= 0. - Pontus von Brömssen, Sep 06 2021
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LINKS
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FORMULA
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a(n) = Sum_{0 <= i <= 6} 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) = ceiling((1+sqrt(3))^(n+2)/12). - Mitch Harris, Apr 26 2006
G.f.: (1-x-x^2)/((1-x)*(1-2*x-2*x^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). (End)
More generally, it appears that a(base,n) = a(base-1,n) + 3^(n-1) for base >= n; a(base,n) = a(base-1,n) + 3^(n-1)-2 when base = n-1. - R. H. Hardin, Dec 26 2006
a(n) = 2*a(n-1) + 2*a(n-2) - 1 for n >= 2, a(0) = 1, a(1) = 2. - Philippe Deléham, Mar 01 2024
E.g.f.: exp(x)*(1 + 2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Mar 02 2024
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EXAMPLE
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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
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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+2), n=0..28); # Zerinvary Lajos, Mar 08 2008
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MATHEMATICA
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CoefficientList[Series[(1-x-x^2)/((1-x)*(1-2*x-2*x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 13 2012 *)
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PROG
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(S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>1)+($[i+1]`-$[i]`>1)) # R. H. Hardin, Dec 26 2006
(Python)
from functools import cache
@cache
def B(n, j):
if not 0 <= j < 5:
return 0
if n == 0:
return j == 0
return B(n - 1, j - 1) + B(n - 1, j) + B(n - 1, j + 1)
return sum(B(n, j) for j in range(5))
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CROSSREFS
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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.
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KEYWORD
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nonn,base,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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