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A057960 Number of base 5 n-digit numbers with adjacent digits differing by one or less. 14
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; text; internal format)
OFFSET

0,2

COMMENTS

Or, number of three-choice paths along a corridor of width 5, starting from one side.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

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, Apr 26 2006

a(n) = floor(a(n-1)*(a(n-1)+1/2)/a(n-2). - Franklin T. Adams-Watters and Max Alekseyev, Apr 25 2006

a(n) = floor(a(n-1)*(1+3^0, 5)). - Philippe Deléham, Jul 25 2003

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). - Paul Barry, Sep 16 2003

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

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.

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, Mar 08 2008

MATHEMATICA

Join[{a=1, b=2}, Table[c=(a+b)*2-1; a=b; b=c, {n, 0, 50}]] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)

CoefficientList[Series[(1-x-x^2)/((1-x)*(1-2*x-2*x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 13 2012 *)

PROG

(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

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.

Cf. A155020 (first differences).

Sequence in context: A240609 A054657 A024576 * A227045 A007075 A000107

Adjacent sequences:  A057957 A057958 A057959 * A057961 A057962 A057963

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, May 18 2001

EXTENSIONS

This is the result of merging two identical entries submitted by Henry Bottomley and R. H. Hardin. - N. J. A. Sloane, Aug 14 2012

STATUS

approved

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Last modified February 23 23:00 EST 2018. Contains 299595 sequences. (Running on oeis4.)