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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 Jean-Luc Baril, Sergey Kirgizov, Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018. 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 May 23 05:25 EDT 2019. Contains 323508 sequences. (Running on oeis4.)