OFFSET
0,3
COMMENTS
Case for base 2 see A000225: 2^n - 1.
If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).
FORMULA
G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012
Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022
a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022
MAPLE
nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);
# End program 1
with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);
# End program 2
# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012
# third Maple program:
a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:
seq (a(n), n=0..60); # Alois P. Heinz, Sep 23 2012
MATHEMATICA
Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)
PROG
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -3, 0, 4, 0]^n*[0; 1; 2; 4])[1, 1] \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Patrick De Geest, Feb 15 1999
STATUS
approved