1,2

a(n)-1 counts ternary numbers with no 0 digit (A007931) and at least one 2 digit, where the total of ternary digits is <= n. E.g. a(4)-1 = 7: 2 12 21 22 112 121 211. - Frank Ellermann (frank.ellermann(AT)t-online.de), Dec 02, 2001

A107909(a(n-1)) = A000079(n-1) = 2^(n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005

a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 iff i=1 or j=n or |i-j|<=1. For example, a(5)=15 is per([[1, 1, 1, 1, 1], [1, 1, 1, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]]). - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006

Conjecture. Let S(1)={1} and, for n>1, let S(n) be the smallest set containing x+1 and 2x+1 for each element x in S(n-1). Then a(n) is the sum of the elements in S(n). (See A122554 for a sequence defined in this way.) - John W. Layman (layman(AT)math.vt.edu), Nov 21 2007

a(n+1) indexes the corner blocks on the Fibonacci spiral built from blocks of unit area (using F(1) and F(2) as the sides of the first block). - Paul Barry (pbarry(AT)wit.ie), Mar 06 2008

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

T. D. Noe, Table of n, a(n) for n=1..201

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

G.f.: -(1 - x + x^3 ) / (( x^2 + x - 1 )*( x - 1 )^2 ).

a(n) = Fib(n+4)-(n+1) = a(n-1)+a(n-2)+n-2 = A001924(n-1)+1 = A065220(n+3)+2. - Henry Bottomley (se16(AT)btinternet.com), Oct 22 2001

a(n)=2*a(n-1)-a(n-3)+1 - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 13 2006

a(n+1)=1+sum{k=0..n, F(k+2)-1}=sum{k=0..n, F(k+2)}-n=F(n+4)-n-2; - Paul Barry (pbarry(AT)wit.ie), Mar 06 2008

A000126:=-(1-z+z**3)/(z**2+z-1)/(z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]

a:= n-> (Matrix([[1, 1, 1, 2]]). Matrix(4, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, -2, -1, 1][i] else 0 fi)^n)[1, 2]; seq (a(n), n=1..36); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 26 2008]

Heap-transform of A000071 - John Layman.

Cf. A066067, A001924, A065220.

Cf. A007931: binary strings with leading 0's, or ternary strings without 0's.

Differences are A000071.

Cf. A122554.

Adjacent sequences: A000123 A000124 A000125 this_sequence A000127 A000128 A000129

Sequence in context: A125513 A054174 A001523 this_sequence A143281 A098057 A074029

nonn,easy

njas