0,4

A Lam{\'e} sequence of higher order.

Could have begun 1,0,0,1,1,1,2,3,4,6,9,... but that would spoil many nice properties.

Number of tilings of a 3 X n rectangle with straight trominoes.

Number of ways to arrange n-1 tatami mats in a 2 X (n-1) room such that no 4 meet at a point. For example, there are 6 ways to cover a 2 X 5 room, described by 11111, 2111, 1211, 1121, 1112, 212.

Equivalently, number of ordered partitions of n-1 into parts 1 and 2 with no two 2's adjacent. E.g. there are 6 such ways to partition 5, namely 11111, 2111, 1211, 1121, 1112, 212, so a(9) = 6.

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = sum(binomial(n-(m-1)*i, i), i=0..n/m). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.

a(n-4) = number of n-bit sequences that start and end with 0 but avoid both 010 and 0110. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004

a(n+2) = number of n-bit 0-1 sequences that avoid both 00 and 010. - David Callan (callan(AT)stat.wisc.edu), Mar 25 2004

Also number of compositions of n+1 into parts congruent to 1 mod m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 09 2005

Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - Paul Barry (pbarry(AT)wit.ie), Feb 25 2005

Row sums of Riordan array (1,x(1+x^2)). - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 78,80.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.

M. Feinberg, New slants, Fib. Quart. 2 (1964), 223-227.

T. M. Green, Recurrent sequences and Pascal's triangle, Math. Mag., 41 (1968), 13-21.

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13.

T. D. Noe, Table of n, a(n) for n=0..500

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 14

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 376

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.

T. Sillke, The binary form of Conway's Sequence

E. Wilson, The Scales of Mt. Meru

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

a(n) = sum(binomial(n-2*i, i), i=0..n/3).

a(n) = a(n-2) + a(n-3) + a(n-4) for n>3.

a(n) = floor( d*c^n + 1/2) where c is the real root of x^3-x^2-1 and d is the real root of 31*x^3-31*x^2+9*x-1 ( c=1.465571231876768... and d= 0.611491991950812...) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 30 2002

a(n)=sum{k=0..n, binomial(floor((n+2k-2)/3), k)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004

a(n)=sum{k=0..n, C(k, floor((n-k)/2))(1+(-1)^(n-k))/2}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006

a(n)=sum{k=0..n, C((n+2k)/3,(n-k)/3)*(2*cos(2*pi*(n-k)/3)+1)/3}; - Paul Barry (pbarry(AT)wit.ie), Dec 15 2006

a(n) = term (1,1) in matrix [1,1,0; 0,0,1; 1,0,0]^n - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 20 2008

f := proc(r) local t1, i; t1 := []; for i from 1 to r do t1 := [op(t1), 0]; od: for i from 1 to r+1 do t1 := [op(t1), 1]; od: for i from 2*r+2 to 50 do t1 := [op(t1), t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order

with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 2)}, unlabeled]: seq(count(SeqSetU, size=j), j=3..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 10 2006

seq(add(binomial(n-2*k, k), k=0..floor(n/3)), n=0..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2007

BB :=-x/(-1+x^2+x^6); series(BB, x, 77); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2007

A000930:=-1/(-1+z+z**3); [S. Plouffe in his 1992 dissertation.]

a := n -> (Matrix([[1, 1, 0], [0, 0, 1], [1, 0, 0]])^n)[1, 1]; seq ((a(n)), n=0..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 20 2008

a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ]

CoefficientList[Series[1/(1-x-x^3), {x, 0, 45}], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007

For Lam{\'e} sequences of orders 1 through 9 see A000045, this one, A017898-A017904.

Cf. A048715, A069241.

See also A000079, A003269, A003520, A005708, A005709, A005710.

Essentially the same as A068921 and A078012.

Sequence in context: A139077 A017825 A017826 this_sequence A068921 A078012 A135851

Adjacent sequences: A000927 A000928 A000929 this_sequence A000931 A000932 A000933

nonn,easy,nice

njas