1,2

See A001835 for another version.

Greedy frac multiples of sqrt(3): a(1)=1, sum(n>0,frac(a(n)*x)) < 1 at x=sqrt(3).

The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,0,0,...]. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

For n>0, a(n)= ceil( (2+sqrt(3))^n/(3+sqrt(3)) ).

G.f.: (1-x)/(1-4x+x^2); E.g.f.: exp(2x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); a(n)=(1/2+sqrt(3)/6)(2+sqrt(3))^n+(1/2-sqrt(3)/6)(2-sqrt(3))^n (offset 0). Binomial transform of A002605. - Paul Barry (pbarry(AT)wit.ie), Sep 17 2003

a(n)=sum{k=0..n, binomial(2n-k, k)2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), Jan 22 2005

a(n)=(-1)^n*U(2n, I*sqrt(2)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

a(n)=Jacobi_P(n,-1/2,1/2,2)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006

a(4) = 41 since frac(1x) + frac(3x) + frac(11x) + frac(41x) < 1, while frac(1x) + frac(3x) + frac(11x) + frac(k*x) > 1 for all k>11 and k<41.

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}]] (from Robert G. Wilson v Jan 13 2005)

(Other) sage: [lucas_number1(n, 4, 1)-lucas_number1(n-1, 4, 1) for n in xrange(1, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2009]

Cf. A002530 (denominators of convergents to sqrt(3)), A079934, A079936, A001353.

Cf. A001835 (same except for the first term).

Row 4 of array A094954.

Sequence in context: A077831 A032952 A001835 this_sequence A113437 A076540 A129637

Adjacent sequences: A079932 A079933 A079934 this_sequence A079936 A079937 A079938

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2003