0,3

Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. Then the sequence 0,1,1,5,... or (3^(n-1)-(-1)^n)/2+0^n/3 with g.f. x(1-x)/(1-2x-3x^2) corresponds to the (1,2) term of A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

3*a(n+1) + a(n) = 4*A060925(n); a(n+1) = A015518(n) + A060925(n); a(n+1) - 6*A015518(n) = (-1)^n. The floretion -.5'i + 'k - .5i' - .5k' + .5'ii' + 1.5'jj' - .5'kk' - .5'ij' + 'ki' - .5e is a generator of the above formula. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 15 2004

The sequence corresponds to the (1,1) term of the matrix [1,2;2,1]^n. - Simone Severini (ss54(AT)york.ac.uk), Dec 04 2004

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

a(n)^2 + (2*A015518(n))^2 = a(2n). E.g. a(3) = 13, 2*A015518(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

Index entries for sequences related to linear recurrences with constant coefficients

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

a(n)=Sum{k=0..n, Binomial(n, 2k)2^(2k)} - Paul Barry (pbarry(AT)wit.ie), Feb 26 2003

Binomial transform of A000302 (powers of 4) with interpolated zeros. Inverse binomial transform of A081294. - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003

E.g.f.: exp(x)cosh(2x). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003

a(n)=ceiling(3^n/4)+floor(3^n/4)=ceiling(3^n/4)^2-floor(3^n/4)^2. - Paul Barry (pbarry(AT)wit.ie), Jan 17 2005

a(n)=sum{k=0..n, sum{j=0..n, C(n,j)C(n-j,k)*(1+(-1)^(j-k))/2}}; - Paul Barry (pbarry(AT)wit.ie), May 21 2006

a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*4^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007

a(n) = (3^n+(-1)^n)/2. - M. F. Hasler, Mar 20 2008

a(n) = 2 A015518(n) + (-1)^n ; for n>0, a(n) = A080925(n). - M. F. Hasler, Mar 20 2008

((1+sqrt4)^n+(1-sqrt4)^n)/2. The offset is 0. a(3)=13. [From Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008]

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2] od: seq(a[n], n=0..33); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]

Expand[Table[((3)^n + (-1)^n)/(2), {n, 0, 30}]] - Artur Jasinski (grafix(AT)csl.pl), Dec 10 2006

(PARI) if(n<0, 0, (3^n+(-1)^n)/2)

(Other) sage: [lucas_number2(n, 2, -3)/2 for n in xrange(0, 27)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]

The first difference sequence of A015518.

Row sums of triangle A080928.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Cf. A015518.

Sequence in context: A100210 A080267 A034735 this_sequence A080925 A164907 A085601

Adjacent sequences: A046714 A046715 A046716 this_sequence A046718 A046719 A046720

nonn,easy

G. Deroo (gervais.deroo(AT)lemel.fr) and M. Deroo.

Description corrected by and more terms from Michael Somos