0,2

W(n){1,3;2,-1,1} = Sum[ i=1 to n ]W(i){1,2;2,-1,0}; where W(n){a,b; p,q,r} implies x(n)=p*x(n-1) - q*x(n-2) + r; x(0)= a, x(1)= b.

Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - Ron Hardin (rhh(AT)cadence.com), Mar 16 2002

Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 16 2004

M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.

M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.

A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.

T. D. Noe, Table of n, a(n) for n=0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1065

a(n)=2*a(n-1)+a(n-2)+1; a(0)=1, a(1)=3.

a(0)=1, a(n+1)=ceil(x*a(n)) for n>0 where x=1+sqrt(2). - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 22 2003

a(n)=3a(n-1)-a(n-2)-a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n)=sum{k=0..floor((n+2)/2), comb(n+2, 2k+2)2^k }. - Paul Barry (pbarry(AT)wit.ie), Aug 16 2003

Binomial transform of A029744. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2004

E.g.f.: exp(x)(cosh(x/sqrt(2))+sqrt(2)sinh(x/sqrt(2)))^2.

a(n)={[ (2+(3*sqrt(2))/2)(1+sqrt(2))^n - (2-(3*sqrt(2))/2)(1-sqrt(2))^n ]/ 2*sqrt(2)} - 1/2.

a:=n->sum(fibonacci(i, 2), i=0..n): seq(a(n), n=1..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

(PARI) a(n)=local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n

Partial sums of Pell numbers A000129.

First row of table A083087.

With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).

Cf. A001333, A048654, A048655, A083087, A083044, A083047, A083050.

-a(-3-n)=A077921(n).

Adjacent sequences: A048736 A048737 A048738 this_sequence A048740 A048741 A048742

Sequence in context: A038746 A126876 A090757 this_sequence A054192 A054185 A093963

easy,nice,nonn

Barry E. Williams

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002