0,2

a(n) = A048739(n-1)+1 = 1/2 * (P(n)+P(n-1)+1), with P(n) = Pell numbers (A000129).

Number of (3412,#)-avoiding involutions in S_{n+1}, where # can be one of 22 patterns, see Egge reference.

Number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+1, s(0) = 1, s(n+1) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004

Define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1} > a_{n+1}/a_n for n >= 0 . This is S(2,4). (For proof, see the Alekseyev link.) - R. K. Guy (rkg(AT)cpsc.ucalgary.ca)

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993;.

Max Alekseyev, Notes on A024537

E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8

a(n) = 2a(n-1)+a(n-2)-1 (from Christian G. Bower, bowerc(AT)usa.net).

a(n) = 3a(n-1)-a(n-2)-a(n-3).

G.f.: (1-x-x^2)/((1-x)(1-2x-x^2))=(1-x-x^2)/(1-3x+x^2+x^3); E.g.f.: exp((1+sqrt(2))x)(1+sqrt(2))/4+exp((1-sqrt(2))x)(1-sqrt(2))/4+exp(x)/2; - Paul Barry (pbarry(AT)wit.ie), Dec 25 2003

a(n) = (1/4)*( 2 + (1-Sqrt(2))^(n+1) + (1+Sqrt(2))^(n+1) ) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL2), b=ZL3], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabeled], size=15):seq(count([S, {Q}, unlabeled], size=n), n=1..31); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2008

with (combinat):a:=n->sum(fibonacci(i, 2), i=0..n):seq(a(n)+1, n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2008

s = 1; lst = {s}; Do[s += Fibonacci[n, 2]; AppendTo[lst, s], {n, 1, 30, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009]

Sequence in context: A119967 A052921 A018905 this_sequence A027826 A091964 A092423

Adjacent sequences: A024534 A024535 A024536 this_sequence A024538 A024539 A024540

nonn

Clark Kimberling (ck6(AT)evansville.edu)

Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Max Alekseyev, Aug 24 2007