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A024537
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a(n) = floor( a(n-1)/(sqrt(2) - 1) ), with a(0) = 1.
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10
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1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 57122, 137904, 332929, 803761, 1940450, 4684660, 11309769, 27304197, 65918162, 159140520, 384199201, 927538921, 2239277042, 5406093004, 13051463049, 31509019101, 76069501250, 183648021600
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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)
This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211) [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Jun 28 2010]
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REFERENCES
| 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.
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LINKS
| Max Alekseyev, Notes on A024537
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2) (2000) 175-179. [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Jun 28 2010]
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346. [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Jun 28 2010]
Index to sequences with linear recurrences with constant coefficients, signature (3,-1,-1).
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FORMULA
| a(n) = 2*a(n-1)+a(n-2)-1 (from Christian G. Bower, bowerc(AT)usa.net).
a(n) = 3*a(n-1)-a(n-2)-a(n-3).
G.f.: (1-x-x^2)/((1-x)*(1-2*x-x^2))=(1-x-x^2)/(1-3*x+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
Let M = a tridiagonal matrix with all 1's in the super and main diagonals and [1,1,0,0,0,...] in the subdiagonal; with V = vector [1,0,0,0,...], and the rest zeros. The sequence is generated as leftmost column from iterates of M*V. [from Gary W. Adamson, (qntmpkt(AT)yahoo.com), Jun 07 2011]
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MAPLE
| 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
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MATHEMATICA
| 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]
t={0, 1, 3}; Do[AppendTo[t, t[[-2]]+2*t[[-1]]+1], {n, 40}]; t+1 (* From Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)
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PROG
| (PARI) a=vector(99); a[1]=1; for(n=2, #a, a[n]=a[n-1]\(sqrt(2) - 1)); a \\ Charles R Greathouse IV, Jun 14 2011
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CROSSREFS
| Cf. A048211, A153588, A174283 - A174286, A176499 - A176502. [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Jun 28 2010]
Sequence in context: A119967 A052921 A018905 * A171842 A027826 A091964
Adjacent sequences: A024534 A024535 A024536 * A024538 A024539 A024540
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KEYWORD
| nonn,easy
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Max Alekseyev, Aug 24 2007
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