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 A143608 A005319 and A002315 interleaved. 16
 0, 1, 4, 7, 24, 41, 140, 239, 816, 1393, 4756, 8119, 27720, 47321, 161564, 275807, 941664, 1607521, 5488420, 9369319, 31988856, 54608393, 186444716, 318281039, 1086679440, 1855077841, 6333631924, 10812186007, 36915112104, 63018038201, 215157040700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also, numerators of the lower principal and intermediate convergents to 2^(1/2). The lower principal and intermediate convergents to 2^(1/2), beginning with 1/1, 4/3, 7/5, 24/17, 41/29, form a strictly increasing sequence; essentially, numerators=A143608 and denominators=A079496. Sequence a(n) such that a(2*n) = sqrt(2*A001108(2*n)) and a(2*n+1) = sqrt(A001108(2*n+1)). For n > 0, a(n) divides A******(k+1,n+1)-A******(k,n+1) where A****** is any one of A182431, A182439, A182440, A182441 and k is any nonnegative integer. If p is a prime of the form 8*r +/- 3 then a(p+1) == 0 (mod p); if p is a prime of the form 8*r +/- 1 then a(p-1) == 0 (mod p). Numbers n such that sqrt(floor(n^2/2 + 1)) is an integer. The integer square roots are given by A079496. - Richard R. Forberg, Aug 01 2013 From Peter Bala, Mar 23 2018: (Start) Define a binary operation o on the real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0. Then we have a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and a(2*n) = sqrt(2)*(1 o 1 o ... o 1) (2*n terms). Cf. A084068. This is a fourth-order divisibility sequence. Indeed, a(2*n) = sqrt(2)*U(2*n) and a(2*n+1) = U(2*n+1), where U(n) is the Lehmer sequence [Lehmer, 1930] defined by the recurrence U(n) = 2*sqrt(2)*U(n-1) - U(n-2) with U(0) = 0 and U(1) = 1. The solution to the recurrence is U(n) = (1/2)*( (sqrt(2) + 1)^n - (sqrt(2) - 1)^n ). (End) REFERENCES Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011. Creighton Kenneth Dement, Comments on A143608 and A143609 C. Kimberling, Best lower and upper approximations to irrational numbers, Elem. Math. vol. 52 iss. 3 (1997) 122-126. D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448. E. W. Weisstein, MathWorld: Lehmer Number Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1). FORMULA a(2*n) = (a(2*n - 1) + a(2*n + 1))/2. a(2*n + 1) = (a(2*n) + a(2*n + 2))/4. a(2*n) = 4*A001109(n). a(2*n + 1) = 4*A001109(n) + A001541(n). From Colin Barker, Jun 29 2012: (Start) a(n) = 6*a(n-2) - a(n-4). G.f.: x*(1 + 4*x + x^2)/((1 + 2*x - x^2)*(1 - 2*x - x^2)) = x*(1 + 4*x + x^2)/(1 - 6*x^2 + x^4). (End) 2*a(n) = A078057(n) - A123335(n-1). - R. J. Mathar, Jul 04 2012 a(2n) = A005319(n); a(2n+1) = A002315(n). - R. J. Mathar, Jul 17 2009 a(n)*a(n+1) + 1 = A001653(n+1). - Charlie Marion, Dec 11 2012 a(n) = (((-2 - sqrt(2) + (-1)^n * (-2+sqrt(2))) * ((-1+sqrt(2))^n - (1+sqrt(2))^n)))/(4*sqrt(2)). - Colin Barker, Mar 27 2016 MAPLE A143608 := proc(n)     option remember;     if n <= 3 then         op(n+1, [0, 1, 4, 7]) ;     else         6*procname(n-2)-procname(n-4) ;     end if; end proc: # R. J. Mathar, Jul 22 2012 MATHEMATICA a = -4; b = -1; Reap[While[b<2000000000, t = 4*b-a; Sow[t]; a=b; b=t; t = 2*b-a; Sow[t]; a=b; b=t]][[2, 1]] CoefficientList[Series[x*(1 + 4*x + x^2)/(1 - 6*x^2 + x^4), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 24 2014 *) LinearRecurrence[{0, 6, 0, -1}, {0, 1, 4, 7}, 31] (* Jean-François Alcover, Sep 21 2017 *) PROG (PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 6, 0]^n*[0; 1; 4; 7])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015 (PARI) concat(0, Vec(x*(1+4*x+x^2)/((1+2*x-x^2)*(1-2*x-x^2)) + O(x^50))) \\ Colin Barker, Mar 27 2016 (MAGMA) I:=[0, 1, 4, 7]; [n le 4 select I[n] else 6*Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Mar 27 2018 CROSSREFS Cf. A002315, A005319, A182431, A001108, A049629, A084068. Sequence in context: A288299 A288763 A203230 * A079441 A129418 A073218 Adjacent sequences:  A143605 A143606 A143607 * A143609 A143610 A143611 KEYWORD nonn,easy AUTHOR Originally submitted by Clark Kimberling, Aug 27 2008. Merged with an essentially identical sequence submitted by Kenneth J Ramsey, Jun 01 2012, by N. J. A. Sloane, Aug 02 2012 STATUS approved

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Last modified April 9 01:33 EDT 2020. Contains 333339 sequences. (Running on oeis4.)