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%I #86 Feb 16 2023 05:40:56
%S 0,1,4,7,24,41,140,239,816,1393,4756,8119,27720,47321,161564,275807,
%T 941664,1607521,5488420,9369319,31988856,54608393,186444716,318281039,
%U 1086679440,1855077841,6333631924,10812186007,36915112104,63018038201,215157040700
%N A005319 and A002315 interleaved.
%C 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.
%C Sequence a(n) such that a(2*n) = sqrt(2*A001108(2*n)) and a(2*n+1) = sqrt(A001108(2*n+1)).
%C 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.
%C 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).
%C 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
%C From _Peter Bala_, Mar 23 2018: (Start)
%C 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
%C a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and
%C a(2*n) = sqrt(2)*(1 o 1 o ... o 1) (2*n terms). Cf. A084068.
%C 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)
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%H Colin Barker, <a href="/A143608/b143608.txt">Table of n, a(n) for n = 0..1000</a>
%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H Creighton Kenneth Dement, <a href="/A143608/a143608.txt">Comments on A143608 and A143609</a>
%H Vincent Granville, <a href="https://math.stackexchange.com/questions/3336135/successive-records-in-mathematical-sequences-surprising-result">Successive records in mathematical sequences: surprising result</a>, Mathematics Stack Exchange, 2019.
%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximations to irrational numbers</a>, Elem. Math. vol. 52 iss. 3 (1997) 122-126.
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968235">An extended theory of Lucas' functions</a>, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F a(2*n) = (a(2*n - 1) + a(2*n + 1))/2.
%F a(2*n + 1) = (a(2*n) + a(2*n + 2))/4.
%F a(2*n) = 4*A001109(n).
%F a(2*n + 1) = 4*A001109(n) + A001541(n).
%F From _Colin Barker_, Jun 29 2012: (Start)
%F a(n) = 6*a(n-2) - a(n-4).
%F 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)
%F 2*a(n) = A078057(n) - A123335(n-1). - _R. J. Mathar_, Jul 04 2012
%F a(2n) = A005319(n); a(2n+1) = A002315(n). - _R. J. Mathar_, Jul 17 2009
%F a(n)*a(n+1) + 1 = A001653(n+1). - _Charlie Marion_, Dec 11 2012
%F 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
%F a(n) = A084068(n) - A079496(n). - _César Aguilera_, Feb 14 2023
%p A143608 := proc(n)
%p option remember;
%p if n <= 3 then
%p op(n+1,[0,1,4,7]) ;
%p else
%p 6*procname(n-2)-procname(n-4) ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 22 2012
%t 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]]
%t CoefficientList[Series[x*(1 + 4*x + x^2)/(1 - 6*x^2 + x^4), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Aug 24 2014 *)
%t LinearRecurrence[{0, 6, 0, -1}, {0, 1, 4, 7}, 31] (* _Jean-François Alcover_, Sep 21 2017 *)
%o (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
%o (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
%o (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
%Y Cf. A002315, A005319, A182431, A001108, A049629, A084068.
%K nonn,easy
%O 0,3
%A 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
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