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A143608 A005319 and A002315 interleaved. 15
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

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.

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 *)

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

CROSSREFS

Cf. A002315, A005319, A182431, A001108.

Sequence in context: A242315 A275313 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 May 27 16:49 EDT 2017. Contains 287207 sequences.