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A008843 Squares of NSW numbers (A002315): x^2 such that x^2 - 2y^2 = -1 for some y. 7
1, 49, 1681, 57121, 1940449, 65918161, 2239277041, 76069501249, 2584123765441, 87784138523761, 2982076586042449, 101302819786919521, 3441313796169221281, 116903366249966604049, 3971273138702695316401 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also indices of triangular numbers (A000217) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 16 2015

a(n)-th triangular number is a square; subsequence of A001108. - Jaroslav Krizek, Aug 05 2016

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.

P. F. Teilhet, Note #2094, L'Intermed. Math., 10 (1903), pp. 235-238.

LINKS

Table of n, a(n) for n=0..14.

M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.

D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.

Index entries for sequences related to Bernoulli numbers.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

a(n) = 34a(n-1)-a(n-2)+16 = A002315(n)^2 = 2*A001653(n)^2-1 = 2*A008844(n)-1 = [A046176(n)*sqrt(2) ] = 6*A055792(n+1)-a(n-1)+4 = (6*A055792(n+2)+a(n-1)-20)/35. - Henry Bottomley, Nov 13 2001

a(n) = A001108(2n+1). - Ira M. Gessel, Nov 05 2014

a(n) = sum(k=1, 2*n+1, 2^(k-1)*binomial(4*n+2, 2*k) ). - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 03 2003

O.g.f.: = -(1+14*x+x^2)/((-1+x)*(1-34*x+x^2)). - R. J. Mathar, Nov 23 2007

a(n) = -1/2+(1/2)*sqrt(2)*[17+12*sqrt(2)]^n+(3/4)*[17-12*sqrt(2)]^n-(1/2)*[17-12*sqrt(2)]^n *sqrt(2)+(3/4)*[17+12*sqrt(2)]^n, with n>=0. - Paolo P. Lava, Jun 17 2008

a(n) = -Cosh[(2*n - 1) ArcTanh[Sqrt[2]]])^2 = -1 + (Sinh[(2*n - 1) ArcTanh[Sqrt[2]]])^2. - Artur Jasinski, Oct 30 2008

MATHEMATICA

Table[Round[N[ -(Cosh[(2 n - 1) ArcTanh[Sqrt[2]]])^2, 100]], {n, 1, 20}] (* Artur Jasinski, Oct 30 2008 *)

CROSSREFS

Cf. A000217, A002315, A016754, A146313, A253826.

Sequence in context: A069327 A088068 A140394 * A145848 A014942 A260856

Adjacent sequences:  A008840 A008841 A008842 * A008844 A008845 A008846

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 21 01:33 EST 2019. Contains 329349 sequences. (Running on oeis4.)