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A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n).
(Formerly M4423 N1869)
88
1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319, 54608393, 318281039, 1855077841, 10812186007, 63018038201, 367296043199, 2140758220993, 12477253282759, 72722761475561, 423859315570607, 2470433131948081 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Named after the Newman-Shanks-Williams reference.

Also numbers n such that A125650(3*n^2) is an odd perfect square. Such numbers 3*n^2 form a bisection of A125651. - Alexander Adamchuk, Nov 30 2006

For positive n, a(n) corresponds to the sum of legs of near-isosceles primitive Pythagorean triangles (with consecutive legs). - Lekraj Beedassy, Feb 06 2007

Also numbers n such that n^2 is a centered 16-gonal number; or a number of the form 8k(k+1)+1, where k = A053141(n) = {0, 2, 14, 84, 492, 2870, ...}. - Alexander Adamchuk, Apr 21 2007

A002315(n)=A001333(2*n+1) [From Ctibor O. Zizka, Aug 13 2008]

The lower principal convergents to 2^(1/2), beginning with 1/1, 7/5, 41/29, 239/169, comprise a strictly increasing sequence; numerators=A002315 and denominators=A001653. - Clark Kimberling, Aug 27 2008

The upper intermediate convergents to 2^(1/2) beginning with 10/7, 58/41, 338/239, 1970/1393 form a strictly decreasing sequence; essentially, numerators=A075870, denominators=A002315. - Clark Kimberling, Aug 27 2008

General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka, Sep 02 2008]

Numbers n such that (ceiling(sqrt(n*n/2)))^2 = (1+n*n)/2 [From Ctibor O. Zizka, Nov 09 2009]

A001109(n)/a(n) converges to cos^2(Pi/8) = 1/2 + 2^(1/2)/4 [From Gary Detlefs, Nov 25 2009]

a(n) represents all positive integers K for which 2(K^2+1) is a perfect square. [From Neelesh Bodas (neelesh.bodas(AT)gmail.com), Aug 13 2010]

For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(8)'s along the main diagonal, and i's along the superdiagonal and subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]

Integers n such that A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) is a square (cf. A202391). [From Max Alekseyev, Dec 19 2011]

Integer square roots of floor(n^2/2 - 1) or A047838. - Richard R. Forberg, Aug 01 2013

REFERENCES

E. Barcucci et al., A combinatorial interpretation of the recurrence f_{n+1} = 6 f_n - f_{n-1}, Discrete Math., 190 (1998), 235-240.

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012

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

J. Bonin, L. Shapiro and R. Simion, Some q-analogues of the Schroeder numbers arising from combinatorial statistics on lattice paths, H. Statistical Planning and Inference, 16,1993,35-55 (p. 50).

Melissa Emory, The Diophantine equation X^4 + Y^4 = D^2 Z^4 in quadratic fields, INTEGERS 12 (2012), #A65. - From N. J. A. Sloane, Feb 06 2013

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.

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

M. Newman, D. Shanks and H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith. 38 (1980/81), no. 2, 129-140. MR82b:20022

Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.

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

Santana, S. F. and Diaz-Barrero, J. L. (2006). Some properties of sums involving Pell numbers. Missouri Journal of Mathematical Sciences 18(1).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

R. A. Sulanke, Bijective recurrences concerning Schroeder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.

P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003.

A. S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

The Prime Glossary, NSW number.

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Eric Weisstein's World of Mathematics, NSW Number.

Eric Weisstein's World of Mathematics, Centered Polygonal Number.

Index entries for sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

a(n) = (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).

a(n)=(1+sqrt(2))/2*(3+sqrt(8))^n+(1-sqrt(2))/2*(3-sqrt(8))^n. - Ralf Stephan, Feb 23 2003

a(n) = sqrt(2*(A001653(n))^2-1)

G.f.: (1+x)/(1-6*x+x^2)

a(n) = S(n, 6)+S(n-1, 6) = S(2*n, sqrt(8)), S(n, x) = U(n, x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n, 6)= A001109(n+1).

a(n) ~ 1/2*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002

Lim n -> inf. a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson, Oct 06 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -8)=a(n) - Benoit Cloitre, Nov 10 2002

With a=3+2sqrt(2), b=3-2sqrt(2): a(n)=(a^((2n+1)/2)-b^((2n+1)/2))/2. a(n)=A077444(n)/2. - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003

a(n)=sum(k=0, n, 2^k*binomial(2*n+1, 2*k)) - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 08 2003

Same as: i such that Mod(sigma(i^2+1, 2), 2)=1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

a(n) = L(n, -6)*(-1)^n, where L is defined as in A108299; see also A001653 for L(n, +6). - Reinhard Zumkeller, Jun 01 2005

a(n)=A001652(n)+A046090(n); e.g. 239=119+120 - Charlie Marion, Nov 20 2003

A001541(n)*a(n+k)=A001652(2n+k)+A001652(k)+1; e.g. 3*1393=4069+119+1; for k>0, A001541(n+k)*a(n)=A001652(2n+k)-A001652(k-1); e.g. 99*7=696-3 - Charlie Marion, Mar 17 2003

a(n)=Jacobi_P(n,1/2,-1/2,3)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry, Feb 03 2006

P_{2n}+P_{2n+1} where P_i are the Pell numbers (A000129). Also the square root of the partial sums of Pell numbers: P_{2n}+P_{2n+1} = sqrt(sum_{i=0}^{4n+1} P_i) (Santana and Diaz-Barrero, 2006). - David Eppstein, Jan 28 2007

a(n) = 2*A001652(n) + 1 = 2*A046729(n) + (-1)^n. - Lekraj Beedassy, Feb 06 2007

a(n) = sqrt[A001108(2*n+1)]. - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

a(n) = Sqrt[ 8*A053141(n)*(A053141(n) + 1) + 1 ]. - Alexander Adamchuk, Apr 21 2007

a(n+1) = 3*a(n)+(8*a(n)^2+8)^0.5, a(1)=1. - Richard Choulet, Sep 18 2007

a(n)= third binomial transform of 1,4,8,32,64,256,512 [From Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]

a(n)= (-1)^(n - 1) (1/sqrt[ -1]) cos[(2 n - 1) arcsin(sqrt(2)) [From Artur Jasinski, Feb 17 2010]

a(n+k)=A001541(k)*a(n)+4*A001109(k)*A001653(n); e.g., 8119=17*239+4*6*169. - Charlie Marion, Feb 04 2011

In general, a(n+k) = A001541(k)*a(n)) + (A001108(2k)*(a(n)^2+1))^0.5.  See formula for Sep 18 2007. - Charlie Marion, Dec 07 2011

a(n) = floor((1+sqrt(2))^(2n+1))/2. [Thomas Ordowski, Jun 12 2012]

MAPLE

a[0]:=1: a[1]:=7: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); - Zerinvary Lajos, Jul 26 2006

A002315:=(1+z)/(1-6*z+z**2); [Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

a[0] = 1; a[1] = 7; a[n_] := a[n] = 6a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 20}] (from Robert G. Wilson v, Jun 09 2004)

q=16; s=0; lst={}; Do[s+=n; If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]], AppendTo[lst, Sqrt[q*s+1]]], {n, 0, 8!}]; lst [From Vladimir Joseph Stephan Orlovsky, Apr 02 2009]

Round[Table[(-1)^(n - 1) (1/Sqrt[1 - 2]) Cos[(2 n - 1) ArcSin[Sqrt[2]]], {n, 1, 10}]] (*Artur Jasinski*) [From Artur Jasinski, Feb 17 2010]

Transpose[NestList[Flatten[{Rest[#], ListCorrelate[{-1, 6}, #]}]&, {1, 7}, 20]][[1]]  (* From Harvey P. Dale, Mar 23 2011 *)

Table[ If[n>0, a=b; b=c; c=6b-a, b=-1; c=1], {n, 0, 20}] (* Jean-François Alcover, Oct 19 2012 *)

PROG

(PARI) a(n)=subst(poltchebi(abs(n+1))-poltchebi(abs(n)), x, 3)/2

(PARI) a(n)=if(n<0, -a(-1-n), polsym(x^2-2*x-1, 2*n+1)[2*n+2]/2)

(PARI) a(n)=local(w=3+quadgen(32)); imag((1+w)*w^n)

(PARI) for (i=1, 10000, if(Mod(sigma(i^2+1, 2), 2)==1, print1(i, ", ")))

(Sage) [(lucas_number2(n, 6, 1)-lucas_number2(n-1, 6, 1))/4 for n in xrange(1, 22)]# [From Zerinvary Lajos, Nov 10 2009]

(Haskell)

a002315 n = a002315_list !! n

a002315_list = 1 : 7 : zipWith (-) (map (* 6) (tail a002315_list)) a002315_list

-- Reinhard Zumkeller, Jan 10 2012

CROSSREFS

Bisection of A001333. Cf. A001109, A001653. A065513(n)=a(n)-1.

First differences of A001108 and A055997. Bisection of A084068 and A088014. Pairwise sums of A001109. Cf. A077444.

Cf. A125650, A125651, A125652.

Row sums of unsigned triangle A127675.

Cf. A053141.

The values 2(a(n)^2+1) are all perfect squares, whose square root is given by A075870 [From Neelesh Bodas (neelesh.bodas(AT)gmail.com), Aug 13 2010]

Sequence in context: A173409 A057009 A140480 * A141813 A088165 A108983

Adjacent sequences:  A002312 A002313 A002314 * A002316 A002317 A002318

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Feb 16 2000

STATUS

approved

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Last modified April 17 20:01 EDT 2014. Contains 240655 sequences.