0,2

Named after the Newman-Shanks-Williams reference.

Also numbers n such that A125650[ 3*n^2 ] is an odd perfect square, where A124650(n) is a numerator of n(n+3)/(4(n+1)(n+2)) = Sum[ 1/(k(k+1)(k+2)), {k,1,n} ]. Sequence of numbers 3*n^2 is a bisection of A125651(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), 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 (blekraj(AT)yahoo.com), 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 (alex(AT)kolmogorov.com), Apr 21 2007

A002315(n)=A001333(2*n+1) [From Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), 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 (ck6(AT)evansville.edu), 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 (ck6(AT)evansville.edu), 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 (ctibor.zizka(AT)seznam.cz), Sep 02 2008]

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

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.

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).

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.

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

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

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, Link to a section of The World of Mathematics.

Eric Weisstein, Link to a section of The World of Mathematics, Centered Polygonal Number.

Index entries for sequences related to Chebyshev polynomials.

Missouri J. Mathematical Sciences (New web address)

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 (omomom(AT)hotmail.com), 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 (benoit7848c(AT)orange.fr), 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 (reinhard.zumkeller(AT)gmail.com), Jun 01 2005

a(n)=A001652(n)+A046090(n); e.g. 239=119+120 - Charlie Marion (charliem(AT)bestweb.net), 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 (charliemath(AT)verizon.net), Mar 17 2003

a(n)=Jacobi_P(n,1/2,-1/2,3)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), 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 (eppstein(AT)ics.uci.edu), Jan 28 2007

a(n) = 2*A001652(n) + 1 = 2*A046729(n) + (-1)^n. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (alex(AT)kolmogorov.com), Apr 21 2007

a(n+1) = 3*a(n)+(8*a(n)^2+8)^0.5, a(1)=1. - Richard Choulet (richardchoulet(AT)yahoo.fr), 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[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 (zerinvarylajos(AT)yahoo.com), Jul 26 2006

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

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 Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

(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, ", ")))

(Other) sage: [(lucas_number2(n, 6, 1)-lucas_number2(n-1, 6, 1))/4 for n in xrange(1, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]

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.

Sequence in context: A026002 A057009 A140480 this_sequence A141813 A088165 A108983

Adjacent sequences: A002312 A002313 A002314 this_sequence A002316 A002317 A002318

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 16 2000

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009