Search: id:A002315 Results 1-1 of 1 results found. %I A002315 M4423 N1869 %S A002315 1,7,41,239,1393,8119,47321,275807,1607521,9369319,54608393,318281039, %T A002315 1855077841,10812186007,63018038201,367296043199,2140758220993, %U A002315 12477253282759,72722761475561,423859315570607,2470433131948081 %N 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). %C A002315 Named after the Newman-Shanks-Williams reference. %C A002315 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 %C A002315 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 %C A002315 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 %C A002315 A002315(n)=A001333(2*n+1) [From Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Aug 13 2008] %C A002315 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 %C A002315 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 %C A002315 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] %C A002315 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] %D A002315 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. %D A002315 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256. %D A002315 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). %D A002315 A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288. %D A002315 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649. %D A002315 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 %D A002315 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28. %D A002315 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288. %D A002315 Santana, S. F. and Diaz-Barrero, J. L. (2006). Some properties of sums involving Pell numbers. Missouri Journal of Mathematical Sciences 18(1). %D A002315 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002315 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A002315 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. %D A002315 R. A. Sulanke, Bijective recurrences concerning Schroeder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp. %D A002315 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238. %H A002315 T. D. Noe, Table of n, a(n) for n = 0..200 %H A002315 Index entries for two-way infinite sequences %H A002315 Index entries for sequences related to linear recurrences with constant coefficients %H A002315 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. %H A002315 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A002315 Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003. %H A002315 A. S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284. %H A002315 Tanya Khovanova, Recursive Sequences %H A002315 The Prime Glossary, NSW number. %H A002315 R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1. %H A002315 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A002315 Eric Weisstein, Link to a section of The World of Mathematics, Centered Polygonal Number. %H A002315 Index entries for sequences related to Chebyshev polynomials. %H A002315 Missouri J. Mathematical Sciences (New web address) %F A002315 a(n) = (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)). %F A002315 a(n)=(1+sqrt(2))/2*(3+sqrt(8))^n+(1-sqrt(2))/2*(3-sqrt(8))^n. - Ralf Stephan, Feb 23 2003 %F A002315 a(n) = sqrt(2*(A001653(n))^2-1) %F A002315 G.f.: (1+x)/(1-6*x+x^2) %F A002315 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). %F A002315 a(n) ~ 1/2*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002 %F A002315 Lim n -> inf. a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 06 2002 %F A002315 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 %F A002315 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 %F A002315 a(n)=sum(k=0, n, 2^k*binomial(2*n+1, 2*k)) - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 08 2003 %F A002315 Same as: i such that Mod(sigma(i^2+1, 2), 2)=1 - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004 %F A002315 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 %F A002315 a(n)=A001652(n)+A046090(n); e.g. 239=119+120 - Charlie Marion (charliem(AT)bestweb.net), Nov 20 2003 %F A002315 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 %F A002315 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 %F A002315 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 %F A002315 a(n) = 2*A001652(n) + 1 = 2*A046729(n) + (-1)^n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 06 2007 %F A002315 a(n) = sqrt[A001108(2*n+1)] - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007 %F A002315 a(n) = Sqrt[ 8*A053141(n)*(A053141(n) + 1) + 1 ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 21 2007 %F A002315 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 %F A002315 a(n)= third binomial transform of 1,4,8,32,64,256,512 [From Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009] %p A002315 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 %p A002315 A002315:=(1+z)/(1-6*z+z**2); [S. Plouffe in his 1992 dissertation.] %t A002315 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) %t A002315 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] %o A002315 (PARI) a(n)=subst(poltchebi(abs(n+1))-poltchebi(abs(n)),x,3)/2 %o A002315 (PARI) a(n)=if(n<0,-a(-1-n),polsym(x^2-2*x-1,2*n+1)[2*n+2]/2) %o A002315 (PARI) a(n)=local(w=3+quadgen(32)); imag((1+w)*w^n) %o A002315 (PARI) for (i=1,10000,if(Mod(sigma(i^2+1,2),2)==1,print1(i,","))) %o A002315 (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] %Y A002315 Bisection of A001333. Cf. A001109, A001653. A065513(n)=a(n)-1. %Y A002315 First differences of A001108 and A055997. Bisection of A084068 and A088014. Pairwise sums of A001109. Cf. A077444. %Y A002315 Cf. A125650, A125651, A125652. %Y A002315 Row sums of unsigned triangle A127675. %Y A002315 Cf. A053141. %Y A002315 Sequence in context: A026002 A057009 A140480 this_sequence A141813 A088165 A108983 %Y A002315 Adjacent sequences: A002312 A002313 A002314 this_sequence A002316 A002317 A002318 %K A002315 nonn,easy,nice,new %O A002315 0,2 %A A002315 N. J. A. Sloane (njas(AT)research.att.com). %E A002315 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 16 2000 %E A002315 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009 Search completed in 0.003 seconds