Search: id:A028230 Results 1-1 of 1 results found. %I A028230 %S A028230 1,15,209,2911,40545,564719,7865521,109552575,1525870529,21252634831, %T A028230 296011017105,4122901604639,57424611447841,799821658665135, %U A028230 11140078609864049,155161278879431551,2161117825702177665 %N A028230 Bisection of A001353. Indices of square numbers which are also octagonal. %C A028230 Chebyshev S-sequence with Diophantine property. %C A028230 4*b(n)^2 - 3*a(n)^2 = 1 with b(n)=A001570(n), n>=0. %C A028230 y satisfying the Pellian x^2 - 3*y^2=1, for even x given by A094347(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004 %C A028230 a(n) = L(n,-14)*(-1)^n, where L is defined as in A108299; see also A001570 for L(n,+14). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005 %C A028230 Product x*y, where the pair (x, y) solves for x^2 - 3y^2 = -2, i.e., a(n)=A001834(n)*A001835(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 13 2006 %C A028230 Numbers n such that RootMeanSquare(1,3,...,2*A001570(k)-1) = n. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Sep 04 2008] %D A028230 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329. %D A028230 T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-. %D A028230 W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced knots, Math. Comp., 25 (1971), 797-801. %D A028230 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104. %D A028230 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83. %H A028230 Tanya Khovanova, Recursive Sequences %H A028230 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A028230 Index entries for sequences related to linear recurrences with constant coefficients %H A028230 Index entries for sequences related to Chebyshev polynomials. %F A028230 a(n)=2*A001921(n)+1. %F A028230 a(n) = 14*a(n-1) - a(n-2), a(-1)=-1, a(0)=1. %F A028230 a(n) = S(n, 14) + S(n-1, 14) = S(2*n, 4) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 14)=A007655(n+1) and S(n, 4)=A001353(n+1). %F A028230 G.f.: x*(1+x)/(1-14*x+x^2). %F A028230 a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := 2+sqrt(3) and am := 2-sqrt(3). %F A028230 a(n+1)= sum(((-1)^k)*binomial(2*n-k, k)*16^(n-k), k=0..n), n>=0. %F A028230 a(n) = sqrt((4*A001570(n-1)^2 - 1)/3). %F A028230 a(n) ~ 1/6*sqrt(3)*(2 + sqrt(3))^(2*n-1) - Joe Keane (jgk(AT)jgk.org), May 15 2002 %F A028230 4*a(n+1) = (A001834(n))^2 + 4*(A001835(n+1))^2 - (A001835(n))^2. E.g. 4*a(3) = 4*209 = 19^2 + 4*11^2 - 3^2 = (A001834(2))^2 + 4*(A001835(3))^2 - A001835(2))^2. Generating floretion: 'i + 2'j + 3'k + i' + 2j' + 3k' + 4'ii' + 3'jj' + 4'kk' + 3'ij' + 3'ji' + 'jk' + 'kj' + 4e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 04 2004 %F A028230 Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[a(n-1),7] + f[a(n-2),7]. - Marcos Carreira, Dec 27 2006 %F A028230 a(n+1)=35*a(n)- 35*a(n-1)+ a(n-2) a(1)=1,a(2)=35,a(3)=1189 [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 29 2009] %o A028230 (Other) sage: [(lucas_number2(n,14,1)-lucas_number2(n-1,14,1))/12 for n in xrange(1, 18)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009] %Y A028230 Cf. A036428, A046184. %Y A028230 Cf. A077416 with companion A077417. %Y A028230 Sequence in context: A078265 A089138 A051813 this_sequence A122572 A067560 A019553 %Y A028230 Adjacent sequences: A028227 A028228 A028229 this_sequence A028231 A028232 A028233 %K A028230 nonn %O A028230 1,2 %A A028230 N. J. A. Sloane (njas(AT)research.att.com). %E A028230 More terms from Eric Weisstein (eric(AT)weisstein.com) %E A028230 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002 Search completed in 0.002 seconds