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A028230 Bisection of A001353. Indices of square numbers which are also octagonal.

%I A028230 #102 May 22 2025 09:18:18
%S A028230 1,15,209,2911,40545,564719,7865521,109552575,1525870529,21252634831,
%T A028230 296011017105,4122901604639,57424611447841,799821658665135,
%U A028230 11140078609864049,155161278879431551,2161117825702177665,30100488280951055759,419245718107612602961,5839339565225625385695
%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_, 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_, 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_, Jul 13 2006
%C A028230 Numbers n such that RootMeanSquare(1,3,...,2*A001570(k)-1) = n. - _Ctibor O. Zizka_, Sep 04 2008
%C A028230 As n increases, this sequence is approximately geometric with common ratio r = lim(n -> oo, a(n)/a(n-1)) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). - _Ant King_, Nov 15 2011
%D A028230 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
%D A028230 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
%H A028230 Vincenzo Librandi, <a href="/A028230/b028230.txt">Table of n, a(n) for n = 1..890</a>
%H A028230 K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H A028230 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H A028230 T. N. E. Greville, <a href="http://dx.doi.org/10.1090/S0025-5718-1970-0258238-1">Table for third-degree spline interpolations with equally spaced arguments</a>, Math. Comp., 24 (1970), 179-183.
%H A028230 W. D. Hoskins, <a href="http://dx.doi.org/10.1090/S0025-5718-1971-0298873-9">Table for third-degree spline interpolation using equi-spaced knots</a>, Math. Comp., 25 (1971), 797-801.
%H A028230 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A028230 E. Kilic, Y. T. Ulutas, and N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=2.
%H A028230 Dino Lorenzini, and Z. Xiang, <a href="http://alpha.math.uga.edu/~lorenz/IntegralPoints.pdf">Integral points on variable separated curves</a>, Preprint 2016.
%H A028230 F. V. Waugh and M. W. Maxfield, <a href="http://www.jstor.org/stable/2688511">Side-and-diagonal numbers</a>, Math. Mag., 40 (1967), 74-83.
%H A028230 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalSquareNumber.html">Octagonal Square Number.</a>
%H A028230 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).
%H A028230 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A028230 a(n) = 2*A001921(n)+1.
%F A028230 a(n) = 14*a(n-1) - a(n-2) for n>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_{k=0..n} (-1)^k*binomial(2*n-k, k)*16^(n-k), 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_, Dec 04 2004
%F A028230 a(n) = f(a(n-1),7) + f(a(n-2),7), where f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. - _Marcos Carreira_, Dec 27 2006
%F A028230 From _Ant King_, Nov 15 2011: (Start)
%F A028230 a(n) = 1/6 * sqrt(3) * ( (tan(5*Pi/12)) ^ (2n-1) - (tan(Pi/12)) ^ (2n-1) ).
%F A028230 a(n) = floor (1/6 * sqrt(3) * (tan(5*Pi/12)) ^ (2n-1)). (End)
%F A028230 a(n) = A001353(n)^2-A001353(n-1)^2. - _Antonio Alberto Olivares_, Apr 06 2020
%F A028230 E.g.f.: 1 - exp(7*x)*(3*cosh(4*sqrt(3)*x) - 2*sqrt(3)*sinh(4*sqrt(3)*x))/3. - _Stefano Spezia_, Dec 12 2022
%F A028230 a(n) = sqrt(A036428(n)). - _Bernard Schott_, Dec 19 2022
%p A028230 seq(coeff(series((1+x)/(1-14*x+x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Dec 06 2019
%t A028230 LinearRecurrence[{14, - 1}, {1, 15}, 17] (* _Ant King_, Nov 15 2011 *)
%t A028230 CoefficientList[Series[(1+x)/(1-14x+x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 17 2014 *)
%o A028230 (Sage) [(lucas_number2(n,14,1)-lucas_number2(n-1,14,1))/12 for n in range(1, 18)] # _Zerinvary Lajos_, Nov 10 2009
%o A028230 (PARI) Vec((1+x)/(1-14*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 16 2014
%o A028230 (PARI) isok(n) = ispolygonal(n^2, 8); \\ _Michel Marcus_, Jul 09 2017
%o A028230 (Magma) I:=[1,15]; [n le 2 select I[n] else 14*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 06 2019
%o A028230 (GAP) a:=[1,15];; for n in [3..30] do a[n]:=14*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 06 2019
%Y A028230 Cf. A001353, A001570, A001834, A001835, A001921, A007655, A036428, A046184, A049310, A094347.
%Y A028230 Cf. A077416 with companion A077417.
%K A028230 nonn,easy
%O A028230 1,2
%A A028230 _N. J. A. Sloane_
%E A028230 Additional comments from _Wolfdieter Lang_, Nov 29 2002
%E A028230 Incorrect recurrence relation deleted by _Ant King_, Nov 15 2011
%E A028230 Minor edits by _Vaclav Kotesovec_, Jan 28 2015