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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+1).

%I A002315 M4423 N1869 #476 May 26 2025 04:32:20
%S A002315 1,7,41,239,1393,8119,47321,275807,1607521,9369319,54608393,318281039,
%T A002315 1855077841,10812186007,63018038201,367296043199,2140758220993,
%U A002315 12477253282759,72722761475561,423859315570607,2470433131948081,14398739476117879,83922003724759193
%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+1).
%C A002315 Named after the Newman-Shanks-Williams reference.
%C A002315 Also numbers k such that A125650(3*k^2) is an odd perfect square. Such numbers 3*k^2 form a bisection of A125651. - _Alexander Adamchuk_, 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_, Feb 06 2007
%C A002315 Also numbers m such that m^2 is a centered 16-gonal number; or a number of the form 8k(k+1)+1, where k = A053141(m) = {0, 2, 14, 84, 492, 2870, ...}. - _Alexander Adamchuk_, Apr 21 2007
%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_, 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_, Aug 27 2008
%C A002315 General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->oo} 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. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - _Ctibor O. Zizka_, Sep 02 2008
%C A002315 Numbers k such that (ceiling(sqrt(k*k/2)))^2 = (1+k*k)/2. - _Ctibor O. Zizka_, Nov 09 2009
%C A002315 A001109(n)/a(n) converges to cos^2(Pi/8) = 1/2 + 2^(1/2)/4. - _Gary Detlefs_, Nov 25 2009
%C A002315 The values 2(a(n)^2+1) are all perfect squares, whose square root is given by A075870. - Neelesh Bodas (neelesh.bodas(AT)gmail.com), Aug 13 2010
%C A002315 a(n) represents all positive integers K for which 2(K^2+1) is a perfect square. - Neelesh Bodas (neelesh.bodas(AT)gmail.com), Aug 13 2010
%C A002315 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). - _John M. Campbell_, Jul 08 2011
%C A002315 Integers k such that A000217(k-2) + A000217(k-1) + A000217(k) + A000217(k+1) is a square (cf. A202391). - _Max Alekseyev_, Dec 19 2011
%C A002315 Integer square roots of floor(k^2/2 - 1) or A047838. - _Richard R. Forberg_, Aug 01 2013
%C A002315 Remark: x^2 - 2*y^2 = +2*k^2, with positive k, and X^2 - 2*Y^2 = +2 reduce to the present Pell equation a^2 - 2*b^2 = -1 with x = k*X = 2*k*b and y = k*Y = k*a. (After a proposed solution for k = 3 by _Alexander Samokrutov_.) - _Wolfdieter Lang_, Aug 21 2015
%C A002315 If p is an odd prime, a((p-1)/2) == 1 (mod p). - _Altug Alkan_, Mar 17 2016
%C A002315 a(n)^2 + 1 = 2*b(n)^2, with b(n) = A001653(n), is the necessary and sufficient condition for a(n) to be a number k for which the diagonal of a 1 X k rectangle is an integer multiple of the diagonal of a 1 X 1 square. If squares are laid out thus along one diagonal of a horizontal 1 X a(n) rectangle, from the lower left corner to the upper right, the number of squares is b(n), and there will always be a square whose top corner lies exactly within the top edge of the rectangle. Numbering the squares 1 to b(n) from left to right, the number of the one square that has a corner in the top edge of the rectangle is c(n) = (2*b(n) - a(n) + 1)/2, which is A055997(n). The horizontal component of the corner of the square in the edge of the rectangle is also an integer, namely d(n) = a(n) - b(n), which is A001542(n). - _David Pasino_, Jun 30 2016
%C A002315 (a(n)^2)-th triangular number is a square; a(n)^2 = A008843(n) is a subsequence of A001108. - _Jaroslav Krizek_, Aug 05 2016
%C A002315 a(n-1)/A001653(n) is the closest rational approximation of sqrt(2) with a numerator not larger than a(n-1). These rational approximations together with those obtained from the sequences A001541 and A001542 give a complete set of closest rational approximations of sqrt(2) with restricted numerator or denominator. a(n-1)/A001653(n) < sqrt(2). - _A.H.M. Smeets_, May 28 2017
%C A002315 Consider the quadrant of a circle with center (0,0) bounded by the positive x and y axes. Now consider, as the start of a series, the circle contained within this quadrant which kisses both axes and the outer bounding circle. Consider further a succession of circles, each kissing the x-axis, the outer bounding circle, and the previous circle in the series. See Holmes link. The center of the n-th circle in this series is ((A001653(n)*sqrt(2)-1)/a(n-1), (A001653(n)*sqrt(2)-1)/a(n-1)^2), the y-coordinate also being its radius. It follows that a(n-1) is the cotangent of the angle subtended at point (0,0) by the center of the n-th circle in the series with respect to the x-axis. - _Graham Holmes_, Aug 31 2019
%C A002315 There is a link between the two sequences present at the numerator and at the denominator of the fractions that give the coordinates of the center of the kissing circles. A001653 is the sequence of numbers k such that 2*k^2 - 1 is a square, and here, we have 2*A001653(n)^2 - 1 = a(n-1)^2. - _Bernard Schott_, Sep 02 2019
%C A002315 Let G be a sequence satisfying G(i) = 2*G(i-1) + G(i-2) for arbitrary integers i and without regard to the initial values of G. Then a(n) = (G(i+4*n+2) - G(i))/(2*G(i+2*n+1)) as long as G(i+2*n+1) != 0. - _Klaus Purath_, Mar 25 2021
%C A002315 All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0 < a < b < c are given by a=A001542(n), b=A005319(n), c=A001542(n+1), x=A001541(n), y=A001653(n+1), z=A002315(n) with 0 < n. - _Michael Somos_, Jun 26 2022
%C A002315 3*a(n-1) is the n-th almost Lucas-cobalancing number of second type (see Tekcan and Erdem). - _Stefano Spezia_, Nov 26 2022
%C A002315 In Moret-Blanc (1881) on page 259 some solution of m^2 - 2n^2 = -1 are listed. The values of m give this sequence, and the values of n give A001653. - _Michael Somos_, Oct 25 2023
%C A002315 From _Klaus Purath_, May 11 2024: (Start)
%C A002315 For any two consecutive terms (a(n), a(n+1)) = (x,y): x^2 - 6xy + y^2 = 8 = A028884(1). In general, the following applies to all sequences (t) satisfying t(i) = 6t(i-1) - t(i-2) with t(0) = 1 and two consecutive terms (x,y): x^2 - 6xy + y^2 = A028884(t(1)-6). This includes and interprets the Feb 04 2014 comment on A001541 by _Colin Barker_ as well as the Mar 17 2021 comment on A054489 by _John O. Oladokun_ and the Sep 28 2008 formula on A038723 by _Michael Somos_. By analogy to this, for three consecutive terms (x,y,z) y^2 - xz = A028884(t(1)-6) always applies.
%C A002315 If (t) is a sequence satisfying t(k) = 7t(k-1) - 7t(k-2) + t(k-3) or t(k) = 6t(k-1) - t(k-2) without regard to initial values and including this sequence itself, then a(n) = (t(k+2n+1) - t(k))/(t(k+n+1) - t(k+n)) always applies, as long as t(k+n+1) - t(k+n) != 0 for integer k and n >= 0. (End)
%D A002315 Julio R. Bastida, 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)
%D A002315 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
%D A002315 Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
%D A002315 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 247.
%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 P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.
%D A002315 P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - _N. J. A. Sloane_, Mar 08 2022
%H A002315 Indranil Ghosh, <a href="/A002315/b002315.txt">Table of n, a(n) for n = 0..1303</a> (terms 0..200 from T. D. Noe)
%H A002315 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A002315 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 A002315 E. Barcucci et al., <a href="http://dx.doi.org/10.1016/S0012-365X(98)80008-3">A combinatorial interpretation of the recurrence f_{n+1} = 6 f_n - f_{n-1}</a>, Discrete Math., 190 (1998), 235-240.
%H A002315 Elena Barcucci, Antonio Bernini, and Renzo Pinzani, <a href="http://ceur-ws.org/Vol-2113/paper8.pdf">A Gray code for a regular language</a>, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018) Vol. 2113.
%H A002315 Hacène Belbachir and Yassine Otmani, <a href="http://math.colgate.edu/~integers/x27/x27.pdf">Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences</a>, Integers (2023) Vol. 23.
%H A002315 J. Bonin, L. Shapiro and R. Simion, <a href="http://dx.doi.org/10.1016/0378-3758(93)90032-2">Some q-analogues of the Schroeder numbers arising from combinatorial statistics on lattice paths</a>, H. Statistical Planning and Inference, 16, 1993, 35-55 (p. 50).
%H A002315 P. Catarino, H. Campos, and P. Vasco, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_45_from11to24.pdf">On some identities for balancing and cobalancing numbers</a>, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.
%H A002315 Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Duchi/duchi4.html">A Note on Rational Succession Rules</a>, J. Integer Seqs., Vol. 6, 2003.
%H A002315 Melissa Emory, <a href="https://www.emis.de/journals/INTEGERS/papers/m65/m65.Abstract.html">The Diophantine equation X^4 + Y^4 = D^2 Z^4 in quadratic fields</a>, INTEGERS 12 (2012), #A65. - _N. J. A. Sloane_, Feb 06 2013
%H A002315 S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.
%H A002315 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://cdm.ucalgary.ca/article/view/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 A002315 A. S. Fraenkel, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00138-2">On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications</a>, Discrete Mathematics 224 (2000), pp. 273-279.
%H A002315 A. S. Fraenkel, <a href="http://dx.doi.org/10.1016/S0304-3975(00)00062-1">Recent results and questions in combinatorial game complexities</a>, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
%H A002315 A. S. Fraenkel, <a href="http://dx.doi.org/10.1016/S0304-3975(01)00070-6">Arrays, numeration systems and Frankenstein games</a>, Theoret. Comput. Sci. 282 (2002), 271-284.
%H A002315 Bernard Frénicle de Bessy, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k3187867/f23.item">Solutio duorum problematum circa numeros cubos et quadratos</a>, (1657), page 9. Bibliothèque Nationale de Paris.
%H A002315 M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, <a href="http://www.jstor.org/stable/2968551">Problem 47</a>, Amer. Math. Monthly, 4 (1897), 25-28.
%H A002315 R. J. Hetherington, <a href="/A000129/a000129.pdf">Letter to N. J. A. Sloane, Oct 26 1974</a>.
%H A002315 Graham Holmes, <a href="/A002315/a002315.jpg">Kissing circles and cotangents</a>.
%H A002315 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A002315 Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H A002315 D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.
%H A002315 Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H A002315 aBa Mbirika, Janeè Schrader, and Jürgen Spilker, <a href="https://arxiv.org/abs/2301.05758">Pell and associated Pell braid sequences as GCDs of sums of k consecutive Pell, balancing, and related numbers</a>, arXiv:2301.05758 [math.NT], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Mbirika/mbir5.html">J. Int. Seq.</a> (2023) Vol. 26, Art. 23.6.4.
%H A002315 Donatella Merlini and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174.
%H A002315 Claude Moret-Blanc, <a href="http://www.numdam.org/item/NAM_1881_2_20__253_0.pdf">Questions Nouvelles d'Arithmétique Supérieure Proposées Par M. Edouard Lucas</a>, Nouvelles annales de mathématiques 2^e serie, tome 20 (1881), 253-265.
%H A002315 Morris Newman, Daniel Shanks, and H. C. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3826.pdf">Simple groups of square order and an interesting sequence of primes</a>, Acta Arith., 38 (1980/1981) 129-140.
%H A002315 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002315 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A002315 The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=NSWNumber">NSW number</a>.
%H A002315 S. F. Santana and J. L. Diaz-Barrero, <a href="http://cs.ucmo.edu/~mjms/2006.1/diazbar.pdf">Some properties of sums involving Pell numbers</a>, Missouri Journal of Mathematical Sciences 18(1), 2006.
%H A002315 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A002315 R. A. Sulanke, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r47">Bijective recurrences concerning Schroeder paths</a>, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
%H A002315 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SULANKE/sulanke.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002315 Ahmet Tekcan and Alper Erdem, <a href="https://arxiv.org/abs/2211.08907">General Terms of All Almost Balancing Numbers of First and Second Type</a>, arXiv:2211.08907 [math.NT], 2022.
%H A002315 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NSWNumber.html">NSW Number</a>.
%H A002315 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>.
%H A002315 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A002315 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A002315 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A002315 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H A002315 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A002315 a(n) = (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).
%F A002315 a(n) = A001109(n)+A001109(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+1))^2-1), n >= 0. [Pell equation a(n)^2 - 2*Pell(2*n+1)^2 = -1. - _Wolfdieter Lang_, Jul 11 2018]
%F A002315 G.f.: (1 + x)/(1 - 6*x + x^2). - _Simon Plouffe_ in his 1992 dissertation
%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 Limit_{n->oo} a(n)/a(n-1) = 3 + 2*sqrt(2). - _Gregory V. Richardson_, 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_, Nov 10 2002
%F A002315 With a=3+2*sqrt(2), b=3-2*sqrt(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 sigma(i^2+1, 2) mod 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_, Jun 01 2005
%F A002315 a(n) = A001652(n)+A046090(n); e.g., 239=119+120. - _Charlie Marion_, 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_, 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_, 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_, Jan 28 2007
%F A002315 a(n) = 2*A001652(n) + 1 = 2*A046729(n) + (-1)^n. - _Lekraj Beedassy_, 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_, Apr 21 2007
%F A002315 a(n+1) = 3*a(n) + sqrt(8*a(n)^2 + 8), a(1)=1. - _Richard Choulet_, Sep 18 2007
%F A002315 a(n) = A001333(2*n+1). - _Ctibor O. Zizka_, Aug 13 2008
%F A002315 a(n) = third binomial transform of 1, 4, 8, 32, 64, 256, 512, ... . - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%F A002315 a(n) = (-1)^(n-1)*(1/sqrt(-1))*cos((2*n - 1)*arcsin(sqrt(2)). - _Artur Jasinski_, Feb 17 2010 *WRONG*
%F A002315 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
%F A002315 In general, a(n+k) = A001541(k)*a(n)) + sqrt(A001108(2k)*(a(n)^2+1)). See Sep 18 2007 entry above. - _Charlie Marion_, Dec 07 2011
%F A002315 a(n) = floor((1+sqrt(2))^(2n+1))/2. - _Thomas Ordowski_, Jun 12 2012
%F A002315 (a(2n-1) + a(2n) + 8)/(8*a(n)) = A001653(n). - _Ignacio Larrosa Cañestro_, Jan 02 2015
%F A002315 (a(2n) + a(2n-1))/a(n) = 2*sqrt(2)*( (1 + sqrt(2))^(4*n) - (1 - sqrt(2))^(4*n))/((1 + sqrt(2))^(2*n+1) + (1 - sqrt(2))^(2*n+1)). [This was my solution to problem 5325, School Science and Mathematics 114 (No. 8, Dec 2014).] - _Henry Ricardo_, Feb 05 2015
%F A002315 From _Peter Bala_, Mar 22 2015: (Start)
%F A002315 The aerated sequence (b(n))n>=1 = [1, 0, 7, 0, 41, 0, 239, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -4, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047.
%F A002315 b(n) = 1/2*((-1)^n - 1)*Pell(n) + 1/2*(1 + (-1)^(n+1))*Pell(n+1). The o.g.f. is x*(1 + x^2)/(1 - 6*x^2 + x^4).
%F A002315 Exp( Sum_{n >= 1} 2*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*A026003(n-1)*x^n.
%F A002315 Exp( Sum_{n >= 1} (-2)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*A026003(n-1)*(-x)^n.
%F A002315 Exp( Sum_{n >= 1} 4*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*Pell(n)*x^n.
%F A002315 Exp( Sum_{n >= 1} (-4)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*Pell(n)*(-x)^n.
%F A002315 Exp( Sum_{n >= 1} 8*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 8*A119915(n)*x^n.
%F A002315 Exp( Sum_{n >= 1} (-8)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 8*A119915(n)*(-x)^n. Cf. A002878, A004146, A113224, and A192425. (End)
%F A002315 E.g.f.: (sqrt(2)*sinh(2*sqrt(2)*x) + cosh(2*sqrt(2)*x))*exp(3*x). - _Ilya Gutkovskiy_, Jun 30 2016
%F A002315 a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 2^k * 2^ceiling(k/2). - _David Pasino_, Jul 09 2016
%F A002315 a(n) = A001541(n) + 2*A001542(n). - _A.H.M. Smeets_, May 28 2017
%F A002315 a(n+1) = 3*a(n) + 4*b(n), b(n+1) = 2*a(n) + 3*b(n), with b(n)=A001653(n). - _Zak Seidov_, Jul 13 2017
%F A002315 a(n) = |Im(T(2n-1,i))|, i=sqrt(-1), T(n,x) is the Chebyshev polynomial of the first kind, Im is the imaginary part of a complex number, || is the absolute value. - _Leonid Bedratyuk_, Dec 17 2017
%F A002315 a(n) = sinh((2*n + 1)*arcsinh(1)). - _Bruno Berselli_, Apr 03 2018
%F A002315 a(n) = 5*a(n-1) + A003499(n-1), a(0) = 1. - _Ivan N. Ianakiev_, Aug 09 2019
%F A002315 From _Klaus Purath_, Mar 25 2021: (Start)
%F A002315 a(n) = A046090(2*n)/A001541(n).
%F A002315 a(n+1)*a(n+2) = a(n)*a(n+3) + 48.
%F A002315 a(n)^2 + a(n+1)^2 = 6*a(n)*a(n+1) + 8.
%F A002315 a(n+1)^2 = a(n)*a(n+2) + 8.
%F A002315 a(n+1) = a(n) + 2*A001541(n+1).
%F A002315 a(n) = 2*A046090(n) - 1. (End)
%F A002315 3*a(n-1) = sqrt(8*b(n)^2 + 8*b(n) - 7), where b(n) = A358682(n). - _Stefano Spezia_, Nov 26 2022
%F A002315 a(n) = -(-1)^n - 2 + Sum_{i=0..n} A002203(i)^2. - _Adam Mohamed_, Aug 22 2024
%F A002315 From _Peter Bala_, May 09 2025: (Start)
%F A002315 a(n) = Dir(n, 3), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.
%F A002315 For arbitrary x, a(n+x)^2 - 6*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 8 with a(n) := (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)) as above. The particular case x = 0 is noted above,
%F A002315 a(n+1/2) = sqrt(2) * A001542(n+1).
%F A002315 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/8 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A081554(n) + 1/A081554(n+1)).
%F A002315 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(2) (telescoping product: Product_{n = 1..k} ((a(n) + 1)/(a(n) - 1))^2 = 2*(1 - 1/A055997(k+2))). (End)
%e A002315 G.f. = 1 + 7*x + 41*x^2 + 239*x^3 + 1393*x^4 + 8119*x^5 + 17321*x^6 + ... - _Michael Somos_, Jun 26 2022
%p A002315 A002315 := proc(n)
%p A002315     option remember;
%p A002315     if n = 0 then
%p A002315         1 ;
%p A002315     elif n = 1 then
%p A002315         7;
%p A002315     else
%p A002315         6*procname(n-1)-procname(n-2) ;
%p A002315     end if;
%p A002315 end proc: # _Zerinvary Lajos_, Jul 26 2006, modified _R. J. Mathar_, Apr 30 2017
%p A002315 a:=n->abs(Im(simplify(ChebyshevT(2*n+1,I)))):seq(a(n),n=0..20); # _Leonid Bedratyuk_, Dec 17 2017
%p A002315 # third Maple program:
%p A002315 a:= n-> (<<0|1>, <-1|6>>^n. <<1, 7>>)[1, 1]:
%p A002315 seq(a(n), n=0..22);  # _Alois P. Heinz_, Aug 25 2024
%t A002315 a[0] = 1; a[1] = 7; a[n_] := a[n] = 6a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 20}] (* _Robert G. Wilson v_, Jun 09 2004 *)
%t A002315 Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{-1,6},#]}]&, {1,7},20]][[1]]  (* _Harvey P. Dale_, Mar 23 2011 *)
%t A002315 Table[ If[n>0, a=b; b=c; c=6b-a, b=-1; c=1], {n, 0, 20}] (* _Jean-François Alcover_, Oct 19 2012 *)
%t A002315 LinearRecurrence[{6, -1}, {1, 7}, 20] (* _Bruno Berselli_, Apr 03 2018 *)
%t A002315 a[ n_] := -I*(-1)^n*ChebyshevT[2*n + 1, I]; (* _Michael Somos_, Jun 26 2022 *)
%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) = my(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 (PARI) {a(n) = -I*(-1)^n*polchebyshev(2*n+1, 1, I)}; /* _Michael Somos_, Jun 26 2022 */
%o A002315 (Haskell)
%o A002315 a002315 n = a002315_list !! n
%o A002315 a002315_list = 1 : 7 : zipWith (-) (map (* 6) (tail a002315_list)) a002315_list
%o A002315 -- _Reinhard Zumkeller_, Jan 10 2012
%o A002315 (Magma) I:=[1,7]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 22 2015
%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. Cf. A077444.
%Y A002315 Cf. A125650, A125651, A125652.
%Y A002315 Row sums of unsigned triangle A127675.
%Y A002315 Cf. A053141, A075870. Cf. A000045, A002878, A004146, A026003, A100047, A119915, A192425, A088165 (prime subsequence), A057084 (binomial transform), A108051 (inverse binomial transform).
%Y A002315 See comments in A301383.
%Y A002315 Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%Y A002315 Cf. A000129, A001653, A003499, A358682, A002203.
%K A002315 nonn,easy,nice
%O A002315 0,2
%A A002315 _N. J. A. Sloane_