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%I M4423 N1869 #407 Apr 18 2024 11:47:34
%S 1,7,41,239,1393,8119,47321,275807,1607521,9369319,54608393,318281039,
%T 1855077841,10812186007,63018038201,367296043199,2140758220993,
%U 12477253282759,72722761475561,423859315570607,2470433131948081,14398739476117879,83922003724759193
%N 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 Named after the Newman-Shanks-Williams reference.
%C Also numbers n such that A125650(3*n^2) is an odd perfect square. Such numbers 3*n^2 form a bisection of A125651. - _Alexander Adamchuk_, Nov 30 2006
%C 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 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_, Apr 21 2007
%C 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 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 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. 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 Numbers n such that (ceiling(sqrt(n*n/2)))^2 = (1+n*n)/2. - _Ctibor O. Zizka_, Nov 09 2009
%C A001109(n)/a(n) converges to cos^2(Pi/8) = 1/2 + 2^(1/2)/4. - _Gary Detlefs_, Nov 25 2009
%C 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 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 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 Integers n such that A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) is a square (cf. A202391). - _Max Alekseyev_, Dec 19 2011
%C Integer square roots of floor(n^2/2 - 1) or A047838. - _Richard R. Forberg_, Aug 01 2013
%C 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 If p is an odd prime, a((p-1)/2) == 1 (mod p). - _Altug Alkan_, Mar 17 2016
%C 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 (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 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 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 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 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 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 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 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
%D 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 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.
%D P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - _N. J. A. Sloane_, Mar 08 2022
%H 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 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 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 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 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 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 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 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 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 Melissa Emory, <a href="http://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. - From N. J. A. Sloane, Feb 06 2013
%H 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 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 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 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 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 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 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 R. J. Hetherington, <a href="/A000129/a000129.pdf">Letter to N. J. A. Sloane, Oct 26 1974</a>
%H Graham Holmes, <a href="/A002315/a002315.jpg">Kissing circles and cotangents</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H 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 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 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 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 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 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 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 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=NSWNumber">NSW number.</a>
%H 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 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NSWNumber.html">NSW Number.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>.
%H 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 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 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n) = (1/2)*((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).
%F a(n) = A001109(n)+A001109(n+1).
%F a(n) = (1+sqrt(2))/2*(3+sqrt(8))^n+(1-sqrt(2))/2*(3-sqrt(8))^n. - _Ralf Stephan_, Feb 23 2003
%F 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 G.f.: (1 + x)/(1 - 6*x + x^2). - _Simon Plouffe_ in his 1992 dissertation
%F 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 a(n) ~ (1/2)*(sqrt(2) + 1)^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F Limit_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - _Gregory V. Richardson_, Oct 06 2002
%F 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 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 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 Same as: i such that sigma(i^2+1, 2) mod 2 = 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
%F 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 a(n) = A001652(n)+A046090(n); e.g., 239=119+120. - _Charlie Marion_, Nov 20 2003
%F 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 a(n) = Jacobi_P(n,1/2,-1/2,3)/Jacobi_P(n,-1/2,1/2,1). - _Paul Barry_, Feb 03 2006
%F 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 a(n) = 2*A001652(n) + 1 = 2*A046729(n) + (-1)^n. - _Lekraj Beedassy_, Feb 06 2007
%F a(n) = sqrt(A001108(2*n+1)). - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007
%F a(n) = sqrt(8*A053141(n)*(A053141(n) + 1) + 1). - _Alexander Adamchuk_, Apr 21 2007
%F a(n+1) = 3*a(n) + sqrt(8*a(n)^2 + 8), a(1)=1. - _Richard Choulet_, Sep 18 2007
%F a(n) = A001333(2*n+1). - _Ctibor O. Zizka_, Aug 13 2008
%F a(n) = third binomial transform of 1, 4, 8, 32, 64, 256, 512, ... . - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009
%F a(n) = (-1)^(n-1)*(1/sqrt(-1))*cos((2*n - 1)*arcsin(sqrt(2)). - _Artur Jasinski_, Feb 17 2010
%F 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 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 a(n) = floor((1+sqrt(2))^(2n+1))/2. - _Thomas Ordowski_, Jun 12 2012
%F (a(2n-1) + a(2n) + 8)/(8*a(n)) = A001653(n). - _Ignacio Larrosa Cañestro_, Jan 02 2015
%F (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 From _Peter Bala_, Mar 22 2015: (Start)
%F 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 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 Exp( Sum_{n >= 1} 2*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*A026003(n-1)*x^n.
%F Exp( Sum_{n >= 1} (-2)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*A026003(n-1)*(-x)^n.
%F Exp( Sum_{n >= 1} 4*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*Pell(n)*x^n.
%F Exp( Sum_{n >= 1} (-4)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*Pell(n)*(-x)^n.
%F Exp( Sum_{n >= 1} 8*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 8*A119915(n)*x^n.
%F 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 E.g.f.: (sqrt(2)*sinh(2*sqrt(2)*x) + cosh(2*sqrt(2)*x))*exp(3*x). - _Ilya Gutkovskiy_, Jun 30 2016
%F a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 2^k * 2^ceiling(k/2). - _David Pasino_, Jul 09 2016
%F a(n) = A001541(n) + 2*A001542(n). - _A.H.M. Smeets_, May 28 2017
%F 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 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 a(n) = sinh((2*n + 1)*arcsinh(1)). - _Bruno Berselli_, Apr 03 2018
%F a(n) = 5*a(n-1) + A003499(n-1), a(0) = 1. - _Ivan N. Ianakiev_, Aug 09 2019
%F From _Klaus Purath_, Mar 25 2021: (Start)
%F a(n) = A046090(2*n)/A001541(n).
%F a(n+1)*a(n+2) = a(n)*a(n+3) + 48.
%F a(n)^2 + a(n+1)^2 = 6*a(n)*a(n+1) + 8.
%F a(n+1)^2 = a(n)*a(n+2) + 8.
%F a(n+1) = a(n) + 2*A001541(n+1).
%F a(n) = 2*A046090(n) - 1. (End)
%F 3*a(n-1) = sqrt(8*b(n)^2 + 8*b(n) - 7), where b(n) = A358682(n). - _Stefano Spezia_, Nov 26 2022
%e 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 := proc(n)
%p option remember;
%p if n = 0 then
%p 1 ;
%p elif n = 1 then
%p 7;
%p else
%p 6*procname(n-1)-procname(n-2) ;
%p end if;
%p end proc: # _Zerinvary Lajos_, Jul 26 2006, modified _R. J. Mathar_, Apr 30 2017
%p a:=n->abs(Im(simplify(ChebyshevT(2*n+1,I)))):seq(a(n),n=0..20); # _Leonid Bedratyuk_, Dec 17 2017
%t 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 Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{-1,6},#]}]&, {1,7},20]][[1]] (* _Harvey P. Dale_, Mar 23 2011 *)
%t 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 LinearRecurrence[{6, -1}, {1, 7}, 20] (* _Bruno Berselli_, Apr 03 2018 *)
%t a[ n_] := -I*(-1)^n*ChebyshevT[2*n + 1, I]; (* _Michael Somos_, Jun 26 2022 *)
%o (PARI) {a(n) = subst(poltchebi(abs(n+1)) - poltchebi(abs(n)), x, 3)/2};
%o (PARI) {a(n) = if(n<0, -a(-1-n), polsym(x^2-2*x-1, 2*n+1)[2*n+2]/2)};
%o (PARI) {a(n) = my(w=3+quadgen(32)); imag((1+w)*w^n)};
%o (PARI) for (i=1,10000,if(Mod(sigma(i^2+1,2),2)==1,print1(i,",")))
%o (PARI) {a(n) = -I*(-1)^n*polchebyshev(2*n+1, 1, I)}; /* _Michael Somos_, Jun 26 2022 */
%o (Haskell)
%o a002315 n = a002315_list !! n
%o a002315_list = 1 : 7 : zipWith (-) (map (* 6) (tail a002315_list)) a002315_list
%o -- _Reinhard Zumkeller_, Jan 10 2012
%o (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 Bisection of A001333. Cf. A001109, A001653. A065513(n)=a(n)-1.
%Y First differences of A001108 and A055997. Bisection of A084068 and A088014. Cf. A077444.
%Y Cf. A125650, A125651, A125652.
%Y Row sums of unsigned triangle A127675.
%Y Cf. A053141, A075870. Cf. A000045, A002878, A004146, A026003, A100047, A119915, A192425, A088165 (prime subsequence), A057084 (binomial transform), A108051 (inverse binomial transform).
%Y See comments in A301383.
%Y Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.
%Y Cf. A000129, A001653, A003499, A358682.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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