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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). (Formerly M4423 N1869) 101

%I M4423 N1869

%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

%D Bastida, Julio R. 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 Fink, Alex, Richard Guy, and Mark Krusemeyer. "Partitions with parts occurring at most thrice." Contributions to Discrete Mathematics 3.2 (2008), 76-114. See Section 13.

%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.

%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, 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 Andersen, K., Carbone, L. and Penta, D., <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, 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 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, 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, Applied Mathematics</a>, 2014, 5, 2226-2234.

%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 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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%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 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 Morris Newman, Daniel Shanks, 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="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H The Prime Glossary, <a href="http://primes.utm.edu/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 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) = (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 Lim_{n->inf.} 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+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 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

%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 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 (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)

%t Round[Table[(-1)^(n - 1) (1/Sqrt[1 - 2]) Cos[(2 n - 1) ArcSin[Sqrt[2]]], {n, 1, 10}]] (* _Artur Jasinski_, Feb 17 2010 *)

%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 *)

%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) for (i=1,10000,if(Mod(sigma(i^2+1,2),2)==1,print1(i,",")))

%o (Sage) [(lucas_number2(n,6,1)-lucas_number2(n-1,6,1))/4 for n in xrange(1, 22)]# _Zerinvary Lajos_, Nov 10 2009

%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. Pairwise sums of A001109. 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 Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

%K nonn,easy,nice,changed

%O 0,2

%A _N. J. A. Sloane_

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)