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%I #86 Mar 28 2024 21:54:16
%S 1,6,29,139,666,3191,15289,73254,350981,1681651,8057274,38604719,
%T 184966321,886226886,4246168109,20344613659,97476900186,467039887271,
%U 2237722536169,10721572793574,51370141431701,246129134364931,1179275530392954,5650248517599839
%N Chebyshev even-indexed U-polynomials evaluated at sqrt(7)/2.
%C a(n) = L(n,-5)*(-1)^n, where L is defined as in A108299; see also A004253 for L(n,+5). - _Reinhard Zumkeller_, Jun 01 2005
%C 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 the present sequence. 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 The primes in this sequence are 29, 139, 3191, 15289, 350981, 1681651, ... - _Ctibor O. Zizka_, Sep 02 2008
%C Inverse binomial transform of A030240. - _Philippe Deléham_, Nov 19 2009
%C For positive n, a(n) equals the permanent of the (2n)X(2n) matrix with sqrt(7)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C The aerated sequence (b(n))n>=1 = [1, 0, 6, 0, 29, 0, 139, 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 = -3, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for a connection with Chebyshev polynomials. - _Peter Bala_, Mar 22 2015
%C From _Wolfdieter Lang_, Oct 26 2020: (Start)
%C ((-1)^n)*a(n) = X(n) = ((-1)^n)*(S(n, 5) + S(n-1, 5)) and Y(n) = X(n-1) gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 5*X*Y = +7, for n = -oo..+oo, with Chebyshev S polynomials (see A049310), with S(-1, x) = 0, and S(-n, x) = - S(n-2, x), for n >= 2.
%C This binary indefinite quadratic form of discriminant 21, representing 7, has only this family of proper solutions (modulo sign flip), and no improper ones.
%C This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)
%H G. C. Greubel, <a href="/A030221/b030221.txt">Table of n, a(n) for n = 0..1000</a>
%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 K. Dilcher and K. B. Stolarsky, <a href="http://www.maa.org/programs/maa-awards/writing-awards/a-pascal-type-triangle-characterizing-twin-primes">A Pascal-type triangle characterizing twin primes</a>, Amer. Math. Monthly, 112 (2005), 673-681. (see page 678)
%H 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 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 18.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=6.
%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 Donatella Merlini and Renzo Sprugnoli, <a href="https://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 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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F a(n) = 5*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
%F a(n) = U(2*n, sqrt(7)/2).
%F G.f.: (1+x)/(x^2-5*x+1).
%F a(n) = A004254(n) + A004254(n+1).
%F a(n) ~ (1/2 + (1/6)*sqrt(21))*((1/2)*(5 + sqrt(21)))^n. - Joe Keane (jgk(AT)jgk.org), May 16 2002
%F Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then a(n) = (-1)^n*q(n, -7). - _Benoit Cloitre_, Nov 10 2002
%F A054493(2*n) = a(n)^2 for all n in Z. - _Michael Somos_, Jan 22 2017
%F a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Jan 22 2017
%F 0 = -7 + a(n)*(+a(n) - 5*a(n+1)) + a(n+1)*(+a(n+1)) for all n in Z. - _Michael Somos_, Jan 22 2017
%F a(n) = S(n, 5) + S(n-1, 5) = S(2*n, sqrt(7) (see above in terms of U), for n >= 0 with S(-1, 5) = 0, where the coefficients of the Chebyshev S polynomials are given in A049310. - _Wolfdieter Lang_, Oct 26 2020
%e G.f. = 1 + 6*x + 29*x^2 + 139*x^3 + 666*x^4 + 3191*x^5 + 15289*x^6 + ...
%p A030221 := proc(n)
%p option remember;
%p if n <= 1 then
%p op(n+1,[1,6]);
%p else
%p 5*procname(n-1)-procname(n-2) ;
%p end if;
%p end proc: # _R. J. Mathar_, Apr 30 2017
%t t[n_, k_?EvenQ] := I^k*Binomial[n-k/2, k/2]; t[n_, k_?OddQ] := -I^(k-1)*Binomial[n+(1-k)/2-1, (k-1)/2]; l[n_, x_] := Sum[t[n, k]*x^(n-k), {k, 0, n}]; a[n_] := (-1)^n*l[n, -5]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 05 2013, after _Reinhard Zumkeller_ *)
%t a[ n_] := ChebyshevU[2 n, Sqrt[7]/2]; (* _Michael Somos_, Jan 22 2017 *)
%o (Sage) [(lucas_number2(n,5,1)-lucas_number2(n-1,5,1))/3 for n in range(1,22)] # _Zerinvary Lajos_, Nov 10 2009
%o (Magma) I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 22 2015
%o (PARI) {a(n) = simplify(polchebyshev(2*n, 2, quadgen(28)/2))}; /* _Michael Somos_, Jan 22 2017 */
%Y Cf. A004253, A004254, A100047, A054493 (partial sums), A049310, A003501 (first differences), A299109 (subsequence of primes).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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