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A030221 Chebyshev even indexed U-polynomials evaluated at sqrt(7)/2. 30
1, 6, 29, 139, 666, 3191, 15289, 73254, 350981, 1681651, 8057274, 38604719, 184966321, 886226886, 4246168109, 20344613659, 97476900186, 467039887271, 2237722536169, 10721572793574, 51370141431701, 246129134364931, 1179275530392954, 5650248517599839 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [Ctibor O. Zizka, Sep 02 2008]

Inverse binomial transform of A030240. [Philippe Deléham, Nov 19 2009]

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]

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

REFERENCES

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Merlini, Donatella, and Renzo Sprugnoli. "Arithmetic into geometric progressions through Riordan arrays." Discrete Mathematics 340.2 (2017): 160-174.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681. (see page 678)

Tanya Khovanova, Recursive Sequences

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=6.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,-1).

FORMULA

a(n) = 5*a(n-1)-a(n-2), a(-1)=-1, a(0)=1.

a(n) = U(2*n, sqrt(7)/2).

g.f.: (1+x)/(x^2-5*x+1).

a(n) = A004254(n) + A004254(n+1).

a(n) ~ (1/2 + 1/6*sqrt(21))*(1/2*(5 + sqrt(21)))^n. - Joe Keane (jgk(AT)jgk.org), May 16 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -7)=a(n). - Benoit Cloitre, Nov 10 2002

A054493(2*n) = a(n)^2 for all n in Z. - Michael Somos, Jan 22 2017

a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jan 22 2017

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

EXAMPLE

G.f. = 1 + 6*x + 29*x^2 + 139*x^3 + 666*x^4 + 3191*x^5 + 15289*x^6 + ...

MATHEMATICA

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 *)

a[ n_] := ChebyshevU[2 n, Sqrt[7]/2]; (* Michael Somos, Jan 22 2017 *)

PROG

(Sage) [(lucas_number2(n, 5, 1)-lucas_number2(n-1, 5, 1))/3 for n in xrange(1, 22)] # Zerinvary Lajos, Nov 10 2009

(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

(PARI) {a(n) = simplify(polchebyshev(2*n, 2, quadgen(28)/2))}; /* Michael Somos, Jan 22 2017 */

CROSSREFS

Cf. A004253, A004254, A100047, A054493.

Sequence in context: A045445 A026884 A110311 * A271753 A009153 A012325

Adjacent sequences:  A030218 A030219 A030220 * A030222 A030223 A030224

KEYWORD

nonn,easy,changed

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified April 25 23:17 EDT 2017. Contains 285426 sequences.