A182432 Recurrence a(n)*a(n-2) = a(n-1)*(a(n-1)+3) with a(0) = 1, a(1) = 4.

1, 4, 28, 217, 1705, 13420, 105652, 831793, 6548689, 51557716, 405913036, 3195746569, 25160059513, 198084729532, 1559517776740, 12278057484385, 96664942098337, 761041479302308, 5991666892320124, 47172293659258681, 371386682381749321

Offset

0,2

Comments

The non-linear recurrence equation a(n)*a(n-2) = a(n-1)*(a(n-1)+r) with initial conditions a(0) = 1, a(1) = 1+r has the solution a(n) = 1/2 + 1/2*sum_{k=0..n} (2*r)^k*binomial(n+k,2*k) = 1/2 + b(n,2*r)/2, where b(n,x) are the Morgan-Voyce polynomials of A085478. The recurrence produces sequences A101265 (r = 1), A011900 (r = 2) and A054318 (r = 4), as well as signed versions of A133872 (r = -1), A109613(r = -2), A146983 (r = -3) and A084159(r = -4).

Also the indices of centered pentagonal numbers (A005891) which are also centered triangular numbers (A005448). - Colin Barker, Jan 01 2015

Also positive integers y in the solutions to 3*x^2 - 5*y^2 - 3*x + 5*y = 0. - Colin Barker, Jan 01 2015

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1116 (first 201 terms from Vincenzo Librandi)

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.

Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Formula

a(n) = 1/2 + 1/2*sum_{k = 0..n} 6^k*binomial(n+k,2*k).

a(n) = R(n,3) where R(n,x) denotes the row polynomials of A211955.

a(n) = 1/u*T(n,u)*T(n+1,u) with u = sqrt(5/2) and T(n,x) the Chebyshev polynomial of the first kind.

Recurrence equation: a(n) = 8*a(n-1) - a(n-2)-3 with a(0) = 1, a(1) = 4.

O.g.f.: (1 - 5*x + x^2)/((1 - x)*(1 - 8*x + x^2)) = 1 + 4*x + 28*x^2 + ....

Sum_{n>=0} 1/a(n) = sqrt(5/3); 5 - 3*(sum_{n=0..2*n} 1/a(k))^2 = 2/A070997(n)^2.

a(0)=1, a(1)=4, a(2)=28, a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3). - Harvey P. Dale, May 14 2012

Mathematica

RecurrenceTable[{a[0]==1, a[1]==4, a[n]==(a[-1+n] (3+a[-1+n]))/a [-2+n]}, a[n], {n, 30}] (* or *) LinearRecurrence[{9, -9, 1}, {1, 4, 28}, 30] (* Harvey P. Dale, May 14 2012 *)

Prog

(Magma) I:=[1, 4, 28]; [n le 3 select I[n] else 9*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..25]]; // Vincenzo Librandi, May 18 2012
(PARI) Vec((1-5*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

Crossrefs

Cf. A011900, A054318, A070997, A101265, A109613, A133872, A146983, A211955.

Sequence in context: A213232 A039741 A130185 * A026020 A243116 A026033

Adjacent sequences: A182429 A182430 A182431 * A182433 A182434 A182435

Keyword

nonn,easy

Author

Peter Bala, Apr 30 2012

Status

approved