A084159 Pell oblongs.

1, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041

Offset

0,2

Comments

Essentially the same as A046727.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. E. Trier, "Almost Isosceles" Right-Angled Triangles, Eureka, No. 4, May 1940, pp. 9 - 11.

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

Formula

a(n) = ((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) + 2*(-1)^n)/4.

a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). - Paul Curtz, May 17 2008

G.f.: (1-x)^2/((1+x)*(1-6*x+x^2)). - R. J. Mathar, Sep 17 2008

a(n) = A078057(n)*A001333(n). - R. J. Mathar, Jul 08 2009

a(n) = A001333(n)*A001333(n+1).

From Peter Bala, May 01 2012: (Start)

a(n) = (-1)^n*R(n,-4), where R(n,x) is the n-th row polynomial of A211955.

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

a(n) = (-1)^n + 4*Sum_{k = 1..n} (-1)^(n-k)*8^(k-1)*binomial(n+k,2*k).

Recurrence equations: a(n) = 6*a(n-1) - a(n-2) + 4*(-1)^n, with a(0) = 1 and a(1) = 3; a(n)*a(n-2) = a(n-1)*(a(n-1)+4*(-1)^n).

Sum_{k >= 0} (-1)^k/a(k) = 1/sqrt(2).

1 - 2*(Sum_{k = 0..n} (-1)^k/a(k))^2 = (-1)^(n+1)/A090390(n+1). (End)

a(n) = (A001333(2*n+1) + (-1)^n)/2. - G. C. Greubel, Oct 11 2022

Mathematica

b[n_]:= Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]];
Join[{1}, Table[b[n+1], {n, 50}]*Table[b[n], {n, 50}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
LinearRecurrence[{5, 5, -1}, {1, 3, 21}, 30] (* Harvey P. Dale, Aug 04 2019 *)

Prog

(Magma) [Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)+2*(-1)^n)/4): n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
(SageMath) [(lucas_number2(2*n+1, 2, -1) + 2*(-1)^n)/4 for n in range(31)] # G. C. Greubel, Oct 11 2022

Crossrefs

Cf. A046727 (same sequence except for first term).

Cf. A001333, A001654, A078057, A084158, A084175, A090390, A182432, A211955.

Sequence in context: A005057 A092634 A178537 * A046727 A283421 A117512

Adjacent sequences: A084156 A084157 A084158 * A084160 A084161 A084162

Keyword

easy,nonn

Author

Paul Barry, May 18 2003

Status

approved