A084159 Pell oblongs.

1, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041, 7199369738058939, 41961001862379597, 244566641436218639

Offset: 0

Author

Paul Barry, May 18 2003

Keywords

• easy
• nonn

Comments

Essentially the same as A046727.

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 60 at p. 123.

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).

Crossrefs

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

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

Programs

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

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 *)

SageMath

[(lucas_number2(2*n+1, 2, -1) + 2*(-1)^n)/4 for n in range(31)] # G. C. Greubel, Oct 11 2022

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

E.g.f.: exp(-x)*(1 + exp(4*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/2. - Stefano Spezia, Aug 03 2024