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A084159 Pell oblongs. 14
1, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Essentially the same as A046727.

LINKS

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

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)). [From R. J. Mathar, Sep 17 2008]

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

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

Contribution 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..inf} (-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)

MATHEMATICA

Join[{1}, Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 2, 50}] * Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 1, 49}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011*)

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

CROSSREFS

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

Cf. A084158, A084175, A001654.

Cf. A090390, A182432, A211955.

Sequence in context: A005057 A092634 A178537 * A046727 A117512 A068127

Adjacent sequences:  A084156 A084157 A084158 * A084160 A084161 A084162

KEYWORD

easy,nonn

AUTHOR

Paul Barry, May 18 2003

STATUS

approved

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Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.