A079097 Mix odd numbers and squares.

1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, 36, 13, 49, 15, 64, 17, 81, 19, 100, 21, 121, 23, 144, 25, 169, 27, 196, 29, 225, 31, 256, 33, 289, 35, 324, 37, 361, 39, 400, 41, 441, 43, 484, 45, 529, 47, 576, 49, 625, 51, 676, 53, 729, 55, 784, 57, 841, 59, 900, 61, 961, 63, 1024, 65

Offset

0,3

Comments

The old entry with this sequence number was a duplicate of A010892.

This sequence is visible in the identity 4/Pi = 1 + 1/(3 + 4/(5 + 9/(7 + 16/(9 + 25/(11 + ...))))) (see the Wolfram Research web site).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Wolfram Research, Pi

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: (1+x^2)*(x^2-x-1)/((x-1)^3*(1+x)^3). - R. J. Mathar, Jan 05 2009

From Colin Barker, Jan 27 2016: (Start)

a(n) = (-1)*((1+n)*(-5-3*(-1)^n+(-1+(-1)^n)*n))/8.

a(n) = n-1 for n even.

a(n) = (n^2+2*n+1)/4 for n odd.

(End)

Maple

f:=n->if n mod 2 = 0 then n+1 else ((n+1)/2)^2; fi;

Mathematica

Riffle[2 Range@ Floor[#/2] - 1, Range[#]^2] &@66 (* or *) CoefficientList[Series[(1 + x^2) (x^2 - x - 1)/((x - 1)^3*(1 + x)^3), {x, 0, 64}], x] (* Michael De Vlieger, Jan 27 2016 *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 1, 3, 4, 5, 9}, 70] (* Vincenzo Librandi, Jan 28 2016 *)

Prog

(PARI) Vec((1+x^2)*(x^2-x-1)/((x-1)^3*(1+x)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016
(Magma) [(-1)*((1+n)*(-5-3*(-1)^n+(-1+(-1)^n)*n))/8: n in [0..70]]; // Vincenzo Librandi, Jan 28 2016

Crossrefs

Cf. A086377.

Cf. A088538 for decimal expansion of 4/Pi.

Sequence in context: A324506 A342772 A065336 * A202475 A036710 A098801

Adjacent sequences: A079094 A079095 A079096 * A079098 A079099 A079100

Keyword

nonn,easy

Author

N. J. A. Sloane, Sep 20 2008

Status

approved