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

Author

N. J. A. Sloane, Sep 20 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A086377.

Cf. A088538 for decimal expansion of 4/Pi.

Programs

Magma

[(-1)*((1+n)*(-5-3*(-1)^n+(-1+(-1)^n)*n))/8: n in [0..70]]; // Vincenzo Librandi, Jan 28 2016

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 *)

PARI

Vec((1+x^2)*(x^2-x-1)/((x-1)^3*(1+x)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

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)