login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A201207 Half-convolution of sequence A000032 (Lucas) with itself. 2
4, 2, 7, 11, 27, 41, 84, 137, 270, 435, 826, 1338, 2488, 4024, 7353, 11899, 21461, 34723, 61960, 100255, 177344, 286947, 503892, 815316, 1422892, 2302286, 3996619, 6466667, 11173935, 18079805, 31114236 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

For the definition of the half-convolution of a sequence with itself see a comment on A201204. There the rule for the o.g.f. is given. Here the o.g.f. is (L(x)^2 + L2(x^2))/2, with the o.g.f. L(x)=(2-x)/(1-x-x^2) of A000032, and L2(x)= (4-7*x-x^2)/((1+x)*(1-3*x+x^2)) the o.g.f. of A001254.

  This leads to the o.g.f given in the formula section.

For the bisection of this sequence see A203570 and A203574.

LINKS

Table of n, a(n) for n=0..30.

FORMULA

a(n) = sum(L(k)*L(n-k),n=0..floor(n/2)),n>=0, with the Lucas numbers L(n)=A000032(n).

O.g.f.: (4-2*x-7*x^2+6*x^3-x^4+3*x^5)/((1-3*x^2+x^4)*(1+x^2)*(1-x-x^2)). See a comment above.

a(n) = (1/4)*(2*(2*n+5+(-1)^n)*F(n+1)-(2*n+3+(-1)^n)*F(n)) +(i^n+(-i)^n)/2, n>=0, with the Fibonacci numbers F(n)=A000045(n) and the imaginary unit i=sqrt(-1). From the partial fraction decomposition of the o.g.f. and the Fibonacci recurrence.

CROSSREFS

Cf. A000032, A000045, A201204, A203570, A203574.

Sequence in context: A120871 A019689 A072009 * A151890 A227352 A108167

Adjacent sequences:  A201204 A201205 A201206 * A201208 A201209 A201210

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 03 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 21 10:46 EST 2014. Contains 252305 sequences.