The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. 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 FORMULA a(n) = Sum_{n=0..floor(n/2)} (L(k)*L(n-k), 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: A332651 A072009 A257502 * A151890 A227352 A255140 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 | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 17 02:16 EDT 2021. Contains 343059 sequences. (Running on oeis4.)