|
| |
|
|
A024495
|
|
C(n,2) + C(n,5) + ... + C(n,3*[n/3]+2).
|
|
37
| |
|
|
0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846, 43691, 87381, 174762, 349525, 699051, 1398102, 2796203, 5592405, 11184810, 22369621, 44739243, 89478486, 178956971, 357913941, 715827882, 1431655765, 2863311531, 5726623062
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Trisections give A082365, A132804, A132805. - Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2007
This is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a_n=a_{n-1}+n-3, a(1)=0; a(2)=0; a(3)=1; and so on. - Srikanth.K.S (sriperso(AT)gmail.com), Jan 23 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix
[1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] =
[11, 11, 10], sum = 2^5 = 32. (End)
For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2-3*i/2+1 different types of i, (i=1,2,...). [From Milan R. Janjic (agnus(AT)blic.net), Sep 24 2010]
|
|
|
REFERENCES
| D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.
|
|
|
LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,2).
Eric Weisstein's World of Mathematics, Plane division by lines
|
|
|
FORMULA
| a(n) = (1/3)*(2^n+2*cos( (n-4)*Pi/3 )).
a(n) = 2*a(n-1)+A010892(n-2) = a(n-1)+A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley (se16(AT)btinternet.com), Jun 04 2001
a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 04 2002
a(n) = sum{k=0..n, 2^k*2*sin(pi*(n-k)/3+pi/3)/sqrt(3)} (offset 0). - Paul Barry (pbarry(AT)wit.ie), May 18 2004
G.f.: 1/(x*(1-(x+x^2+x^3+...)^3)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 04 2004
G.f.: x^2/((1-x)^3-x^3).
a(n) = 3*a(n-1)-3*a(n-2)+2*a(n-3). - Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2007
a(n) + A024493(n-1) = 0, 1, 2, 4, 8 = A131577. Note 0, 1, 3, 6, 11, 21, 42, ... + A024493 = A000079. - Paul Curtz (bpcrtz(AT)free.fr), Jan 24 2008
From Paul Curtz, May 29 2011. (Start)
a(n)+a(n+3)= 3*2^n = A007283(n).
a(n+6)-a(n)= 21*2^n = A175805(n).
a(n)+a(n+9)= 171*2^n.
a(n+12)-a(n)= 1365*2^n. (End)
a(n)= A113405(n) + A113405(n+1). - Paul Curtz, Jun 05 2011
|
|
|
MATHEMATICA
| Join[{a = 0, b = 0}, Table[c = 2^n - a + b; a = b; b = c, {n, 0, 100}]] (* From Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
|
|
|
PROG
| (PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k+2)) /* Michael Somos Feb 14 2006 */
(PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[3, 1]) /* Michael Somos Feb 14 2006 */
|
|
|
CROSSREFS
| Cf. A010892, A083322.
Sequence in context: A050951 A202012 A132658 * A104253 A191581 A192896
Adjacent sequences: A024492 A024493 A024494 * A024496 A024497 A024498
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
|
| |
|
|
