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Search: id:A024495
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| A024495 |
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C(n,2) + C(n,5) + ... + C(n,3[n/3]+2). |
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+0
33
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| 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
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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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)
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REFERENCES
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D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
Eric Weisstein's World of Mathematics, Plane division by lines
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FORMULA
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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*2sin(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+1)-2a(n) = A010892(n-1) if n>0.
a(n) = 3a(n-1)-3a(n-2)+2a(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
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A010892.
Sequence in context: A018174 A050951 A132658 this_sequence A104253 A115030 A018177
Adjacent sequences: A024492 A024493 A024494 this_sequence A024496 A024497 A024498
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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