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A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2). 42
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; text; internal format)
OFFSET

0,4

COMMENTS

Trisections give A082365, A132804, A132805. - Paul Curtz, Nov 18 2007

If the offset is changed to 1, 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, Jan 23 2008

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. Gary W. Adamson, Mar 13 2009

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,...). - Milan Janjic, Sep 24 2010

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - Stanislav Sykora, Jun 10 2012

REFERENCES

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

LINKS

Table of n, a(n) for n=0..34.

P. Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2, example 9.

Eric Weisstein's World of Mathematics, Plane division by lines

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

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, Jun 04 2001

a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre, 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, May 18 2004

G.f.: 1/(x*(1-(x + x^2 + x^3 + ...)^3)) - Jon Perry, Jul 04 2004

G.f.: x^2/((1-x)^3 - x^3) =  -x^2 / ( (2*x-1)*(x^2-x+1) ).

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Paul Curtz, 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, 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

Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - Stanislav Sykora, Jun 10 2012

G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013

a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - John M. Campbell, Dec 23 2016

a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n + 3)/6) + 16*floor((n + 4)/6) - 24*floor((n + 5)/6) - 22*floor((n + 7)/6) + 21*floor((n + 8)/6) + 10*floor((n + 9)/6) + 5*floor((n + 10)/6) + 3*floor((n + 11)/6) + floor((n + 13)/6)). - John M. Campbell, Dec 24 2016

MATHEMATICA

Join[{a = 0, b = 0}, Table[c = 2^n - a + b; a = b; b = c, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)

LinearRecurrence[{3, -3, 2}, {0, 0, 1}, 40] (* Harvey P. Dale, Sep 20 2016 *)

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: A261392 A251655 A132658 * A104253 A283668 A191581

Adjacent sequences:  A024492 A024493 A024494 * A024496 A024497 A024498

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 25 22:15 EDT 2017. Contains 288730 sequences.