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A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2). 45
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

{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017

This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

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.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

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) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n -A057079(n))/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.: 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) = A131577(n). - 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

a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - Vladimir Shevelev, Aug 01 2017

MATHEMATICA

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: A251655 A132658 A293320 * A293066 A104253 A283668

Adjacent sequences:  A024492 A024493 A024494 * A024496 A024497 A024498

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 17 16:34 EDT 2017. Contains 293471 sequences.