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Search: id:A024494
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| A024494 |
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C(n,1) + C(n,4) + ... + C(n,3[n/3]+1). |
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+0
13
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| 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811, 22369622, 44739243, 89478485, 178956970, 357913941, 715827883, 1431655766, 2863311531, 5726623061, 11453246122
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n * [1,0,0] = [A024493(n), A024495(n), a(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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FORMULA
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a(n) = (1/3)*(2^n+2*cos( (n-2)*Pi/3 )).
G.f.: x(1-x)/((1-2x)(1-x+x^2)) - Paul Barry (pbarry(AT)wit.ie), Feb 11 2004
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.: (x*(1-x^2)*(1-x^3)/(1-x^6))/(1-2*x) . - Michael Somos Feb 14 2006
a(n+1)-2a(n)=A010892(n+1). - Michael Somos Feb 14 2006
a(n)=3a(n-1)-3a(n-2)+2a(n-3). - Paul Curtz (bpcrtz(AT)free.fr), Nov 20 2007
Equals binomial transform of (1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2008
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PROGRAM
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(PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k+1)) /* Michael Somos Feb 14 2006 */
(PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[2, 1]) /* Michael Somos Feb 14 2006 */
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CROSSREFS
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Cf. A010892. See A131708 for another version.
Sequence in context: A125312 A014626 A132418 this_sequence A131708 A002991 A022861
Adjacent sequences: A024491 A024492 A024493 this_sequence A024495 A024496 A024497
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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