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A024493
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C(n,0) + C(n,3) + ... + C(n,3[n/3]).
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22
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1, 1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| First differences of A131708. First differences give A024495. - Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2007
a(n) = upper left term of X^n, where X = the 4 X 4 matrix [1,0,1,0; 1,1,0,0; 0,1,1,1; 0,0,0,1]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 01 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n * [1,0,0] = [a(n), A024495(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
| 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
| a(n) = (1/3)*(2^n+2*cos( n*Pi/3 )).
G.f.: (1-x)^2/((1-2x)(1-x+x^2))=(1-2x+x^2)/(1-3x+3x^2-2x^3) - Paul Barry (pbarry(AT)wit.ie), Feb 11 2004
a(n)=(1/3)*(2^n+b(n)) where b(n) is the 6-periodic sequence {2, 1, -1, -2, -1, 1}. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 23 2004
Binomial transform of 1/(1-x^3). G.f. : (1-x)^2/((1-x)^3-x^3)=x/(1-x-2x^2)+1/(1+x^3); a(n)=sum{k=0..floor(n/3), binomial(n, 3k)}; a(n)=sum{k=0..n, binomial(n, k)(cos(2*pi*k/3+pi/3)/3+sin(2*pi*k/3+pi/3)/sqrt(3)+1/3)}; a(n)=A001045(n)+sqrt(3)cos(pi*n/3+pi/6)/3+sin(pi*n/3+pi*/6)/3+(-1)^n/3. - Paul Barry (pbarry(AT)wit.ie), Jul 25 2004
a(n)=sum{k=0..n, binomial(n, 3(n-k))} - Paul Barry (pbarry(AT)wit.ie), Aug 30 2004
G.f.: ((1-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). - Michael Somos Feb 14 2006
E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A079978. - Geoffrey Critzer, Dec 27 2011.
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MATHEMATICA
| nn = 18; a = Sum[x^(3 i)/(3 i)!, {i, 0, nn}]; b = Exp[x]; Range[0, nn]! CoefficientList[Series[a b , {x, 0, nn}], x] (*Geoffrey Critzer, Dec 27 2011*)
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PROG
| (PARI) a(n)=sum(i=0, n, sum(j=0, n, if(n-i-3*j, 0, n!/(i)!/(3*j)!)))
(PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k)) /* Michael Somos Feb 14 2006 */
(PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[1, 1]) /* Michael Somos Feb 14 2006 */
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CROSSREFS
| Row sums of A098172.
Cf. A024494, A094715, A094717.
Sequence in context: A091357 A129715 * A130781 A071015 A084188 A044432
Adjacent sequences: A024490 A024491 A024492 * A024494 A024495 A024496
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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