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A101265
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a(1) = 1, a(2) = 2, a(3) = 6; a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n>3.
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6
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1, 2, 6, 21, 77, 286, 1066, 3977, 14841, 55386, 206702, 771421, 2878981, 10744502, 40099026, 149651601, 558507377, 2084377906, 7779004246, 29031639077, 108347552061, 404358569166, 1509086724602, 5631988329241, 21018866592361
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Let M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; then [1,0,0]*M^n = [a(n), A001353(n), A061278(n-1)] for n>1; Further a(n) = A061278(n) + 1; A001353(n) is first difference of a(n) and first difference of A061278(n). Let v(n) = [1,0,0]*M^n, then sum(v_i(n)) = A001075(n) for n>=0; and v_1(n) + v_3(n) = A001835(n) for n>=0; Characteristic polynomial of M = x^3 - 5x^2 + 5x - 1; a(n)/a(n-1) tends to 2 + sqrt(3) = 3.732.... (see A019973) (a root of the polynomial and an eigenvalue of the matrix).
a(n) = 1 + A061278(n). Also numbers n such that the RootMeanSquare(1,...,6*n-5) is an integer. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Dec 17 2008]
Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jul 01 2010: (Start)
a(n) blue and b(n) red balls in an urn; draw 3 balls without return;
Probability(3 red balls)=Probability(1 red and 2 blue balls);
binomial(b(n),3)=binomial(b(n),1)*binomial(a(n),2); b(n)=A179167(n);
(End)
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FORMULA
| a(n) = A005246(n)*A005246(n+1). a(n+1) = a(n)*(a(n)+1)/a(n-1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 24 2006
a(n) = (A001835(n) + 1) / 2. - Ralf Stephan, May 16 2007
O.g.f.: x(1-3x+x^2)/((1-x)(1-4x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 22 2008]
a(n) = 4a(n - 1) - a(n - 2) - 1 [From Naoki Sato (nsato7(AT)yahoo.ca), Jan 21 2010]
Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jul 01 2010: (Start)
a(n)=(6+(3+r)*(2+r)^(n-1)+(3-r)*(2-r)^(n-1))/12; r=sqrt(3);
(End)
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EXAMPLE
| Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jul 01 2010: (Start)
for n=5 a(5)=77; b(5)=134; binomial(134,3)=392084;
binomial(134,1)*binomial(77,2)=134*2926=392084;
(End)
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MAPLE
| Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Jul 01 2010: (Start)
r:=sqrt(3): for n from 1 to 100 do
a[n]:=(6+(3+r)*(2+r)^(n-1)+(3-r)*(2-r)^(n-1))/12: end do:
a[n]:=round((6+(3+r)*(2+r)^(n-1))/12):
(End)
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PROG
| (PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1, 30, print1(([1, 0, 0]*M^i)[1], ", "))
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CROSSREFS
| Cf. A001353, A061278, A001835, A001075, A019973.
Cf. A005246.
Sequence in context: A144904 A151287 * A101879 A063023 A150188 A150189
Adjacent sequences: A101262 A101263 A101264 * A101266 A101267 A101268
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KEYWORD
| nonn
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AUTHOR
| Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 25 2005
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