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A050253
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G.f.: ( 1 - x^2 - Sqrt[ 1 - 2 x^2 - 4 x^3 - 3 x^4 ] ) / ( 2 x^3 ).
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1
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1, 1, 1, 2, 3, 5, 9, 16, 29, 54, 101, 191, 365, 702, 1359, 2647, 5181, 10187, 20113, 39856, 79243, 158036, 316053, 633689, 1273559, 2565136, 5177043, 10468199, 21204379, 43022215, 87423573, 177906552, 362531425, 739700055, 1511091377
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=number of Motzkin (n-1)-paths (A001006) containing no three consecutive weakly-rising steps (n>=1). A weakly-rising step is an upstep or flatstep. For example, a(5)=5 counts FUDF, UDFF, UDUD, UFDF, UUDD while the path FUFD, say, is not counted because the first 3 steps are weakly-rising. - David Callan (callan(AT)stat.wisc.edu), Oct 25 2004
Hankel transform is A010892(n+1). [From Paul Barry (pbarry(AT)wit.ie), Jul 29 2010]
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FORMULA
| a(n)=A108296(n+2)-A108296(n). - Paul Barry (pbarry(AT)wit.ie), May 31 2005
G.f.: 1/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 25 2009]
G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Jul 29 2010]
Conjecture: (n+3)*a(n) +(n+2)*a(n-1) -2n*a(n-2) +2*(4-3n)*a(n-3) +(19-7n)*a(n-4) +3*(4-n)*a(n-5) =0. - R. J. Mathar, Nov 15 2011
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CROSSREFS
| Sequence in context: A103285 A000049 A000050 * A198518 A107250 A050168
Adjacent sequences: A050250 A050251 A050252 * A050254 A050255 A050256
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KEYWORD
| easy,nonn
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AUTHOR
| Emanuele Munarini (munarini(AT)mate.polimi.it), May 09 2003
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