The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A026135

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A026135

Number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T defined in A026120.
(history; published version)

#17 by Bruno Berselli at Mon May 22 02:38:40 EDT 2017

STATUS

proposed

approved

#16 by G. C. Greubel at Mon May 22 00:28:33 EDT 2017

STATUS

editing

proposed

#15 by G. C. Greubel at Mon May 22 00:28:22 EDT 2017

LINKS

G. C. Greubel, <a href="/A026135/b026135.txt">Table of n, a(n) for n = 0..1000</a>

FORMULA

G.f. : ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - David Callan, Aug 16 2006

MATHEMATICA

CoefficientList[Series[((x - 1)^2*((1 + x)/(1 - 3 x))^(1/2) + x^2 - 1)/(2*x^2), {x, 0, 50}], x] (* G. C. Greubel, May 22 2017 *)

PROG

(PARI) x='x+O('x^50); Vec(((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2)) \\ G. C. Greubel, May 22 2017

STATUS

approved

editing

#14 by R. J. Mathar at Sun Jun 23 11:04:34 EDT 2013

STATUS

editing

approved

#13 by R. J. Mathar at Sun Jun 23 11:04:27 EDT 2013

FORMULA

Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n-2)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

STATUS

approved

editing

#12 by Charles R Greathouse IV at Fri May 10 12:44:05 EDT 2013

FORMULA

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - _Vladeta Jovovic (vladeta(AT)eunet.rs), _, Sep 18 2003

Discussion

Fri May 10

12:44

OEIS Server: https://oeis.org/edit/global/1911

#11 by Russ Cox at Sat Mar 31 10:22:41 EDT 2012

COMMENTS

a(n) is the total number of rows of consecutive peaks in all Motzkin (n+2)-paths. For example, with U=upstep, D=downstep, F=flatstep, the path FU(UD)FU(UDUDUD)DD(UD) contains 3 rows of peaks (in parentheses). The 9 Motzkin 4-paths are FFFF, FF(UD), F(UD)F, FUFD, (UD)FF, (UDUD), UFDF, UFFD, U(UD)D, containing a total of 5 rows of peaks and so a(2)=5. - _David Callan (callan(AT)stat.wisc.edu), _, Aug 16 2006

FORMULA

G.f. ((x-1)^2*((1+x)/(1-3x))^(1/2) + x^2 - 1)/(2*x^2). - _David Callan (callan(AT)stat.wisc.edu), _, Aug 16 2006

EXTENSIONS

More terms from _David Callan (callan(AT)stat.wisc.edu), _, Aug 16 2006

Discussion

Sat Mar 31

10:22

OEIS Server: https://oeis.org/edit/global/348

#10 by Russ Cox at Fri Mar 30 18:56:04 EDT 2012

AUTHOR

Clark Kimberling (ck6(AT)evansville.edu)

Clark Kimberling

Discussion

Fri Mar 30

18:56

OEIS Server: https://oeis.org/edit/global/285

#9 by Russ Cox at Fri Mar 30 17:38:32 EDT 2012

FORMULA

G.f. = (1+z)*(1+z^2)/(1-z) where z=x*A001006(x). [From _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Jul 07 2009]

EXTENSIONS

Typo in a(19) corrected by _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Jul 07 2009

Discussion

Fri Mar 30

17:38

OEIS Server: https://oeis.org/edit/global/190

#8 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

DATA

1, 2, 5, 14, 39, 110, 312, 890, 2550, 7334, 21161, 61226, 177575, 516114, 1502867, 4383462, 12804429, 37452870, 109682319, 321564658, 321563658, 943701141, 2772060618, 8149661730, 23978203662, 70600640796, 208014215066, 613266903927

FORMULA

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*binomial(k+1, floor((k+1)/2)). - Vladeta Jovovic (vladeta(AT)Euneteunet.yurs), Sep 18 2003

G.f. = (1+z)*(1+z^2)/(1-z) where z=x*A001006(x). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009]

KEYWORD

nonn,new

nonn

EXTENSIONS

Typo in a(19) corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009