A025177 -id:A025177 - cp's OEIS frontend

A025179 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A025177.

1, 4, 10, 29, 81, 231, 659, 1891, 5443, 15718, 45508, 132067, 384047, 1118820, 3264642, 9539787, 27913083, 81769236, 239794422, 703906719, 2068153899, 6081507831, 17896695831, 52703944965, 155310270101, 457956633826, 1351132539604

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000 (terms 2..200 from T. D. Noe)

Cf. A024997, A026135, A359364.

Rest[Rest[CoefficientList[Series[((1-x)^2-(1-x)*Sqrt[1-2*x-3*x^2]) /(2*x*Sqrt[1-2*x-3*x^2]), {x, 0, 20}], x]]] (* Vaclav Kotesovec, Feb 13 2014 *)

my(x='x+O('x^50)); Vec(((1-x)^2-(1-x +2*x^2)*sqrt(1-2*x-3*x^2)) /(2*x*sqrt(1 - 2*x -3*x^2))) \\ G. C. Greubel, Mar 01 2017

Equals (1/2) * A024997(n+1).
From Vladeta Jovovic, Jan 01 2004: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k+1, k+1).
E.g.f.: exp(x)*(BesselI(0, 2*x)+BesselI(2, 2*x)). (End)
From Paul Barry, Sep 17 2005: (Start)
G.f.: ((1-x)^2 - (1-x)*sqrt(1-2*x-3*x^2))/(2*x*sqrt(1-2*x-3*x^2)).
a(n+1) = Sum_{k=0..n} C(n, k)*C(k+1, k/2+1)*(1+(-1)^k)/2. (End)
D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +(-n-5)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 26 2012
a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 13 2014
Prepend 1 to the data, assume offset 0, and denote the resulting sequence alpha. Then alpha(n) = Sum_{k=0..n} Sum_{j=0..k} A359364(n, n - j). - Peter Luschny, Jan 10 2023

A027261 a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).

Original entry on oeis.org

1, 4, 18, 72, 270, 972, 3402, 11664, 39366, 131220, 433026, 1417176, 4605822, 14880348, 47829690, 153055008, 487862838, 1549681956, 4907326194, 15496819560, 48814981614, 153418513644, 481176247338, 1506290861232, 4707158941350, 14686335897012, 45753584909922

Offset: 0

Clark Kimberling

nonn

Cf. A014915, A064017, A079272.

Cf. A006234.

a(n) = 2(n+1)*3^(n-1), for n>1 (conjectured). - Ralf Stephan, Feb 02 2004
From Colin Barker, Jul 28 2012: (Start)
Conjecture: a(n) = 6*a(n-1)-9*a(n-2), for n>3.
G.f.: (1-9*x^2+18*x^3)/(1-3*x)^2. (End)

a(1) corrected and more terms from Sean A. Irvine, Oct 26 2019

A025181 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A025177.

Original entry on oeis.org

1, 3, 11, 35, 111, 343, 1050, 3186, 9615, 28897, 86592, 258908, 772863, 2304225, 6863496, 20429784, 60779403, 180751617, 537386595, 1597372371, 4747537641, 14108988509, 41928203694, 124598731750, 370279082745, 1100428538391, 3270534249843

Offset: 3

Clark Kimberling

nonn

Cf. A025568.

First differences of A014532. First differences are in A026070.

Conjecture: +(n+3)*a(n) +(-5*n-7)*a(n-1) +(3*n-7)*a(n-2) +(11*n-7)*a(n-3) +4*(-n+6)*a(n-4) +6*(-n+5)*a(n-5)=0. - R. J. Mathar, Feb 25 2015

Conjecture: -(n+3)*(n-3)*(4*n^2-12*n+17)*a(n) +(n-1)*(8*n^3-20*n^2+30*n-81)*a(n-1) +3*(n-1)*(n-2)*(4*n^2-4*n+9)*a(n-2)=0. - R. J. Mathar, Feb 25 2015

A025180 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A025177.

Original entry on oeis.org

1, 2, 7, 20, 60, 176, 518, 1520, 4461, 13090, 38423, 112828, 331487, 974442, 2866125, 8434992, 24838275, 73181142, 215729781, 636275820, 1877569134, 5543095404, 16372140876, 48377825216, 143009973875, 422918975726, 1251154692297

Offset: 2

Clark Kimberling

nonn

First differences of A014531. First differences are in A026069.

Conjecture: -(n+2)*(n-3)*a(n) +2*(2*n+1)*(n-3)*a(n-1) +(-n^2+10*n-18)*a(n-2) -3*(2*n-3)*(n-3)*a(n-3)=0. - R. J. Mathar, Feb 25 2015

A025182 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A025177.

Original entry on oeis.org

1, 4, 16, 56, 189, 616, 1968, 6192, 19272, 59488, 182468, 556920, 1693146, 5131296, 15511344, 46791072, 140905197, 423709956, 1272596136, 3818355464, 11447074309, 34292702840, 102670377120, 307230479920, 918951019155, 2747624937876

Offset: 4

Clark Kimberling

nonn

Apparently first differences of A014533.

Cf. A014533, A025177.

Conjecture: -(n-4)*(n+4)*a(n) +(4*n^2-7*n-29)*a(n-1) +(-2*n^2+17*n-2)*a(n-2) -(4*n+1)*(n-3)*a(n-3) +3*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 25 2015

Conjecture: -(n-4)*(n+4)*(n^2-3*n+6)*a(n) +(n-1)*(2*n^3-5*n^2+11*n-36)*a(n-1) +3*(n-1)*(n-2)*(n^2-n+4)*a(n-2)=0. - R. J. Mathar, Feb 25 2015

A025191 a(n) = Sum_{k=0..n} T(n,k), where T is the array defined in A025177.

Original entry on oeis.org

1, 1, 4, 11, 33, 97, 288, 855, 2544, 7577, 22590, 67399, 201215, 601017, 1795966, 5368659, 16053417, 48015873, 143649102, 429842511, 1286452725, 3850770081, 11528245602, 34517105907, 103360732956, 309543786441, 927106804368, 2776994293355

Offset: 0

Clark Kimberling

nonn

Conjectures: a(n) = A027914(n)-A027914(n-1) = (A081673(n)-A081673(n-1))/2.

a(1) corrected by Jason Yuen, Aug 05 2024

A025183 a(n) = T(2n-1,n), where T is the array defined in A025177.

Original entry on oeis.org

0, 4, 20, 111, 616, 3465, 19668, 112476, 647088, 3741156, 21718520, 126520240, 739225704, 4330188265, 25421864580, 149541718275, 881200519200, 5200715525016, 30736780735752, 181886951796886

Offset: 1

Clark Kimberling

nonn

A025183 := proc(n)
        A025177(2*n-1,n) ;
end proc: # R. J. Mathar, Jul 28 2016

Conjecture: 9*n*(10833*n-26779)*(3*n-4)*(3*n-2)*a(n) -3*(n-1)*(749757*n^3+5096
41*n^2-11324756*n+13648560)*a(n-1) -2*(2*n-5)*(4152209*n^3-25903434*n^2+5658913
6*n-40818816)*a(n-2) -36*(n-3)*(85207*n-129510)*(2*n-5)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Feb 25 2015

First term corrected. - R. J. Mathar, Feb 25 2015

A025184 a(n) = T(2n,n), where T is the array defined in A025177.

Original entry on oeis.org

1, 1, 7, 35, 189, 1038, 5797, 32747, 186615, 1070762, 6177698, 35802935, 208279007, 1215507450, 7113090285, 41724381765, 245258504925, 1444292029818, 8519114704870, 50323176446818, 297654524450998

Offset: 0

Clark Kimberling

nonn

A025184 := proc(n)
        A025177(2*n,n) ;
end proc: # R. J. Mathar, Jul 28 2016

a(n) = (2*(3*n+2)*(3*n+1)*A006605(n)-(19*n+4)*(2*n-1)*A006605(n-1))/(13*n+4) for n>0. [Mark van Hoeij, Jul 02 2010]

Conjecture: 33*n*(3*n-1)*(3*n-2)*a(n) +11*(2047*n^3-10725*n^2+17192*n-8520)*a(n-1) +9*(-4397*n^3-10169*n^2+110500*n-145368)*a(n-2) -54*(2*n-5)*(5353*n^2-33313*n+53904)*a(n-3) -115668*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 25 2015

A025185 a(n) = T(3n,n), where T is the array defined in A025177.

Original entry on oeis.org

1, 2, 16, 120, 946, 7644, 62832, 522804, 4390056, 37126830, 315770676, 2698179488, 23144461368, 199177780102, 1718891656176, 14869867516812, 128909380154562, 1119623594699610, 9740452802251080, 84865206568065000, 740387260555434996, 6467129226550164474

Offset: 0

Clark Kimberling

nonn

A025185 := proc(n)
        A025177(3*n,n) ;
end proc: # R. J. Mathar, Jul 28 2016

A025186 T(4n,n), where T is the array defined in A025177.

Original entry on oeis.org

1, 3, 29, 286, 2955, 31350, 338514, 3701295, 40849661, 454114540, 5077463898, 57038882004, 643269405230, 7278603015180, 82590333059025, 939444489263403, 10708773485661387, 122299562702918348, 1399055759185079316

Offset: 0

Clark Kimberling

nonn

A025186 := proc(n)
        A025177(4*n,n) ;
end proc: # R. J. Mathar, Jul 28 2016

cp's OEIS Frontend

A025179 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A025177.

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A027261 a(n) = Sum_{k=0..2n} (k+1) * A025177(n, k).

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A025181 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A025177.

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A025180 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A025177.

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A025182 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A025177.

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A025191 a(n) = Sum_{k=0..n} T(n,k), where T is the array defined in A025177.

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A025183 a(n) = T(2n-1,n), where T is the array defined in A025177.

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Maple

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A025184 a(n) = T(2n,n), where T is the array defined in A025177.

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Maple

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A025185 a(n) = T(3n,n), where T is the array defined in A025177.

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Maple

A025186 T(4n,n), where T is the array defined in A025177.

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Maple