A025012 -id:A025012 - cp's OEIS frontend

A204208 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 3.

1, 4, 16, 78, 404, 2208, 12492, 72589, 430569, 2596471, 15870357, 98102191, 612222083, 3852015239, 24408653703, 155629858911, 997744376239, 6427757480074, 41590254520410, 270163621543421, 1761179219680657

Offset: 1

R. H. Hardin, Jan 12 2012

nonn

Column 3 of A204213

Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-3,-2,-1,0,1,2,3}. - David Nguyen, Dec 16 2016

			Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....1....3....3....2....2....1....2....0....0....2....3....0....3....1....2
..5....3....2....2....2....3....1....5....3....0....2....4....3....2....0....3
..2....6....3....4....0....1....0....6....5....1....0....6....5....2....2....5
..2....3....3....3....2....3....3....3....2....1....0....3....3....0....3....3
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..210
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, j] Binomial[-7j + 4i - 1, 3i - 7j] (-1)^j, {j, 0, (3i)/7}]) a[n - i], {i, 1, n}]/n];
a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

a(n):=if n=0 then 1 else sum((sum(binomial(i,j)*binomial(-7*j+4*i-1,3*i-7*j)*(-1)^j,j,0,(3*i)/7))*a(n-i),i,1,n)/n; /* Vladimir Kruchinin, Apr 06 2017 */

{A025012(n)=polcoeff((1+x+x^2+x^3+x^4+x^5+x^6 +x*O(x^(3*n)))^n,3*n)}
{a(n)=polcoeff(exp(sum(m=1,n,A025012(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Aug 01 2013

G.f.: exp( Sum_{n>=1} A025012(n)*x^n/n ) - 1, where A025012(n) = central coefficient of (1+x+x^2+x^3+x^4+x^5+x^6)^n. - Paul D. Hanna, Aug 01 2013

a(n) = Sum_{i=1..n}((Sum_{j=0..(3*i)/7}(binomial(i,j)*binomial(-7*j+4*i-1,3*i-7*j)*(-1)^j))*a(n-i))/n. - Vladimir Kruchinin, Apr 06 2017

A208592 Number of n-bead necklaces labeled with numbers -3..3 not allowing reversal, with sum zero.

Original entry on oeis.org

1, 4, 13, 60, 291, 1564, 8671, 49852, 292927, 1753964, 10656757, 65549844, 407347747, 2553684852, 16130539053, 102563204892, 655918173287, 4216358457772, 27227967629683, 176554882805940, 1149099219084877, 7504110622072860, 49155856119036993, 322903351882566436

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

			All solutions for n=3:
.-2...-1...-3...-3...-1...-2...-3...-2...-3...-2...-2...-1....0
..1....0....2....1....1....0....0....3....3...-1....2...-1....0
..1....1....1....2....0....2....3...-1....0....3....0....2....0

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 3 of A208597.

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 3]; Array[a, 24] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A025012(d). - Andrew Howroyd, Mar 02 2017

a(20)-a(24) from Andrew Howroyd, Mar 02 2017

A125316 Number of base 6 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 6, 30, 138, 694, 3526, 18012, 92140, 471566, 2413794, 12356010, 63250314, 323779348, 1657433992, 8484446994, 43432107538, 222330112286, 1138113767838, 5826034724844, 29823627148788, 152667942902854

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1.

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

Conjectures from Colin Barker, Jun 04 2017: (Start)
G.f.: (1 - 3*x^2 - 16*x^3 + 3*x^4 + 8*x^5) / ((1 - x - x^2)*(1 - 5*x - x^2 + 2*x^3)).
a(n) = 6*a(n-1) - 3*a(n-2) - 8*a(n-3) + a(n-4) + 2*a(n-5) for n>5.
(End)

A125342 Number of base 32 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 32, 212, 1100, 6700, 41252, 260276, 1663260, 10741356, 69920372, 458065232, 3016606328, 19952674708, 132459700444, 882141536432, 5890902558440, 39433549800556, 264525710537012, 1777811473603544, 11968319215904376

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

A125317 Number of base 7 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 7, 37, 175, 925, 4977, 27067, 147777, 808165, 4422703, 24210127, 132542813, 725664307, 3973047319, 21752799623, 119098967235, 652080798837, 3570220727531, 19547391880633, 107024353337437, 585971402810715

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1.

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

G.f.: (1 - 6*x^2 - 28*x^3 + 15*x^4 + 28*x^5 - 5*x^6 - 6*x^7) / ((1 - 2*x - x^2 + x^3)*(1 - 5*x - 3*x^2 + 2*x^3 + x^4)) (conjectured). - Colin Barker, Jun 03 2017

A125318 Number of base 8 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 8, 44, 212, 1156, 6428, 36338, 206942, 1183164, 6777860, 38866134, 222980282, 1279588414, 7343932838, 42151639560, 241943513252, 1388738207756, 7971319975484, 45755345035400, 262636025020328, 1507534275820126

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

G.f.: (1 - 10*x^2 - 40*x^3 + 45*x^4 + 48*x^5 - 35*x^6 - 12*x^7 + 7*x^8) / ((1 + x)*(1 - 3*x + x^3)*(1 - 6*x + x^2 + 3*x^3 - x^4)) (conjectured). - Colin Barker, Jun 03 2017

A125319 Number of base 9 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 9, 51, 249, 1387, 7879, 45663, 267367, 1575395, 9316455, 55211811, 327600447, 1945195699, 11554693471, 68652279747, 407952680479, 2424365629955, 14408066169471, 85629681584835, 508919764066431, 3024669695471107

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

G.f.: (1 - x - 14*x^2 - 36*x^3 + 138*x^4 - 58*x^5 - 52*x^6 + 28*x^7) / ((1 - x)*(1 + x)*(1 - 4*x + 2*x^2)*(1 - 6*x + 2*x^3)) (conjectured). - Colin Barker, Jun 02 2017

A125320 Number of base 10 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 10, 58, 286, 1618, 9330, 54994, 328058, 1973026, 11928418, 72360138, 439900746, 2677993234, 16317209282, 99477496666, 606677367546, 3700738695234, 22577766952946, 137756725752298, 840561549454570

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

G.f.: (1 - 21*x^2 - 56*x^3 + 192*x^4 + 112*x^5 - 260*x^6 - 48*x^7 + 84*x^8) / ((1 + x)*(1 - 4*x + 2*x^3)*(1 - 7*x + 4*x^2 + 10*x^3 - 6*x^4)) (conjectured). - Colin Barker, Jun 01 2017

A125321 Number of base 11 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 11, 65, 323, 1849, 10781, 64325, 388749, 2371457, 14559461, 89788425, 555446661, 3443457697, 21379001421, 132867762617, 826329001973, 5141530912065, 32001680946301, 199227543918473, 1240485517109477

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

G.f.: (1 - 28*x^2 - 56*x^3 + 318*x^4 + 112*x^5 - 540*x^6 - 48*x^7 + 224*x^8) / ((1 - 4*x - 2*x^2 + 4*x^3)*(1 - 7*x + 2*x^2 + 18*x^3 - 2*x^4 - 8*x^5)) (conjectured). - Colin Barker, Jun 03 2017

A125322 Number of base 12 circular n-digit numbers with adjacent digits differing by 3 or less.

Original entry on oeis.org

1, 12, 72, 360, 2080, 12232, 73656, 449440, 2770016, 17195112, 107304872, 672224048, 4223186080, 26586825472, 167628931360, 1058056409920, 6683674433920, 42244783230496, 267124015799712, 1689602502335776

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A025012(n+1) for base>=3.int(n/2)+1

OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

cp's OEIS Frontend

A204208 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 3.

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A208592 Number of n-bead necklaces labeled with numbers -3..3 not allowing reversal, with sum zero.

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A125316 Number of base 6 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125342 Number of base 32 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125317 Number of base 7 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125318 Number of base 8 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125319 Number of base 9 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125320 Number of base 10 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125321 Number of base 11 circular n-digit numbers with adjacent digits differing by 3 or less.

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A125322 Number of base 12 circular n-digit numbers with adjacent digits differing by 3 or less.

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