cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

R. H. Hardin, Jan 12 2012

Keywords

Comments

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

Examples

			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

Links

Programs

  • Mathematica
    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 *)
  • Maxima
    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 */
  • PARI
    {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

Formula

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

A204209 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 4.

Original entry on oeis.org

1, 5, 25, 155, 1025, 7167, 51945, 387000, 2944860, 22791189, 178840639, 1419569398, 11377983292, 91957314063, 748575327757, 6132254500856, 50514620902564, 418174191239443, 3477075679541185, 29026557341147912, 243184916545458556
Offset: 1

Views

Author

R. H. Hardin, Jan 12 2012

Keywords

Comments

Column 4 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 {-4,-3,-2,-1,0,1,2,3,4}. - David Nguyen, Dec 16 2016

Examples

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

Links

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, j] Binomial[-9j + 5i - 1, 4i - 9j] (-1)^j, {j, 0, (4i)/9}]) a[n - i], {i, 1, n}]/n];
a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)
  • Maxima
    a(n):=if n=0 then 1 else sum((sum(binomial(i,j)*binomial(-9*j+5*i-1,4*i-9*j)*(-1)^j,j,0,(4*i)/9))*a(n-i),i,1,n)/n; /* Vladimir Kruchinin, Apr 06 2017 */

Formula

a(n) = Sum_{i=1..n} ((Sum_{j=0..(4*i)/9} (binomial(i,j)*binomial(-9*j+5*i-1,4*i-9*j)*(-1)^j))*a(n-i))/n. - Vladimir Kruchinin, Apr 06 2017

A204210 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 5.

Original entry on oeis.org

1, 6, 36, 271, 2181, 18583, 164255, 1493142, 13868334, 131040555, 1255641305, 12172468671, 119167633383, 1176491130191, 11699794149335, 117092364762013, 1178452895921961, 11919463485729722, 121096865173987004
Offset: 1

Views

Author

R. H. Hardin Jan 12 2012

Keywords

Comments

Column 5 of A204213

Examples

			Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....3....5....3....4....0....5....4....2....3....4....3....4....5....4....1
..2....1....9....1....7....4....7....8....5....3....3....6....3....2....1....3
..1....1....7....3....5....3....8....5....8....7....6....1....2....3....0....3
..4....0....3....2....5....2....4....0....4....5....1....0....1....2....2....3
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

A204211 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 6.

Original entry on oeis.org

1, 7, 49, 434, 4116, 41363, 431445, 4629851, 50775347, 566592411, 6412416619, 73428973184, 849204106870, 9904511238159, 116368090860251, 1375981128400665, 16362074062371685, 195540534817518299, 2347344805869798781
Offset: 1

Views

Author

R. H. Hardin Jan 12 2012

Keywords

Comments

Column 6 of A204213

Examples

			Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....3....3....0....5....0....0....4....2....3....6....4....1....6....6....2
..9....1....6....5....1....1....4....5....5....6...12....1....0....2....0....2
..4....3...10....6....4....4....2....1....4....5...11....6....4....3....1....8
..5....1....5....5....1....1....5....0....1....5....6....1....1....5....0....5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

A204212 Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than 7.

Original entry on oeis.org

1, 8, 64, 652, 7120, 82440, 991152, 12262470, 155072356, 1995610260, 26048900628, 344054761656, 4589736845308, 61750810543324, 836935922008844, 11416462124627925, 156613134014878257, 2159264087976308543
Offset: 1

Views

Author

R. H. Hardin Jan 12 2012

Keywords

Comments

Column 7 of A204213

Examples

			Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....2....7....4....7....6....4....6....4....4....2....4....6....3....1....4
..6....0....9....7....9....6....0....4....3....4....6....5....6....9....6...10
..6....2...14....7...12....8....7....2....4....8....2....8....4....5....2...12
..5....7....7....7....6....1....2....7....4....6....4....5....4....2....5....5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

A204214 Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than n.

Original entry on oeis.org

21, 120, 404, 1025, 2181, 4116, 7120, 11529, 17725, 26136, 37236, 51545, 69629, 92100, 119616, 152881, 192645, 239704, 294900, 359121, 433301, 518420, 615504, 725625, 849901, 989496, 1145620, 1319529, 1512525, 1725956, 1961216, 2219745
Offset: 1

Views

Author

R. H. Hardin, Jan 12 2012

Keywords

Comments

Row 5 of A204213.

Examples

			Some solutions for n=5:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....0....2....2....0....5....0....3....5....0....4....5....4....1....1....3
..1....4....2....2....3....2....2....6....1....5....8...10....8....2....3....5
..4....3....7....1....2....0....3....5....5....4....6...10....3....6....3....6
..2....4....5....3....4....3....0....5....2....5....3....5....4....2....4....3
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

Crossrefs

Cf. A204213.

Formula

Empirical: a(n) = (23/12)*n^4 + (37/6)*n^3 + (91/12)*n^2 + (13/3)*n + 1.
Conjectures from Colin Barker, Jun 06 2018: (Start)
G.f.: x*(21 + 15*x + 14*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A204215 Number of length 7 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than n.

Original entry on oeis.org

51, 473, 2208, 7167, 18583, 41363, 82440, 151125, 259459, 422565, 659000, 991107, 1445367, 2052751, 2849072, 3875337, 5178099, 6809809, 8829168, 11301479, 14298999, 17901291, 22195576, 27277085, 33249411, 40224861, 48324808, 57680043
Offset: 1

Views

Author

R. H. Hardin, Jan 12 2012

Keywords

Comments

Row 6 of A204213.

Examples

			Some solutions for n=5:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..3....1....3....3....5....5....4....2....4....4....4....0....4....1....0....3
..5....6....5....4....6....5....2....3....3....8....9....0....3....3....1....5
..4....4....3....3....6....7....4....6....5....7....7....3....6....4....5....5
..5....5....7....2....8....7....4....1....4....4....6....1....5....0....0....2
..3....1....5....0....3....5....0....1....0....0....4....5....2....2....3....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

Crossrefs

Cf. A204213.

Formula

Empirical: a(n) = (44/15)*n^5 + (133/12)*n^4 + 17*n^3 + (161/12)*n^2 + (167/30)*n + 1.
Conjectures from Colin Barker, Jun 06 2018: (Start)
G.f.: x*(51 + 167*x + 135*x^2 - 6*x^3 + 6*x^4 - x^5) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A204216 Number of length 8 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than n.

Original entry on oeis.org

127, 1925, 12492, 51945, 164255, 431445, 991152, 2057553, 3945655, 7098949, 12120428, 19806969, 31187079, 47562005, 70550208, 102135201, 144716751, 201165445, 274880620, 369851657, 490722639, 642860373, 832425776, 1066448625
Offset: 1

Views

Author

R. H. Hardin, Jan 12 2012

Keywords

Comments

Row 7 of A204213.

Examples

			Some solutions for n=4:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....3....3....2....3....2....3....4....1....4....4....4....1....4....3....3
..3....5....0....2....2....0....6....0....1....4....8....8....1....6....5....2
..4....4....1....5....5....2....8....2....1....5....5....4....0....9....7....3
..4....5....3....7....6....3....6....6....4....2....3....2....1....6...10....1
..7....4....6....7....6....2....2....3....3....0....6....1....3....2....6....2
..4....2....4....4....3....0....2....4....2....1....4....0....4....1....3....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Links

Crossrefs

Cf. A204213.

Formula

Empirical: a(n) = (841/180)*n^6 + (101/5)*n^5 + (1325/36)*n^4 + (73/2)*n^3 + (946/45)*n^2 + (34/5)*n + 1.
Conjectures from Colin Barker, Jun 06 2018: (Start)
G.f.: x*(127 + 1036*x + 1684*x^2 + 481*x^3 + 42*x^4 - 7*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)
Showing 1-8 of 8 results.

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