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 1-4 of 4 results.

A207643 a(n) = 1 + (n-1) + (n-1)*[n/2-1] + (n-1)*[n/2-1]*[n/3-1] + (n-1)*[n/2-1]*[n/3-1]*[n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 26, 31, 71, 129, 262, 291, 1222, 1333, 2198, 5139, 11881, 12673, 39594, 41923, 117326, 251841, 354292, 371163, 1870453, 2598577, 3456926, 7103955, 16665859, 17283113, 72923314, 75437911, 165990152, 335534913, 422310802, 695765699, 3589651696
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2012

Keywords

Comments

Radius of convergence of g.f. A(x) is near 0.54783..., with A(1/2) = 7.2672875151872...
Compare the definition of a(n) to the trivial binomial sum:
2^(n-1) = 1 + (n-1) + (n-1)*(n/2-1) + (n-1)*(n/2-1)*(n/3-1) + (n-1)*(n/2-1)*(n/3-1)*(n/4-1) +...

Examples

			a(2) = 1 + 1 = 2; a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 3*[4/2-1] = 7;
a(5) = 1 + 4 + 4*[5/2-1] = 9;
a(6) = 1 + 5 + 5*[6/2-1] + 5*[6/2-1]*[6/3-1] = 26;
a(7) = 1 + 6 + 6*[7/2-1] + 6*[7/2-1]*[7/3-1] = 31;
a(8) = 1 + 7 + 7*[8/2-1] + 7*[8/2-1]*[8/3-1] + 7*[8/2-1]*[8/3-1]*[8/4-1] = 71; ...

Links

Crossrefs

Cf. A075885, A207645, A207644, A207646, A207647.

Programs

  • Mathematica
    a[n_] := 1 + Sum[ Product[ Floor[(n-j)/j], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Mar 06 2013 *)
  • PARI
    {a(n)=1+sum(k=1,n,prod(j=1,k,floor(n/j-1)))}
for(n=0,50,print1(a(n),", "))
  • PARI
    a(n)=my(t=1);1+sum(k=1,n,t*=n\k-1) \\ Charles R Greathouse IV, Feb 20 2012

Formula

a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-j) / j ).
Equals row sums of irregular triangle A207645.

A207644 a(n) = 1 + (n-1) + (n-2)*[(n-3)/2] + (n-3)*[(n-4)/2]*[(n-5)/3] + (n-4)*[(n-5)/2]*[(n-6)/3]*[(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 10, 17, 30, 42, 55, 116, 172, 220, 391, 683, 1024, 1616, 2050, 3675, 6520, 9504, 12505, 22421, 35572, 56918, 85701, 138110, 202765, 326231, 503632, 860497, 1376870, 1927446, 2818531, 4892966, 7784671, 11432772, 17287295, 30423457, 46453786, 71810414
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2012

Keywords

Comments

Radius of convergence of g.f. A(x) is near 0.637..., with A(1/phi) = 23.059250143... where phi = (sqrt(5)+1)/2.
Compare the definition of a(n) with the binomial sum:
Fibonacci(n) = 1 + (n-1) + (n-2)*((n-3)/2) + (n-3)*((n-4)/2)*((n-5)/3) + (n-4)*((n-5)/2)*((n-6)/3)*((n-7)/4) +...
with summation extending over the initial [n/2+1] products only.

Examples

			a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 2*[1/2] = 4;
a(5) = 1 + 4 + 3*[2/2] = 8;
a(6) = 1 + 5 + 4*[3/2] + 3*[2/2]*[1/3] = 10;
a(7) = 1 + 6 + 5*[4/2] + 4*[3/2]*[2/3] = 17;
a(8) = 1 + 7 + 6*[5/2] + 5*[4/2]*[3/3] + 4*[3/2]*[2/3]*[1/4] = 30;
a(9) = 1 + 8 + 7*[6/2] + 6*[5/2]*[4/3] + 5*[4/2]*[3/3]*[2/4] = 42; ...

Links

Crossrefs

Cf. A075885, A207643, A207645, A207646, A207647.

Programs

  • Mathematica
    a[n_] := 1 + Sum[ Product[ Floor[ (n-k-j+1)/j ], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Mar 06 2013 *)
  • PARI
    {a(n)=1+sum(k=1,n\2,prod(j=1,k,floor((n-k-j+1)/j)))}
for(n=0,60,print1(a(n),", "))

Formula

a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-k-j+1) / j ).
Equals the antidiagonal sums of triangle A207645.

A207645 Triangle where T(n,k) = Product_{j=1..k} floor(n/j - 1), as read by rows n>=0, columns k=0..[n/2].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 4, 4, 1, 5, 10, 10, 1, 6, 12, 12, 1, 7, 21, 21, 21, 1, 8, 24, 48, 48, 1, 9, 36, 72, 72, 72, 1, 10, 40, 80, 80, 80, 1, 11, 55, 165, 330, 330, 330, 1, 12, 60, 180, 360, 360, 360, 1, 13, 78, 234, 468, 468, 468, 468, 1, 14, 84, 336, 672, 1344, 1344, 1344
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2012

Keywords

Comments

Compare the definition to that of Pascal's triangle:
binomial(n,k) = Product_{j=1..k} ((n+1)/j - 1).

Examples

			Triangle begins with row n=0 as:
1;
1;
1,  1;
1,  2;
1,  3,   3;
1,  4,   4;
1,  5,  10,  10;
1,  6,  12,  12;
1,  7,  21,  21,   21;
1,  8,  24,  48,   48;
1,  9,  36,  72,   72,   72;
1, 10,  40,  80,   80,   80;
1, 11,  55, 165,  330,  330,  330;
1, 12,  60, 180,  360,  360,  360;
1, 13,  78, 234,  468,  468,  468,  468;
1, 14,  84, 336,  672, 1344, 1344, 1344;
1, 15, 105, 420, 1260, 2520, 2520, 2520, 2520;
1, 16, 112, 448, 1344, 2688, 2688, 2688, 2688; ...

Links

Crossrefs

Cf. A207643, A207644, A207646, A207647.

Programs

  • Mathematica
    t[n_, k_] := Product[Floor[n/j - 1], {j, 1, k}]; Flatten[Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* Jean-François Alcover, Jun 12 2012 *)
  • PARI
    {T(n,k)=if(k==0,1,prod(j=1,k,floor(n/j-1)))}
for(n=0,12,for(k=0,n\2,print1(T(n,k),", "));print(""))

Formula

Row sums equal A207643.
Antidiagonal sums form A207644.
Right border of even-indexed rows equals A207646.
Right border of odd-indexed rows equals A207647.

A207646 Product_{k=1..n} floor(2*n/k - 1).

Original entry on oeis.org

1, 1, 3, 10, 21, 72, 330, 468, 2520, 8160, 20520, 60480, 318780, 504000, 2426112, 10523520, 21092400, 53222400, 452390400, 506373120, 3226728960, 11604902400, 21299241600, 76640256000, 431500608000, 844958822400, 3197988864000, 10492449177600, 38109367296000
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2012

Keywords

Comments

Forms the right border of even-indexed rows in irregular triangle A207645.

Examples

			Illustration of the initial terms:
a(1) = [2/1-1] = 1;
a(2) = [4/1-1]*[4/2-1] = 3;
a(3) = [6/1-1]*[6/2-1]*[6/3-1] = 10;
a(4) = [8/1-1]*[8/2-1]*[8/3-1]*[8/4-1] = 21;
a(5) = [10/1-1]*[10/2-1]*[10/3-1]*[10/4-1]*[10/5-1] = 72; ...
where [x] = floor(x).

Links

Crossrefs

Cf. A207645, A207647, A207643, A207644.

Programs

  • Mathematica
    Table[Product[Floor[(2n)/k-1],{k,n}],{n,0,30}] (* Harvey P. Dale, Aug 27 2017 *)
  • PARI
    {a(n)=prod(k=1,n,floor(2*n/k-1))}
for(n=0,50,print1(a(n),", "))
  • PARI
    a(n)=n*=2;prod(k=1,n/3,n\k-1) \\ Charles R Greathouse IV, Feb 20 2012

Formula

a(n) = A207645(2*n, n).
Showing 1-4 of 4 results.

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