A132371 -id:A132371 - cp's OEIS frontend

A227187 Numbers n whose factorial base representation A007623(n) contains at least one nonleading zero. (Zero is also included as a(0)).

0, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 88, 90, 91

Offset: 0

Antti Karttunen, Jul 04 2013

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial base representation.

Complement: A227157.
The sequence gives all positions n where A208575 is zero and all terms where A257510 (also A257260) are nonzeros.
Cf. A232745 (a subsequence), A232744.
Cf. also A007623, A132371, A153880, A227130, A227132, A256450 (numbers with at least one 1 in their factorial representation).

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; MemberQ[s, 0]]; q[0] = True; Select[Range[0, 100], q] (* Amiram Eldar, Feb 07 2024 *)

a(0) = 0, a(1) = 2, and for n > 1, if a(n-1) is odd or A257510(a(n-1)) > 1, then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1) + 2. - Antti Karttunen, Apr 29 2015
Other identities:
For all n >= 2, a(A132371(n)) = A000142(n) = n! [See comments in A227157.]

A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 5, 3, 1, 87, 20, 8, 4, 1, 567, 107, 28, 12, 5, 1, 4167, 674, 135, 40, 17, 6, 1, 34407, 4841, 809, 175, 57, 23, 7, 1, 316647, 39248, 5650, 984, 232, 80, 30, 8, 1, 3219687, 355895, 44898, 6634, 1216, 312, 110, 38, 9, 1, 35878887, 3575582, 400793, 51532, 7850, 1528, 422, 148, 47, 10, 1

Offset: 1

Emeric Deutsch, Jun 03 2008

A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

			T(2,1)=1 (the vertical domino); T(2,2)=1 (the horizontal domino); T(3,1)=3 because we have (3), (1,2) and (2,1,1), where (a,b,c,...) stands for a polyomino with columns of lengths a,b,c,..., starting at level 0.
Triangle starts:
    1;
    1,   1;
    3,   2,   1;
   15,   5,   3,   1;
   87,  20,   8,   4,   1;
  567, 107,  28,  12,   5,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Cf. A000142, A132371, A140710.

T:=proc(n,k) options operator, arrow: binomial(n-1, k-1)+sum(factorial(j)*(j-1)*binomial(n-1-j, k-1),j=2..n-1) end proc: for n to 11 do seq(T(n, k),k=1..n) end do; # yields sequence in triangular form

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==1, n! -Sum[j!, {j,n-1}], T[n-1, k] + T[n-1, k-1] ]];
Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, May 02 2021 *)

T(n,k) = binomial(n-1, k-1) + sum(j=2, n-1, j!*(j-1)*binomial(n-1-j, k-1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 16 2019

@CachedFunction
def T(n, k):
    if (k < 0 or k > n): return 0
    elif (k==1): return factorial(n) - sum(factorial(j) for j in (1..n-1))
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 02 2021

T(n,k) = binomial(n-1, k-1) + Sum_{j=2..n-1} j!*(j-1)*binomial(n-1-j, k-1).
T(n,k) = T(n-1, k) + T(n-1, k-1) for n,k >= 2.
Sum of entries in row n is n! (A000142).
T(n,1) = A132371(n).
Sum_{k=1..n} k*T(n,k) = A140710(n).

cp's OEIS Frontend

A227187 Numbers n whose factorial base representation A007623(n) contains at least one nonleading zero. (Zero is also included as a(0)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula