A268128 -id:A268128 - cp's OEIS frontend

A268127 a(n) = (A005704(n)-A006047(n))/3.

0, 0, 0, 1, 1, 1, 3, 3, 3, 7, 8, 9, 12, 13, 14, 19, 20, 21, 30, 33, 36, 42, 45, 48, 57, 60, 63, 79, 86, 93, 103, 111, 119, 132, 141, 150, 168, 180, 192, 209, 222, 235, 257, 271, 285, 316, 335, 354, 380, 400, 420, 453, 474, 495, 543, 573, 603, 639, 672, 705, 747

Offset: 0

Tom Edgar, Jan 26 2016

nonn

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005704, A006047, A268128.

b[n_] := b[n] = If[n <= 2, n+1, b[n-1] + b[Floor[n/3]]];
c = Nest[Join[#, 2#, 3#]&, {1}, 4];
a[n_] := (b[n] - c[[n+1]])/3;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 12 2018 *)

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[floor(i/3)])
    return A[n]
[(b(n)-prod(x+1 for x in n.digits(3)))/3 for n in [0..60]]

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/3)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*3^i is the ternary representation of n. Then a(n) = (1/3)*(b(n) - c(n)).

A268443 a(n) = (A005705(n) - A268444(n))/4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 3, 6, 6, 6, 6, 11, 12, 13, 14, 17, 18, 19, 20, 25, 26, 27, 28, 35, 36, 37, 38, 49, 52, 55, 58, 64, 67, 70, 73, 82, 85, 88, 91, 103, 106, 109, 112, 130, 136, 142, 148, 158, 164, 170, 176, 190, 196, 202, 208, 226, 232, 238, 244

Offset: 0

Tom Edgar, Feb 04 2016

nonn

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005705, A268444, A268127, A268128.

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[i//4])
    return A[n]
print([(b(n)-prod(x+1 for x in n.digits(4)))/4 for n in [0..63]])

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/4)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*4^i is the base 4 representation of n. Then a(n) = (1/4)*(b(n) - c(n)).

A270774 a(n) = (A005706(n) - A194459(n))/5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 32, 33, 34, 35, 36, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 73, 76, 79, 82, 85, 91, 94, 97, 100, 103, 112, 115, 118, 121

Offset: 0

Tom Edgar, Mar 22 2016

nonn

A combinatorial interpretation is given in the Edgar link.

G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.

Cf. A005706, A194459, A268127, A268128, A268443.

b[0] = 1; b[n_] := b[n] = b[n-1] + b[Floor[n/5]];
c[n_] := If[OddQ[n], 2 Count[Table[Binomial[n, k], {k, 0, (n-1)/2}], c_ /; !Divisible[c, 5]], 2 Count[Table[Binomial[n, k], {k, 0, (n-2)/2}], c_ /; !Divisible[c, 5]] + Boole[!Divisible[Binomial[n, n/2], 5]]];
a[n_] := (b[n] - c[n])/5;
Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Feb 15 2019 *)

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[i//5])
    return A[n]
print([(b(n)-prod(x+1 for x in n.digits(5)))/5 for n in [0..63]])

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/5)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*5^i is the base 5 representation of n. Then a(n) = (1/5)*(b(n) - c(n)).

cp's OEIS Frontend

A268127 a(n) = (A005704(n)-A006047(n))/3.

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A268443 a(n) = (A005705(n) - A268444(n))/4.

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A270774 a(n) = (A005706(n) - A194459(n))/5.

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Author

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Links

Crossrefs

Programs

Mathematica

Sage

Formula