A114098 -id:A114098 - cp's OEIS frontend

A147706 Number of partitions of n into parts having distinct digital roots (A010888).

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 11, 15, 16, 20, 25, 28, 32, 39, 46, 50, 62, 66, 78, 93, 101, 112, 132, 150, 161, 192, 202, 232, 268, 287, 312, 361, 400, 425, 497, 516, 582, 658, 698, 748, 858, 932, 982, 1135, 1164, 1296, 1443, 1519, 1610, 1845, 1968, 2059, 2360, 2395

Offset: 1

Reinhard Zumkeller, Nov 11 2008

a(n) <= A000009(n).

Likely a duplicate of A114098. [From R. J. Mathar, Dec 13 2008]

			A000009(16) = 32, in which the following 4 partitions
contain parts with common digital roots:
12 + 3 + 1, 11 + 3 + 2, 10 + 5 + 1 and 10 + 3 + 2 + 1,
therefore a(16) = 32 - 4 = 28.

A114102, A116371, A116372, A116373, A116374, A116375, A116376, A116377, A116378, A114099.

A114091 Number of partitions of n into parts that are distinct mod 3.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 3, 3, 7, 4, 4, 11, 5, 5, 16, 6, 6, 22, 7, 7, 29, 8, 8, 37, 9, 9, 46, 10, 10, 56, 11, 11, 67, 12, 12, 79, 13, 13, 92, 14, 14, 106, 15, 15, 121, 16, 16, 137, 17, 17, 154, 18, 18, 172, 19, 19, 191, 20, 20, 211, 21, 21, 232, 22, 22, 254, 23, 23, 277, 24

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

Each partition can have at most three parts if n is a multiple of three and at most two parts otherwise. - Andrew Howroyd, Jan 28 2020

In general, these sequences can be generated by a linear recurrence with a signature that contains k=1..d tuples of the form (d-1 zeros, (-1)^(k-1) * binomial(d, k)), where d = number of distinct parts (here: d=3). - Georg Fischer, Sep 03 2022

			a(5)=2 because there are 2 such partition of 5: {5}, {2,3}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A008619(d=2), A114092(4), A114093(5), A114094(6), A114095(7), A114096(8), A114098(9), A114097(10).

<< DiscreteMath`Combinatorica`; np[n_]:= Length@Select[Mod[ #,3]& /@ Partitions[n],(Length@# != Length@Union@#)&]; lst = Array[np,50] (* or *)
LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 2, 2, 2, 4, 3, 3, 7}, 64] (* Georg Fischer, Sep 03 2022 *)

a(n)={1 + n\3 + if(n%3==0, binomial(n/3,2))} \\ Andrew Howroyd, Jan 28 2020

a(3*n) = 1 + n + binomial(n, 2); a(3*n-1) = a(3*n-2) = n. - Andrew Howroyd, Jan 28 2020

Terms a(51) and beyond from Andrew Howroyd, Jan 28 2020

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A147706 Number of partitions of n into parts having distinct digital roots (A010888).

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A114091 Number of partitions of n into parts that are distinct mod 3.

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