A116373 -id:A116373 - cp's OEIS frontend

A035382 Number of partitions of n into parts congruent to 1 mod 3.

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 26, 29, 33, 38, 42, 48, 54, 61, 68, 77, 85, 96, 107, 119, 132, 148, 163, 181, 201, 223, 245, 272, 299, 330, 363, 400, 438, 483, 529, 580, 635, 697, 760, 832, 908, 992, 1081, 1180, 1283, 1399, 1521

Offset: 0

Olivier Gérard

nonn

a(n) = A116373(3*n). - Reinhard Zumkeller, Feb 15 2006

			a(3) = 1 because we have [1,1,1];
a(4) = 2 because we have [1,1,1,1] and [4];
a(9) = 4 because we have [7,1,1], [4,4,1], [4,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Joerg Arndt, Matters Computational (The Fxtbook), p. 350.

Cf. A035386, A035451, A261612.

g:= 1/product(1-x^(1+3*j), j=0..50): gser:= series(g, x=0, 64): seq(coeff(gser, x, n), n=0..61); # Emeric Deutsch, Mar 30 2006
# second Maple program
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-3) +`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, 3*iquo(n, 3)+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-3] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, 3*Quotient[n, 3]+1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 50; s = Range[0, nmax/3]*3 + 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

a(n) = 1/n*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.
G.f.: 1/prod(j>=0, 1-x^(1+3*j) ). - Emeric Deutsch, Mar 30 2006
From Joerg Arndt, Oct 02 2012: (Start)
G.f.: sum(n>=0, q^n/prod(k=1..n, 1-q^(3*k)) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(R*n)/prod(k=1..n, 1-q^(M*k) ) ) for partitions into parts R mod M (where R!=0).
G.f. sum(n>=0, q^(3*n^2-2*n) / prod(k=0..n-1, (1-q^(3*k+3))*(1-q^(3*k+1))) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(M*n^2+(R-M)*n) / prod(k=0..n-1, (1-q^(M*k+M))*(1-q^(M*k+R))) ) for partitions into parts R mod M (where R!=0). (See Fxtbook link)
(End)
a(n) ~ Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2*sqrt(3) * (2*Pi*n)^(2/3)) * (1 + (Pi/72 - 2/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
Euler transform of period 3 sequence [ 1, 0, 0, ...]. - Kevin T. Acres, Apr 28 2018

A114099 Number of partitions of n into parts with digital root = 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n).

			a(27) = #{27, 18+9, 9+9+9} = 3.

Antti Karttunen, Table of n, a(n) for n = 0..19683
Eric Weisstein's World of Mathematics, Digital Root

Cf. A000041, A010888, A035444, A147706.

f[n_] := PartitionsP[n/9] If[Mod[n, 9] == 0, 1, 0]; Array[f, 105] (* Robert G. Wilson v, Apr 25 2010 *)

A114099(n) = if((n%9), 0, numbpart(n/9)); \\ Antti Karttunen, Jul 22 2018

a(n) = A000041(floor(n/9))*0^(n mod 9).

a(9n) = A000041(n) and for all others a(n) = 0. [Robert G. Wilson v, Apr 25 2010]

A114102 Number of partitions of n such that all parts of a partition have the same digital root.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 3, 7, 4, 5, 7, 6, 5, 8, 5, 8, 11, 8, 7, 16, 9, 10, 13, 12, 10, 22, 11, 15, 23, 16, 16, 26, 16, 18, 32, 22, 21, 41, 24, 27, 40, 28, 26, 55, 30, 36, 59, 40, 38, 65, 41, 45, 77, 48, 51, 95, 57, 60, 97, 66, 63, 119, 68, 80, 131, 89, 85, 150, 91, 96, 166, 104

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A116371(n) + A116372(n) + A116373(n) + A116374(n) + A116375(n) + A116376(n) + A116377(n) + A116378(n) + A114099(n).

			a(10) = #{10, 5+5, 2+2+2+2+2, 10x1} = 4;
a(11) = #{11, 10+1, 11x1} = 3;
a(12) = #{12, 10+1+1, 6+6, 4+4+4, 3+3+3+3, 2+2+2+2+2+2, 12x1} = 7.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
A156144. [From Reinhard Zumkeller, Feb 05 2009]

a114102 n = length $ filter (== 1) $
            map (length . nub . (map a010888)) $ ps 1 n
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Feb 04 2014

A116371 Number of partitions of n into parts with digital root = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 67, 68

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(18) = #{10+8x1, 18x1} = 2;
a(19) = #{19, 10+9x1, 19x1} = 3;
a(20) = #{19+1, 10+10, 10+10x1, 19x1} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
A017173, A156144, A156145. [From Reinhard Zumkeller, Feb 05 2009]

a116371 n = p a017173_list n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 04 2014

A116376 Number of partitions of n into parts with digital root = 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 7, 0, 0, 7, 0, 0, 9, 0, 0, 9, 0, 0, 12, 0, 0, 11, 0, 0, 15, 0, 0, 15, 0, 0, 18, 0, 0, 19, 0, 0, 23, 0, 0, 23, 0, 0, 29, 0, 0, 29, 0, 0, 35, 0, 0, 37

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116377(n) - A116378(n) - A114099(n).

			a(30) = #{24+6, 15+15, 6+6+6+6+6} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

Cf. A147706.

N:= 1000: # to get a(1) to a(N)
g:= mul(1/(1-x^(6+9*j)), j=0..floor((N-6)/9)):
S:= series(g, x, N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 13 2015

a(n) = A035386(floor(n/3))*0^(n mod 3).

G.f.: Product_{j>=0} 1/(1 - x^(6+9*j)). - Robert Israel, Apr 13 2015

A116372 Number of partitions of n into parts with digital root = 2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 2, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 6, 4, 7, 4, 8, 4, 8, 5, 8, 7, 8, 9, 8, 10, 8, 11, 9, 11, 12, 11, 15, 11, 17, 11, 18, 12, 19, 15, 19, 19, 19, 22, 19, 24, 20, 25, 24, 26, 29, 26, 34, 26, 37, 27, 39, 31, 40, 38, 41

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(31) = #{29+2, 20+11, 11+2+2+2+2+2} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116374 Number of partitions of n into parts with digital root = 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 3, 4, 1, 2, 3, 5, 2, 2, 3, 6, 4, 2, 3, 6, 6, 3, 3, 6, 7, 6, 3, 6, 8, 9, 4, 6, 8, 11, 7, 6, 8, 12, 11, 7, 8, 13, 14, 11, 8, 13, 16, 16, 9, 13, 17, 21, 13, 13, 18, 24, 20, 14, 18, 26

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(39) = #{31+4+4, 22+13+4, 13+13+13} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116375 Number of partitions of n into parts with digital root = 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 2, 4, 3, 2, 1, 4, 5, 3, 2, 2, 6, 5, 3, 2, 5, 7, 5, 3, 3, 8, 8, 5, 3, 6, 10, 8, 5, 4, 10, 11, 8, 5, 8, 13, 12, 8, 6, 13, 15, 12, 8, 10, 18, 16, 12, 9, 17, 21, 17

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(42) = #{32+5+5, 23+14+5, 14+14+14} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116377 Number of partitions of n into parts with digital root = 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 0, 2, 0, 2, 1, 1, 1, 0, 2, 0, 3, 1, 3, 1, 1, 2, 0, 3, 1, 4, 1, 3, 2, 1, 3, 1, 5, 1, 5, 2, 4, 3, 2, 5, 1, 6, 2, 7, 3, 5, 5, 2, 7, 2, 9, 3, 9, 5, 6, 7, 3, 10, 3, 12, 5, 11, 7, 7, 11, 4, 14, 5, 16, 7, 14

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116378(n) - A114099(n).

			a(48) = #{34+7+7, 25+16+7, 16+16+16} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116378 Number of partitions of n into parts with digital root = 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 2, 3, 3, 1, 0, 0, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 2, 1, 2, 3, 5, 7, 9, 8, 6, 2, 2, 3, 5, 7, 10, 11, 11, 6, 3, 3, 5, 7, 11, 13

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A114099(n).

			a(51) = #{35+8+8, 26+17+8, 17+17+17} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

cp's OEIS Frontend

A035382 Number of partitions of n into parts congruent to 1 mod 3.

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A114099 Number of partitions of n into parts with digital root = 9.

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A114102 Number of partitions of n such that all parts of a partition have the same digital root.

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A116371 Number of partitions of n into parts with digital root = 1.

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A116376 Number of partitions of n into parts with digital root = 6.

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A116372 Number of partitions of n into parts with digital root = 2.

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A116374 Number of partitions of n into parts with digital root = 4.

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A116375 Number of partitions of n into parts with digital root = 5.

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