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 1-10 of 11 results. Next

A035386 Number of partitions of n into parts congruent to 2 mod 3.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 11, 15, 15, 18, 19, 23, 23, 29, 29, 35, 37, 43, 45, 53, 55, 64, 68, 78, 82, 95, 99, 114, 121, 136, 145, 164, 173, 196, 208, 232, 248, 276, 294, 328, 349, 386, 413, 456, 486, 537, 572, 629, 673, 737, 787
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

a(n) = A116376(3*n). - Reinhard Zumkeller, Feb 15 2006

Links

Crossrefs

Cf. A035382, A035451, A262928.

Programs

  • Maple
    g:=add(x^(n*(3*n-1))/mul((1-x^(3*k))*(1-x^(3*k-1)), k = 1..n), n = 0..6): gser:=series(g,x,101): seq(coeff(gser,x,n), n = 0..100); # Peter Bala, Feb 02 2021
  • Mathematica
    nmax=100; CoefficientList[Series[Product[1/(1-x^(3*k+2)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, 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 + 2;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)
  • PARI
    {a(n)= if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - (k%3==2) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jul 24 2007 */

Formula

a(n) = 1/n*Sum_{k=1..n} A078182(k)*a(n-k), a(0) = 1. - Vladeta Jovovic, Nov 21 2002
Euler transform of period 3 sequence [ 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ Gamma(2/3) * exp(sqrt(2*n)*Pi/3) / (2^(11/6) * sqrt(3) * Pi^(1/3) * n^(5/6)) * (1 + (Pi/72 - 5/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(3*n-1))/Product_{k = 1..n} ((1 - x^(3*k)) *(1 - x^(3*k-1))). (Set z = x^2 and q = x^3 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000041, A010888, A035444, A147706.

Programs

Formula

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Haskell
    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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Haskell
    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

A116373 Number of partitions of n into parts with digital root = 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 10, 0, 0, 11, 0, 0, 13, 0, 0, 15, 0, 0, 17, 0, 0, 19, 0, 0, 23, 0, 0, 26, 0, 0, 29, 0, 0, 33, 0, 0, 38, 0, 0, 42, 0, 0, 48, 0, 0, 54, 0, 0, 61, 0, 0, 68, 0, 0, 77, 0, 0, 85, 0, 0, 96
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

a(n) = A114102(n) - A116371(n) - A116372(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

Examples

			a(21) = #{21, 12+3+3+3, 3+3+3+3+3+3+3} = 3.

Links

Crossrefs

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]

Formula

a(n) = A035382(floor(n/3))*0^(n mod 3).

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
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