A047501 -id:A047501 - cp's OEIS frontend

A100853 Number of partitions of n in which every part occurs 1, 4, or 5 times. Also number of partitions of n in which every part is congruent to {1, 3, 4, 5, 7} mod 8.

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 31, 38, 48, 59, 72, 88, 107, 130, 157, 188, 225, 270, 321, 380, 451, 533, 627, 737, 864, 1011, 1181, 1375, 1599, 1858, 2152, 2488, 2875, 3316, 3816, 4387, 5036, 5773, 6610, 7555, 8626, 9840, 11207, 12748, 14489

Offset: 0

Vladeta Jovovic, Jan 08 2005

Also number of partitions of n in which every even part occurs exactly twice. - Vladeta Jovovic, Oct 06 2007

Cf. A089958.

seq(coeff(mul((1+x^k)*(1+x^(4*k)),k=1..100),x,n),n=0..60); (C. Ronaldo)

np145Q[j_]:=SubsetQ[{1,4,5},Union[Tally[j][[All,2]]]]; Table[Length[ Select[ IntegerPartitions[n],np145Q]],{n,0,51}] (* Harvey P. Dale, Aug 04 2018 *)

Euler transform of period 8 sequence [1, 0, 1, 1, 1, 0, 1, 0, ...]. G.f.: Product_{k>0} (1+x^k)*(1+x^(4*k)) = 1/Product_{k>0} (1-x^A047501(k)).

a(n) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi/2) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 14 2018

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A047418 Numbers that are congruent to {0, 2, 3, 4, 6} mod 8.

Original entry on oeis.org

0, 2, 3, 4, 6, 8, 10, 11, 12, 14, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 40, 42, 43, 44, 46, 48, 50, 51, 52, 54, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 72, 74, 75, 76, 78, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102

Offset: 1

N. J. A. Sloane

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

Cf. A047412, A047434, A047486, A047489, A047501, A047511, A047584.

Filtered([0..103],n->n mod 8 = 0 or n mod 8 = 2 or n mod 8 = 3 or n mod 8 = 4 or n mod 8 = 6); # Muniru A Asiru, Oct 23 2018

[n : n in [0..150] | n mod 8 in [0, 2, 3, 4, 6]]; // Wesley Ivan Hurt, Aug 08 2016

A047418:=n->8*floor(n/5)+[(0, 2, 3, 4, 6)][(n mod 5)+1]: seq(A047418(n), n=0..100); # Wesley Ivan Hurt, Aug 08 2016

Select[Range[0,100], MemberQ[{0,2,3,4,6}, Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,0,1,-1}, {0,2,3,4,6,8}, 70] (* Harvey P. Dale, Oct 01 2015 *)

G.f.: x^2*(2 + x + x^2 + 2*x^3 + 2*x^4)/((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, Aug 08 2016: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6, a(n) = a(n-5) + 8 for n > 5.
a(n) = (40*n - 45 - 2*(n mod 5) + 3*((n + 1) mod 5) + 3*((n + 2) mod 5) - 2*((n + 3) mod 5) - 2*((n + 4) mod 5))/25.
a(5*k) = 8*k - 2, a(5*k-1) = 8*k - 4, a(5*k-2) = 8*k - 5, a(5*k-3) = 8*k - 6, a(5*k-4) = 8*k - 8. (End)
a(n) = (40*n - 45 + 2*cos(2*Pi*(n - 1)/5) - 2*cos(2*Pi*n/5) -  2*cos(4*Pi*n/5) - 6*cos(2*Pi*(n + 1)/5) - 6*cos(Pi*(2*n + 1)/5) + 6*cos(2*Pi*(2*n + 1)/5) - 2*cos(Pi*(4*n + 1)/5) + 6*sin(Pi*(8*n + 3)/10))/25. - Wesley Ivan Hurt, Oct 10 2018

cp's OEIS Frontend

A100853 Number of partitions of n in which every part occurs 1, 4, or 5 times. Also number of partitions of n in which every part is congruent to {1, 3, 4, 5, 7} mod 8.

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