OFFSET
1,2
COMMENTS
Given 3 distinct numbers i, j and k whose prime signatures are exactly n 1's, then a(n) is the number of prime signatures for all permutations of i*j*k.
a(n) is the number of partitions of 3n such that there are no more than n-1 3's and no parts > 3.
a(3n+1) represents the number of prime signature sets whose members are excluded as terms in A026477, as a consequence of being products of three smaller terms whose prime signatures are exactly 3n+1 1's.
First differences are floor(3n/2) + 1 (A001651(n+1)); second differences are 1 when n is even and 2 when n is odd; third differences are 1 when n is even and -1 when n is odd.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
From Colin Barker, Sep 01 2016: (Start)
a(n) = ((-1)^n + 12*n + 6*n^2 - 9)/8 for n > 0.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.
G.f.: x*(1 + 3*x - x^3) / ((1-x)^3*(1+x)). (End)
a(n) = -1 + Sum_{k=1..n} floor((n+k+3)/2). - Wesley Ivan Hurt, Apr 01 2017
a(n) = a(-2-n) for all n in Z. - Michael Somos, Sep 08 2023
a(n) = floor((3*(n+1)^2 - 7)/4). - Michael Somos, Sep 09 2023
EXAMPLE
a(2)=5; the 5 prime signatures / partitions are: {3,1,1,1}, {2,2,2}, {2,2,1,1}, {2,1,1,1} and {1,1,1,1,1,1}.
G.f. = x + 5*x^2 + 10*x^3 + 17*x^4 + 25*x^5 + 35*x^6 + 46*x^7 + ... - Michael Somos, Sep 08 2023
MATHEMATICA
a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[3 n/2] + 1 ; Array[a, 51] (* Michael De Vlieger, Sep 01 2016 *)
a[n_] := Floor[(3*(n+1)^2 - 7)/4]; (* Michael Somos, Sep 08 2023 *)
PROG
(PARI) Vec(x*(1+3*x-x^3)/((1-x)^3*(1+x)) + O(x^60)) \\ Colin Barker, Sep 01 2016
(PARI) {a(n) = (3*(n+1)^2 - 7)\4}; /* Michael Somos, Sep 09 2023 */
(Magma) [((-1)^n+12*n+6*n^2-9)/8: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016
CROSSREFS
Equals one less than A331952(n+1), two less than A077043(n+1), and three less than A084684(n+1). - Greg Dresden, Feb 22 2020
KEYWORD
nonn,easy
AUTHOR
Bob Selcoe, Sep 01 2016
STATUS
approved