A276382 a(1) = 1, and a(n) = a(n-1) + floor(3*n/2) + 1 for n >= 2.

1, 5, 10, 17, 25, 35, 46, 59, 73, 89, 106, 125, 145, 167, 190, 215, 241, 269, 298, 329, 361, 395, 430, 467, 505, 545, 586, 629, 673, 719, 766, 815, 865, 917, 970, 1025, 1081, 1139, 1198, 1259, 1321, 1385, 1450, 1517, 1585, 1655, 1726, 1799, 1873, 1949, 2026

Offset: 1

Author

Bob Selcoe, Sep 01 2016

Keywords

• nonn
• easy

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.

Examples

			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
		

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).

Crossrefs

Cf. A000040 (prime numbers), A001651, A026477.

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

Programs

Magma

[((-1)^n+12*n+6*n^2-9)/8: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

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 *)

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 */

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