A319014 a(n) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 + ... + (up to n).

1, 2, 6, 10, 26, 126, 133, 182, 630, 640, 740, 1950, 1963, 2132, 4680, 4696, 4952, 9576, 9595, 9956, 17556, 17578, 18062, 29700, 29725, 30350, 47250, 47278, 48062, 71610, 71641, 72602, 104346, 104380, 105536, 147186, 147223, 148592, 202020, 202060, 203660

Offset: 1

Author

Wesley Ivan Hurt, Sep 07 2018

Keywords

• nonn
• easy

Comments

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=3.

Examples

			a(1)  = 1;
a(2)  = 1*2 = 2;
a(3)  = 1*2*3 = 6;
a(4)  = 1*2*3 + 4 = 10;
a(5)  = 1*2*3 + 4*5 = 26;
a(6)  = 1*2*3 + 4*5*6 = 126;
a(7)  = 1*2*3 + 4*5*6 + 7 = 133;
a(8)  = 1*2*3 + 4*5*6 + 7*8 = 182;
a(9)  = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(10) = 1*2*3 + 4*5*6 + 7*8*9 + 10 = 640;
a(11) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11 = 740;
a(12) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(13) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13 = 1963;
a(14) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14 = 2132;
a(15) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(16) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16 = 4696;
a(17) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17 = 4952;
a(18) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576;
a(19) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 + 19 = 9595;
etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).

Crossrefs

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) this sequence, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Cf. A049347, A061347, A268685 (trisection).

Programs

Mathematica

CoefficientList[Series[(1 + x + 4*x^2 + 12*x^4 + 84*x^5 - 3*x^6 - 9*x^7 + 72*x^8 + 2*x^9 - 4*x^10 + 2*x^11)/((1 - x)^5*(1 + x + x^2)^4), {x, 0, 50}], x] (* after Colin Barker *)

PARI

Vec(x*(1 + x + 4*x^2 + 12*x^4 + 84*x^5 - 3*x^6 - 9*x^7 + 72*x^8 + 2*x^9 - 4*x^10 + 2*x^11) / ((1 - x)^5*(1 + x + x^2)^4) + O(x^50)) \\ Colin Barker, Sep 08 2018

Formula

a(n) = Sum_{i=1..floor(n/3)} (3*i)!/(3*i-3)! + Sum_{j=1..2} (1-sign((n-j) mod 3)) * (Product_{i=1..j} n-i+1).

From Colin Barker, Sep 08 2018: (Start)

G.f.: x*(1 + x + 4*x^2 + 12*x^4 + 84*x^5 - 3*x^6 - 9*x^7 + 72*x^8 + 2*x^9 - 4*x^10 + 2*x^11) / ((1 - x)^5*(1 + x + x^2)^4).

a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n>13.

(End)

a(3*k) = 3*k*(k+1)*(3*k-2)*(3*k+1)/4, a(3*k+1) = a(3*k) + 3*k + 1, a(3*k+2) = a(3*k) + (3*k+2)*(3*k+1). - Giovanni Resta, Sep 08 2018

a(n) = (3*n^4 - 6*n^3 + 9*n^2 + 6*n - 8 - 2*(3*n^3 - 6*n^2 - 6*n + 2)*A061347(n-3) + 6*(n^3 - 6*n^2 + 6*n + 2)*A049347(n-2))/36. - Stefano Spezia, Apr 23 2023