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A319014 a(n) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 + ... + (up to n). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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).

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

EXAMPLE

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.

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 *)

PROG

(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

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. A268685 (trisection).

Sequence in context: A061547 A218791 A320429 * A190034 A119459 A291463

Adjacent sequences:  A319011 A319012 A319013 * A319015 A319016 A319017

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Sep 07 2018

STATUS

approved

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Last modified September 15 16:28 EDT 2019. Contains 327078 sequences. (Running on oeis4.)