A370880 Natural numbers repeated 3 times are taken in parts of successive lengths 1,2,3,..., and a(n) is the sum of the numbers in the part with length n.

1, 2, 6, 13, 23, 39, 61, 89, 126, 172, 227, 294, 373, 464, 570, 691, 827, 981, 1153, 1343, 1554, 1786, 2039, 2316, 2617, 2942, 3294, 3673, 4079, 4515, 4981, 5477, 6006, 6568, 7163, 7794, 8461, 9164, 9906, 10687, 11507, 12369, 13273, 14219, 15210, 16246

Offset

1,2

Links

Table of n, a(n) for n=1..46.

Nicolay Avilov, Scheme

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Formula

a(n) = Sum_{i=A000217(n-1)+1..A000217(n)} A002264(i+2).

Let r = n mod 3, then we get

a(n) = n*(n^2+3)/6 if r = 0;

(n*(n^2+3)+2)/6 if r = 1;

(n*(n^2+3)-2)/6 if r = 2.

G.f.: x*(x^4-x^3+3*x^2-x+1)/((x^2+x+1)*(x-1)^4).

E.g.f.: (3*exp(x)*x*(4 + 3*x + x^2) + 4*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 19 2025

a(n) = round(n^3/6+n/2). - Hoang Xuan Thanh, Jul 25 2025

Example

The parts and resulting sums begin:
        Part                       Sum = a(n)
  n=1:  1,  .......................   1
  n=2:  1, 1,  ....................   2
  n=3:  2, 2, 2,  .................   6
  n=4:  3, 3, 3, 4,  ..............  13
  n=5:  4, 4, 5, 5, 5,  ...........  23
  n=6:  6, 6, 6, 7, 7, 7,  ........  39
  n=7:  8, 8, 8, 9, 9, 9, 10, .....  61

Mathematica

LinearRecurrence[{3, -3, 2, -3, 3, -1}, {1, 2, 6, 13, 23, 39}, 46] (* James C. McMahon, Apr 22 2024 *)

Crossrefs

Cf. A000217, A002264, A008620.

Sequence in context: A143689 A180773 A011891 * A184533 A338991 A178532

Adjacent sequences: A370877 A370878 A370879 * A370881 A370882 A370883

Keyword

nonn,easy

Author

Nicolay Avilov, Mar 04 2024

Status

approved