A036410 -id:A036410 - cp's OEIS frontend

A175831 Partial sums of ceiling(n^2/12).

0, 1, 2, 3, 5, 8, 11, 16, 22, 29, 38, 49, 61, 76, 93, 112, 134, 159, 186, 217, 251, 288, 329, 374, 422, 475, 532, 593, 659, 730, 805, 886, 972, 1063, 1160, 1263, 1371, 1486, 1607, 1734, 1868, 2009, 2156, 2311, 2473, 2642, 2819, 3004, 3196, 3397, 3606

Offset: 0

Mircea Merca, Dec 05 2010

Partial sums of A036410.

There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).

			a(12) = 0 + 1 + 1 + 1 + 2 + 3 + 3 + 5 + 6 + 7 + 9 + 11 + 12 = 61.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Cf. A036410.

[Round((2*n+1)*(2*n^2+2*n+41)/144): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

seq(floor((n+1)*(2*n^2+n+41)/72),n=0..50)

Accumulate[Ceiling[Range[0,50]^2/12]] (* or *) LinearRecurrence[{2,0,-1,-1,0,2,-1},{0,1,2,3,5,8,11},60] (* Harvey P. Dale, Apr 16 2023 *)

a(n)=(n+1)*(2*n^2+n+41)\72 \\ Charles R Greathouse IV, Jul 06 2017

a(n) = round((2*n+1)*(2*n^2 + 2*n + 41)/144).
a(n) = floor((n+1)*(2*n^2 + n + 41)/72).
a(n) = ceiling((2*n^3 + 3*n^2 + 42*n)/72).
a(n) = a(n-12) + (n+1)*(n-12) + 61.
G.f.: x*(1-x^2+x^4) / ( (1+x)*(1+x+x^2)*(x-1)^4 ). - R. J. Mathar, Jun 22 2011

A340445 Number of partitions of n into 3 parts that are not all the same.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80, 85, 90, 96, 102, 107, 114, 120, 126, 133, 140, 146, 154, 161, 168, 176, 184, 191, 200, 208, 216, 225, 234, 242, 252, 261, 270, 280, 290, 299, 310, 320, 330

Offset: 0

Wesley Ivan Hurt, Jan 07 2021

Conjecturally the same as A230059 (apart from the offset). - R. J. Mathar, Jan 14 2021

			a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
a(9) = 6; [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2] ( [3,3,3] not counted ).

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A036410, A069905.

Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i = n-i-k]), where [ ] is the (generalized) Iverson bracket.
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i] * [2*i = n-k] * [2*k = n-i]), where [ ] is the Iverson bracket.
From Alois P. Heinz, Jan 07 2021: (Start)
G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)
a(n) = A036410(n-1)-1. - Hugo Pfoertner, Jan 09 2021
a(n) + A079978(n) = A069905(n), n>0. - R. J. Mathar, Jan 18 2021
72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - R. J. Mathar, Jun 09 2022

cp's OEIS Frontend

A175831 Partial sums of ceiling(n^2/12).

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