A173722 -id:A173722 - cp's OEIS frontend

A177116 Partial sums of round(n^2/11).

0, 0, 0, 1, 2, 4, 7, 11, 17, 24, 33, 44, 57, 72, 90, 110, 133, 159, 188, 221, 257, 297, 341, 389, 441, 498, 559, 625, 696, 772, 854, 941, 1034, 1133, 1238, 1349, 1467, 1591, 1722, 1860, 2005, 2158, 2318, 2486, 2662, 2846, 3038, 3239, 3448, 3666, 3893

Offset: 0

Mircea Merca, Dec 09 2010

The round function, also called the nearest integer function, is defined here by round(x)=floor(x+1/2).

There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2, ..., 9, 11, 12, 13, 16, 17, 19, 20, 28, 29, 36, 44).

			a(11) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 6 + 7 + 9 + 11 = 44.

Vincenzo Librandi, Table of n, a(n) for n = 0..870
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 (3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1).

Cf. A173690 (k=5), A173691 (k=6), A173722 (k=8), A177100 (k=9), A181120 (k=12).

seq(round((2*n^3+3*n^2-11*n)/66),n=0..50)

Accumulate[Round[Range[0,50]^2/11]] (* or *) LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1},{0,0,0,1,2,4,7,11,17,24,33,44,57,72},60] (* Harvey P. Dale, Dec 10 2014 *)

a(n)=(2*n^3+3*n^2-11*n+18)\66 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = round((n-2)*(n+3)*(2*n+1)/66).
a(n) = floor((2*n^3 + 3*n^2 - 11*n + 18)/66).
a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 30)/66).
a(n) = round(n*(2*n^2 + 3*n - 11)/66).
a(n) = a(n-11) + (n+1)*(n-11) + 44, n > 10.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14). - R. J. Mathar, Dec 10 2010
G.f.: x^3 *(1+x) *(x^2-x+1) *(x^4-x^3+x^2-x+1) / ( (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) *(x-1)^4 ). - R. J. Mathar, Dec 10 2010 [Typo fixed by Colin Barker, Oct 10 2012]

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

Original entry on oeis.org

0, 1, 8, 36, 104, 245, 492, 896, 1504, 2385, 3600, 5236, 7368, 10101, 13524, 17760, 22912, 29121, 36504, 45220, 55400, 67221, 80828, 96416, 114144, 134225, 156832, 182196, 210504, 242005, 276900, 315456, 357888, 404481, 455464, 511140, 571752, 637621, 709004, 786240

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th row sum of the triangle A325655.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A173722, A317614, A322277, A325655.

Flat(List([0..50], n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3)));

[(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): n in [0..50]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..50);

a[n_]:=(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3); Array[a,50,0]

a(n) = (1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3);

O.g.f.: -x*(1 + 5*x + 13*x*2 + 9*x^3 + 4*x^4)/((-1 + x)^5*(1 + x)^2).
E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 78*exp(2*x)*x + 54*exp(2*x)*x^2 + 8*exp(2*x)*x*3).
a(n) = a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
a(n) = (1/12)*n^2*(4*n^2 + 3*n + 2) if n is even.
a(n) = (1/12)*n*(n + 1)*(4*n^2 - n + 3) if n is odd.
a(n) = n*A173722(2*n). - Stefano Spezia, Dec 21 2021

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A177116 Partial sums of round(n^2/11).

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A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

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cp's OEIS Frontend

A177116 Partial sums of round(n^2/11).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A325656 a(n) = (1/24)*n*((4*n + 3)*(2*n^2 + 1) - 3*(-1)^n).

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Author

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Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).