A011867 -id:A011867 - cp's OEIS frontend

A174738 Partial sums of floor(n/7).

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236

Offset: 0

Mircea Merca, Nov 30 2010

Apart from the initial zeros, the same as A011867.

			a(9) = floor(0/7) + floor(1/7) + floor(2/7) + floor(3/7) + floor(4/7) + floor(5/7) + floor(6/7) + floor(7/7) + floor(8/7) + floor(9/7) = 3.

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,-1,0,0,0,0,1,-2,1).

Cf. A001106, A022264, A022265, A024966, A179986, A186029, A218471.

Cf. Similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A118729, A218470, A131242.

List([0..60], n-> Int((n-2)*(n-3)/14)); # G. C. Greubel, Aug 31 2019

[Round(n*(n-5)/14): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

A174738 := proc(n) round(n*(n-5)/14) ; end proc:
seq(A174738(n),n=0..30) ;

Table[Floor[(n - 2)*(n - 3)/14], {n,0,60}] (* G. C. Greubel, Dec 13 2016 *)

a(n)=(n-2)*(n-3)\14 \\ Charles R Greathouse IV, Sep 24 2015

[floor((n-2)*(n-3)/14) for n in (0..60)] # G. C. Greubel, Aug 31 2019

a(n) = round(n*(n-5)/14).
a(n) = floor((n-2)*(n-3)/14).
a(n) = ceiling((n+1)*(n-6)/14).
a(n) = a(n-7) + n - 6, n > 6.
a(n) = +2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - R. J. Mathar, Nov 30 2010
G.f.: x^7/( (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1-x)^3 ). - R. J. Mathar, Nov 30 2010
a(7n) = A001106(n), a(7n+1) = A218471(n), a(7n+2) = A022264(n), a(7n+3) = A022265(n), a(7n+4) = A186029(n), a(7n+5) = A179986(n), a(7n+6) = A024966(n). - Philippe Deléham, Mar 26 2013

A008725 Molien series for 3-dimensional group [2,n] = *22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236, 244, 252, 261, 270, 279, 288, 297, 306

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 1 and 7, where there are two kinds of part 1. - Joerg Arndt, Sep 27 2020

Define a general Somos-4 sequence by b(n) = (p1*b(n-1)*b(n-3) + p2*b(n-2)^2)/b(n-4) with b(0) = b0, b(1) = b1, b(2) = b2, b(3) = b3 and where p1 = (b1^3*b2 - b0^3*b3) / (b0*(b1^3 + b0^2*b2)), p2 = -b1*(b2^2 + b0*b3) / (b1^3 + b0^2*b2). Then b(n) = -b(-1-n) for all n in Z. The denominator of b(n) is a power of b0 times (b1^3 + b0^2*b2)^a(n-4). - Michael Somos, Nov 23 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 190
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A011867, A174738.

a:=[1,2,3,4,5,6,7,9,11];; for n in [10..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # G. C. Greubel, Sep 09 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^7)) )); // G. C. Greubel, Sep 09 2019

1/((1-x)^2*(1-x^7)): seq(coeff(series(%, x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^7)), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,9,11}, 80] (* Harvey P. Dale, Sep 27 2014 *)
a[ n_] := Floor[(n+4)*(n+5)/14]; (* Michael Somos, Nov 23 2023 *)

my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^7))) \\ G. C. Greubel, Sep 09 2019

{a(n) = (n+4)*(n+5)\14}; /* Michael Somos, Nov 23 2023 */

def A008725_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^7))).list()
A008725_list(80) # G. C. Greubel, Sep 09 2019

G.f.: 1/((1-x)^2*(1-x^7)).
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+7} floor(j/7).
a(n-7) = (1/2)*floor(n/7)*(2*n - 5 - 7*floor(n/7)). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - R. J. Mathar, Apr 20 2010
a(n) = A011867(n+5). - Pontus von Brömssen, Sep 27 2020
a(n) = a(-9-n) = A174738(n+7) = floor((n+4)*(n+5)/14) for all n in Z. - Michael Somos, Nov 23 2023

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

cp's OEIS Frontend

A174738 Partial sums of floor(n/7).

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A008725 Molien series for 3-dimensional group [2,n] = *22n.

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