A174738 -id:A174738 - cp's OEIS frontend

A218530 Partial sums of floor(n/11).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008729.

			As square array:
..0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11
.13...15...17...19...21...23...25...27...29...31...33
.36...39...42...45...48...51...54...57...60...63...66
.70...74...78...82...86...90...94...98..102..106..110
115..120..125..130..135..140..145..150..155..160..165
171..177..183..189..195..201..207..213..219..225..231
238..245..252..259..266..273..280..287..294..301..308
316..324..332..340..348..356..364..372..380..388..396
405..414..423..432..441..450..459..468..477..486..495
505..515..525..535..545..555..565..575..585..595..605
...

Cf. similar sequences: A000217, A002820, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242.

a(11n)    = A051865(n).
a(11n+1)  = A180223(n).
a(11n+4)  = A022268(n).
a(11n+5)  = A022269(n).
a(11n+6)  = A254963(n)
a(11n+9)  = A211013(n).
a(11n+10) = A152740(n).
G.f.: x^11/((1-x)^2*(1-x^11)).

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

A134546 Triangle read by rows: T(n, k) = Sum_{j=0..n} floor(j / k).

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 4, 2, 1, 15, 6, 3, 2, 1, 21, 9, 5, 3, 2, 1, 28, 12, 7, 4, 3, 2, 1, 36, 16, 9, 6, 4, 3, 2, 1, 45, 20, 12, 8, 5, 4, 3, 2, 1, 55, 25, 15, 10, 7, 5, 4, 3, 2, 1, 66, 30, 18, 12, 9, 6, 5, 4, 3, 2, 1, 78, 36, 22, 15, 11, 8, 6, 5, 4, 3, 2, 1, 91, 42, 26, 18, 13, 10, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

From Bob Selcoe, Aug 08 2016: (Start)
Columns are partial sums of k-repeating increasing positive integers:
  Column 1 is {1+2+3+4+5+...}         = A000217 (triangular numbers);
  Column 2 is {1+1+2+2+3+3+4+4+...}   = A002620 (quarter-squares);
  Column 3 is {1+1+1+2+2+2+3+3+3+...} = A130518.
  Columns k = 4..7 are A130519, A130520, A174709 and A174738, respectively.
T(n, k) is the number of positive multiples of k which can be expressed as i-j, {i=1..n; j=0..n-1}. So for example, T(5, 2) = 6 because there are 6 ways to express i-j {i<=5} as a multiple of 2: {5-3, 4-2, 3-1, 2-0, 5-1 and 4-0}. (End)
Conjecture: For T(n, k) n >= k^(3/2), there is at least one prime in the interval [T(n-1, k+1), T(n, k)]. - Bob Selcoe, Aug 21 2016
Theorem: For n >= 3*k, T(n, k) is composite. - Daniel Hoying, Jul 08 2020

			The triangle T(n, k) begins:
   n\k  1   2   3   4  5  6  7  8  9  10 ...
   1:   1
   2:   3   1
   3:   6   2   1
   4:  10   4   2   1
   5:  15   6   3   2  1
   6:  21   9   5   3  2  1
   7:  28  12   7   4  3  2  1
   8:  36  16   9   6  4  3  2  1
   9:  45  20  12   8  5  4  3  2  1
  10:  55  25  15  10  7  5  4  3  2   1
... Reformatted. - _Wolfdieter Lang_, Feb 04 2015
T(10,3) = 15: 3*floor(10/3)*floor(13/3)/2 - floor(10/3)*(3-1 - 13 mod 3) = 3*3*4/2 - 3*(3-1-1) = 18 - 3 = 15. - _Bob Selcoe_, Aug 21 2016

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150)

Cf. A078567 (row sums), A000217 (column 1).

Cf. A004736, A051731, A002620, A130518, A130519, A130520, A174709, A174738.

T := proc(n, k) option remember: `if`(n = k, 1, T(n-1, k) + iquo(n,k)) end:
seq(seq(T(n,k), k=1..n),n=1..16); # Peter Luschny, May 26 2020

nn = 12; f[w_] := Map[PadRight[#, nn] &, w]; MapIndexed[Take[#1, First@ #2] &, f@ Table[Reverse@ Range@ n, {n, nn}].f@ Table[Boole@ Divisible[n, #] & /@ Range@ n, {n, nn}]] // Flatten (* Michael De Vlieger, Aug 10 2016 *)

t(n, k) = if (k>n, 0, if (n==1, 1, t(n-1, k) + n\k));
tabl(nn) = {m = matrix(nn, nn, n , k, t(n,k)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Jan 18 2015

trg(nn) = {ma = matrix(nn, nn, n, k, if (k<=n, n-k+1, 0)); mb = matrix(nn, nn, n, k, if (k<=n, !(n%k), 0)); ma*mb;} \\ Michel Marcus, Jan 20 2015

Original definition: T = A004736 * A051731 as infinite lower triangular matrices.
In other words: T(n, k) = Sum_{m=k..n} A004736(n, m)*A051731(m, k).
T(n, k) = 0 if n < k, T(1, 1) = 1, and T(n, k) = T(n-1, k) + floor(n/k), for n >= 2. - Richard R. Forberg, Jan 17 2015
T(n, k) = k*floor(n/k)*floor((n+k)/k)/2 - floor(n/k)*(k-1-(n mod k)). - Bob Selcoe, Aug 21 2016
T(n, k) = k*A000217(b) + (b+1)*[(n +1)-(b + 1)*k] for 1 <= k <= floor[(n + 1) / 2] where b = floor[(n - k + 1) / k], T(n, k) = n-k+1 for floor[(n + 1) / 2] < k <= n and T(n, k) = 0 for k > n. - Henri Gonin, May 12 2020
T(n, k) = (-k/2)*floor(n/k)^2+(n-k/2+1)*floor(n/k). - Daniel Hoying, May 25 2020
From Daniel Hoying, Jul 06 2020: (Start)
T(m + 2*n - 1, m + n) = n for n > 0, m >= 0.
T(3*m + 3*ceiling((n-3)/6) + (n+1)/2, 2*m + 2*ceiling((n-3)/6) + 1) = n for n > 0, n odd, 0 <= m <= floor(n/3).
T(3*m + 3*ceiling(n/6) + n/2 - 1, 2*m + 2*ceiling(n/6)) = n for n > 0, n even, 0 <= m <= floor(n/3). (End)

Edited. Name clarified. Formulas rewritten. - Wolfdieter Lang, Feb 04 2015
Corrected and extended by Michael De Vlieger, Aug 10 2016
Edited and new name from Peter Luschny, Apr 02 2025

A011867 a(n) = floor(n*(n-1)/14).

Original entry on oeis.org

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, 244, 252, 261, 270, 279, 288, 297

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Cf. A174738, A008725.

[Floor(n*(n-1)/14): n in [0..70]]; // Vincenzo Librandi, Aug 29 2011

Table[Floor[n(n - 1)/14], {n, 0, 59}] (* Alonso del Arte, Aug 26 2011 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,0,0,0,1,2,3,4},70] (* Harvey P. Dale, Oct 22 2016 *)

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9).
G.f.: x^5 / ((1-x)^3*(x^6+x^5+x^4+x^3+x^2+x+1) ). (End)

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A269445 a(n) = Sum_{k=0..n} floor(k/13).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

Partial sums of A090620.

More generally, the ordinary generating function for the Sum_{k=0..n} floor(k/m) is x^m/((1 - x^m)*(1 - x)^2).

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A090620.

Cf. similar sequences of Sum_{k=0..n} floor(k/m): A002620 (m=2), A130518 (m=3), A130519 (m=4), A130520 (m=5), A174709 (m=6), A174738 (m=7), A118729 (m=8), A218470 (m=9), A131242 (m=10), A218530 (m=11), A221912 (m=12), this sequence (m=13).

Table[Sum[Floor[k/13], {k, 0, n}], {n, 0, 73}]
LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2}, 74]

G.f.: x^13/((1 - x^13)*(1 - x)^2).

a(n) = 2*a(n-1) - a(n-2) + a(n-13) - 2*a(n-14) + a(n-15).

cp's OEIS Frontend

A218530 Partial sums of floor(n/11).

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

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A134546 Triangle read by rows: T(n, k) = Sum_{j=0..n} floor(j / k).

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A011867 a(n) = floor(n*(n-1)/14).

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A221912 Partial sums of floor(n/12).

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A269445 a(n) = Sum_{k=0..n} floor(k/13).

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