A130520 -id:A130520 - cp's OEIS frontend

A008732 Molien series for 3-dimensional group [2,n] = *22n.

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 189, 198, 207, 216, 225, 235, 245, 255, 265

Offset: 0

N. J. A. Sloane

			From _Philippe Deléham_, Apr 05 2013: (Start)
Stored in five columns:
    1   2   3   4   5
    7   9  11  13  15
   18  21  24  27  30
   34  38  42  46  50
   55  60  65  70  75
   81  87  93  99 105
  112 119 126 133 140
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 188
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=5]
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A130520.

List([0..50], n-> Int((n+3)*(n+4)/10)); # G. C. Greubel, Jul 30 2019

[Floor((n+3)*(n+4)/10): n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

A092202 := proc(n) op(1+(n mod 5),[0,1,0,-1,0]) ; end proc:
A010891 := proc(n) op(1+(n mod 5),[1,-1,0,0,0]) ; end proc:
A008732 := proc(n) (n+2)*(n+5)/10+(A010891(n-1)+2*A092202(n-1))/5 ; end proc:

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 7, 9}, 50] (* Jean-François Alcover, Jan 18 2018 *)

a(n)=(n+3)*(n+4)\10 \\ Charles R Greathouse IV, Oct 07 2015

[floor((n+3)*(n+4)/10) for n in (0..50)] # G. C. Greubel, Jul 30 2019

a(n) = floor( (n+3)*(n+4)/10 ) = (n+2)*(n+5)/10 + b(n)/5 where b(n) = A010891(n-2) + 2*A092202(n-1) = 0, 1, 1, 0, -2, ... with period length 5.
G.f.: 1/((1-x)^2*(1-x^5)).
a(n) = a(n-5) + n + 1. - Paul Barry, Jul 14 2004
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+5} floor(j/5).
a(n-5) = (1/2)floor(n/5)*(2*n - 3 - 5*floor(n/5)). (End)
a(n) = A130520(n+5). - Philippe Deléham, Apr 05 2013
a(5n) = A000566(n+1), a(5n+1) = A005476(n+1), a(5n+2) = A005475(n+1), a(5n+3) = A147875(n+2), a(5n+4) = A028895(n+1); these formulas correspond to the 5 columns of the array shown in example. - Philippe Deléham, Apr 05 2013

A011858 a(n) = floor( n*(n-1)/5 ).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 8, 11, 14, 18, 22, 26, 31, 36, 42, 48, 54, 61, 68, 76, 84, 92, 101, 110, 120, 130, 140, 151, 162, 174, 186, 198, 211, 224, 238, 252, 266, 281, 296, 312, 328, 344, 361, 378, 396, 414, 432, 451, 470, 490, 510, 530, 551, 572, 594, 616, 638, 661, 684

Offset: 0

N. J. A. Sloane

a(n-2) is the total degree of the irreducible factor F(n) of the n-th Somos polynomial. - Michael Somos, Jul 06 2011

			G.f. = x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 14*x^9 + 18*x^10 + 22*x^11 + ...
F(5) = y + 1 is of degree a(3) = 1, F(6) = y*z + y + z is of degree a(4) = 2.

Matthew House, Table of n, a(n) for n = 0..10000
Michael Somos, Somos Polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A130520.

[Floor(n*(n-1)/5): n in [0..50]]; // G. C. Greubel, Oct 28 2017

a[ n_] := Quotient[ n (n - 1), 5]; (* Michael Somos, Oct 19 2014 *)
LinearRecurrence[{2,-1,0,0,1,-2,1},{0,0,0,1,2,4,6},60] (* Harvey P. Dale, Dec 21 2024 *)

{a(n) = n * (n - 1) \ 5}; /* Michael Somos, Jul 04 2011 */

G.f.: x^3*(x^2+1)/ ((1-x)^3 * (1+x+x^2+x^3+x^4)). a(n) = +2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7). - R. J. Mathar, Apr 15 2010
Euler transform of length 5 sequence [2, 1, 0, -1, 1]. - Michael Somos, Jul 04 2011
a(1-n) = a(n). a(n) = a(n-5) + 2*n - 6 for all n in Z. - Michael Somos, Jul 04 2011
a(n) = a(n-1) + a(n-5) - a(n-6) + 2 for all n in Z. - Michael Somos, Jul 06 2011
a(n) = (1/5) * ( n^2 - n + [0,0,-2,-1,-2](mod 5) ). - Ralf Stephan, Aug 11 2013
a(n) - 2*a(n+1) + a(n+2) = (n == 1 (mod 5)) + (n == 3 (mod 5)) for all n in Z. - Michael Somos, Oct 19 2014
a(n) = A130520(n) + A130520(n+2). - R. J. Mathar, Aug 11 2021
Sum_{n>=3} 1/a(n) = 50/9 - sqrt(2*(5+sqrt(5)))*Pi/3 + tan(Pi/(2*sqrt(5)))*Pi/sqrt(5). - Amiram Eldar, Oct 01 2022

A218530 Partial sums of floor(n/11).

Original entry on oeis.org

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)).

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

A062781 Number of arithmetic progressions of four terms and any mean which can be extracted from the set of the first n positive integers.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392

Offset: 1

Santi Spadaro, Jul 18 2001

This sequence seems to be a shifted version of the Somos sequence A058937.
Equal to the partial sums of A002264 (cf. A130518) but with initial index 1 instead of 0. - Hieronymus Fischer, Jun 01 2007
Apart from offset, the same as A130518. - R. J. Mathar, Jun 13 2008
Apart from offset, the same as A001840. - Michael Somos, Sep 18 2010

Muniru A Asiru, Table of n, a(n) for n = 1..750
Michael Somos, Somos Polynomials
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A058937, A001840.

Cf. A002620, A130519, A130520.

seq(coeff(series(x^4/((1-x^3)*(1-x)^2),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Nov 25 2018

RecurrenceTable[{a[0]==0, a[n]==Floor[n/3] + a[n-1]}, a, {n, 49}] (* Jon Maiga, Nov 25 2018 *)

[floor(binomial(n,2)/3) for n in range(0,50)] # Zerinvary Lajos, Dec 01 2009

a(n) = P(n,4), where P(n,k) = n*floor(n/(k - 1)) - (1/2)(k - 1)(floor(n/(k - 1))*(floor(n/(k - 1)) + 1)); recursion: a(n) = a(n-3) + n - 3; a(1) = a(2) = a(3) = 0.
From Hieronymus Fischer, Jun 01 2007: (Start)
a(n) = (1/2)*floor((n-1)/3)*(2*n - 3 - 3*floor((n-1)/3)).
G.f.: x^4/((1 - x^3)*(1 - x)^2). (End)
a(n) = floor((n-1)/3) + a(n-1). - Jon Maiga, Nov 25 2018
E.g.f.: ((4 - 6*x + 3*x^2)*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Franck Maminirina Ramaharo, Nov 25 2018

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

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 49, 57, 66, 75, 85, 95, 106, 118, 130, 143, 156, 170, 185, 200, 216, 232, 249, 267, 285, 304, 323, 343, 364, 385, 407, 429, 452, 476, 500, 525, 550, 576, 603, 630, 658, 686, 715, 745, 775, 806, 837, 869, 902, 935

Offset: 1

Clark Kimberling, Jul 08 2013

See A227347.

			a(1) = floor(3/5) = 0; a(2) = floor(6/5) = 1; a(3) = a(2) + floor(9/5) = 2; a(4) = a(3) + floor(12/5) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A227347, A130520, A011858, A033437.

Cf. A057355 (first differences).

z = 150; r = 3/5; k = 1; a[n_] := Sum[Floor[r*x^k], {x, 1, n}]; t = Table[a[n], {n, 1, z}]

a(n) = (3*n^2-n)\10; \\ Kevin Ryde, Mar 15 2022

a = lambda n: n*(3*n-1)//10 # Gennady Eremin, Mar 20 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).
G.f.: (x*(1 + x^2 + x^3))/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
According to Wolfram Alpha, a(n) = floor(Re(E(n^2|Pi))) where E(x|m) is the incomplete elliptic integral of the second kind. - Kritsada Moomuang, Jan 28 2022
a(n) = a(n-1) + floor(3*n/5), n > 1. - Gennady Eremin, Mar 15 2022
a(n) = floor(n*(3*n-1)/10). - Kevin Ryde, Mar 15 2022

A185170 a(n) = floor( (2n^2 - 6n + 9) / 5).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 9, 13, 17, 23, 29, 37, 45, 53, 63, 73, 85, 97, 109, 123, 137, 153, 169, 185, 203, 221, 241, 261, 281, 303, 325, 349, 373, 397, 423, 449, 477, 505, 533, 563, 593, 625, 657, 689, 723, 757, 793, 829, 865, 903, 941, 981, 1021, 1061, 1103, 1145

Offset: 0

Michael Somos, Dec 26 2012

nonn

Hasselblatt and Propp on page 8 mentions the sequence as degrees of iterates of (w, x, y, z) -> (x, y, z, z*(w*z - x*y) / (w*y - x*x)). That is, if b(0) = w, b(1) = x, b(2) = y, b(3) = z, b(n) = b(n-1) * (b(n-1)*b(n-4) - b(n-2)*b(n-3)) / (b(n-2)*b(n-4) - b(n-3)*b(n-3)), then b(n) is a rational function such that the total degree of the numerator is a(n) and the denominator is a(n)-1. Also b(n) is a Laurent monomial in variables {w, x, y, z, wz-xy, wy-xx, xz-yy}.

A quasipolynomial. - Charles R Greathouse IV, Dec 28 2012

			G.f. = 1 + x + x^2 + x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 13*x^7 + 17*x^8 + 23*x^9 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
B. Hasselblatt and J. Propp, Degree-Growth of Monomial Maps
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 1, -2, 1).

Cf. A130520.

[Floor((2*n^2-6*n+9)/5): n in [0..50]]; // G. C. Greubel, Aug 10 2018

Table[Floor[(2 n^2 - 6 n + 9)/5], {n, 0, 60}] (* or *) LinearRecurrence[ {2,-1,0,0,1,-2,1},{1,1,1,1,3,5,9},60] (* Harvey P. Dale, Dec 28 2012 *)
a[ n_] := Quotient[ 2 n^2 - 6 n + 9, 5]; (* Michael Somos, Apr 25 2015 *)

A185170(n):=floor((2*n^2-6*n+9)/5)$ makelist(A185170(n),n,0,30); /* Martin Ettl, Dec 28 2012 */

PARI
```
{a(n) = (2*n^2 - 6*n + 9) \ 5};
```

G.f.: (1 - x + 2*x^4 - x^5 + 3*x^6) / ((1 - x)^2 * (1 - x^5)).
a(n) = a(3-n) for all n in Z.
Second difference has period 5.
a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=3, a(5)=5, a(6)=9, a(n)=2*a(n-1)- a(n-2)+ a(n-5)-2*a(n-6)+a (n-7). - Harvey P. Dale, Dec 28 2012

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

A008732 Molien series for 3-dimensional group [2,n] = *22n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A011858 a(n) = floor( n*(n-1)/5 ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A218530 Partial sums of floor(n/11).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A062781 Number of arithmetic progressions of four terms and any mean which can be extracted from the set of the first n positive integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A221912 Partial sums of floor(n/12).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A227353 Number of lattice points in the closed region bounded by the graphs of y = 3*x/5, x = n, and y = 0, excluding points on the x-axis.

Views

Author

Keywords

Comments

A185170 a(n) = floor( (2n^2 - 6n + 9) / 5).