A010762 -id:A010762 - cp's OEIS frontend

A004526 Nonnegative integers repeated, floor(n/2).

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36

Offset: 0

N. J. A. Sloane

Number of elements in the set {k: 1 <= 2k <= n}.
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(2).
Dimension of the space of weight 1 modular forms for Gamma_1(n+1).
Number of ways 2^n is expressible as r^2 - s^2 with s > 0. Proof: (r+s) and (r-s) both should be powers of 2, even and distinct hence a(2k) = a(2k-1) = (k-1) etc. - Amarnath Murthy, Sep 20 2002
Lengths of sides of Ulam square spiral; i.e., lengths of runs of equal terms in A063826. - Donald S. McDonald, Jan 09 2003
Number of partitions of n into two parts. A008619 gives partitions of n into at most two parts, so A008619(n) = a(n) + 1 for all n >= 0. Partial sums are A002620 (Quarter-squares). - Rick L. Shepherd, Feb 27 2004
a(n+1) is the number of 1's in the binary expansion of the Jacobsthal number A001045(n). - Paul Barry, Jan 13 2005
Number of partitions of n+1 into two distinct (nonzero) parts. Example: a(8) = 4 because we have [8,1],[7,2],[6,3] and [5,4]. - Emeric Deutsch, Apr 14 2006
Complement of A000035, since A000035(n)+2*a(n) = n. Also equal to the partial sums of A000035. - Hieronymus Fischer, Jun 01 2007
Number of binary bracelets of n beads, two of them 0. For n >= 2, a(n-2) is the number of binary bracelets of n beads, two of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n+1) = (-1)^n det(A). - Milan Janjic, Jan 24 2010
From Clark Kimberling, Mar 10 2011: (Start)
Let RT abbreviate rank transform (A187224). Then
RT(this sequence) = A187484;
RT(this sequence without 1st term) = A026371;
RT(this sequence without 1st 2 terms) = A026367;
RT(this sequence without 1st 3 terms) = A026363. (End)
The diameter (longest path) of the n-cycle. - Cade Herron, Apr 14 2011
For n >= 3, a(n-1) is the number of two-color bracelets of n beads, three of them are black, having a diameter of symmetry. - Vladimir Shevelev, May 03 2011
Pelesko (2004) refers erroneously to this sequence instead of A008619. - M. F. Hasler, Jul 19 2012
Number of degree 2 irreducible characters of the dihedral group of order 2(n+1). - Eric M. Schmidt, Feb 12 2013
For n >= 3 the sequence a(n-1) is the number of non-congruent regions with infinite area in the exterior of a regular n-gon with all diagonals drawn. See A217748. - Martin Renner, Mar 23 2013
a(n) is the number of partitions of 2n into exactly 2 even parts. a(n+1) is the number of partitions of 2n into exactly 2 odd parts. This just rephrases the comment of E. Deutsch above. - Wesley Ivan Hurt, Jun 08 2013
Number of the distinct rectangles and square in a regular n-gon is a(n/2) for even n and n >= 4. For odd n, such number is zero, see illustration in link. - Kival Ngaokrajang, Jun 25 2013
x-coordinate from the image of the point (0,-1) after n reflections across the lines y = n and y = x respectively (alternating so that one reflection is applied on each step): (0,-1) -> (0,1) -> (1,0) -> (1,2) -> (2,1) -> (2,3) -> ... . - Wesley Ivan Hurt, Jul 12 2013
a(n) is the number of partitions of 2n into exactly two distinct odd parts. a(n-1) is the number of partitions of 2n into exactly two distinct even parts, n > 0. - Wesley Ivan Hurt, Jul 21 2013
a(n) is the number of permutations of length n avoiding 213, 231 and 312, or avoiding 213, 312 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Also a(n) is the number of different patterns of 2-color, 2-partition of n. - Ctibor O. Zizka, Nov 19 2014
Minimum in- and out-degree for a directed K_n (see link). - Jon Perry, Nov 22 2014
a(n) is also the independence number of the triangular graph T(n). - Luis Manuel Rivera Martínez, Mar 12 2015
For n >= 3, a(n+4) is the least positive integer m such that every m-element subset of {1,2,...,n} contains distinct i, j, k with i + j = k (equivalently, with i - j = k). - Rick L. Shepherd, Jan 24 2016
More generally, the ordinary generating function for the integers repeated k times is x^k/((1 - x)(1 - x^k)). - Ilya Gutkovskiy, Mar 21 2016
a(n) is the number of numbers of the form F(i)*F(j) between F(n+3) and F(n+4), where 2 < i < j and F = A000045 (Fibonacci numbers). - Clark Kimberling, May 02 2016
The arithmetic function v_2(n,2) as defined in A289187. - Robert Price, Aug 22 2017
a(n) is also the total domination number of the (n-3)-gear graph. - Eric W. Weisstein, Apr 07 2018
Consider the numbers 1, 2, ..., n; a(n) is the largest integer t such that these numbers can be arranged in a row so that all consecutive terms differ by at least t. Example: a(6) = a(7) = 3, because of respectively (4, 1, 5, 2, 6, 3) and (1, 5, 2, 6, 3, 7, 4) (see link BMO - Problem 2). - Bernard Schott, Mar 07 2020
a(n-1) is also the number of integer-sided triangles whose sides a < b < c are in arithmetic progression with a middle side b = n (see A307136). Example, for b = 4, there exists a(3) = 1 such triangle corresponding to Pythagorean triple (3, 4, 5). For the triples, miscellaneous properties and references, see A336750. - Bernard Schott, Oct 15 2020
For n >= 1, a(n-1) is the greatest remainder on division of n by any k in 1..n. - David James Sycamore, Sep 05 2021
Number of incongruent right triangles that can be formed from the vertices of a regular n-gon is given by a(n/2) for n even. For n odd such number is zero. For a regular n-gon, the number of incongruent triangles formed from its vertices is given by A069905(n). The number of incongruent acute triangles is given by A005044(n). The number of incongruent obtuse triangles is given by A008642(n-4) for n > 3 otherwise 0, with offset 0. - Frank M Jackson, Nov 26 2022
The inverse binomial transform is 0, 0, 1, -2, 4, -8, 16, -32, ... (see A122803). - R. J. Mathar, Feb 25 2023

			G.f. = x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...

G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27th Competition.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, P(n,2).
Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1989, page 77 (partitions of n into at most 2 parts).

David Wasserman, Table of n, a(n) for n = 0..1000
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
British Mathematical Olympiad, 2011/2012 - Round 1 - Problem 2.
Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Zachary Hoelscher and Eyvindur Ari Palsson, Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to omega(t), arXiv:2011.14502 [math.NT], 2020.
Kival Ngaokrajang, The distinct rectangles and square in a regular n-gon for n = 4..18.
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.
Jon Perry, Square of a directed graph.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Prime Partition
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for "core" sequences
Index to sequences related to Olympiads
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

a(n+2) = A008619(n). See A008619 for more references.
A001477(n) = a(n+1)+a(n). A000035(n) = a(n+1)-A002456(n).
a(n) = A008284(n, 2), n >= 1.
Zero followed by the partial sums of A000035.
Column 2 of triangle A094953. Second row of A180969.
Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.
Partial sums: A002620. Other related sequences: A010872, A010873, A010874.
Cf. similar sequences of the integers repeated k times: A001477 (k = 1), this sequence (k = 2), A002264 (k = 3), A002265 (k = 4), A002266 (k = 5), A152467 (k = 6), A132270 (k = 7), A132292 (k = 8), A059995 (k = 10).
Cf. A289187, A139756 (binomial transf).
Cf. A307136, A336750.

a004526 = (`div` 2)
a004526_list = concatMap (\x -> [x, x]) [0..]
-- Reinhard Zumkeller, Jul 27 2012

[Floor(n/2): n in [0..100]]; // Vincenzo Librandi, Nov 19 2014

A004526 := n->floor(n/2); seq(floor(i/2),i=0..50);

Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] (* Stefan Steinerberger, Apr 02 2006 *)
f[n_] := If[OddQ[n], (n - 1)/2, n/2]; Array[f, 74, 0] (* Robert G. Wilson v, Apr 20 2012 *)
With[{c=Range[0,40]},Riffle[c,c]] (* Harvey P. Dale, Aug 26 2013 *)
CoefficientList[Series[x^2/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {0, 0, 1}, 75] (* Robert G. Wilson v, Feb 05 2015 *)
Floor[Range[0, 40]/2] (* Eric W. Weisstein, Apr 07 2018 *)

makelist(floor(n/2),n,0,50); /* Martin Ettl, Oct 17 2012 */

a(n)=n\2 /* Jaume Oliver Lafont, Mar 25 2009 */

x='x+O('x^100); concat([0, 0], Vec(x^2/((1+x)*(x-1)^2))) \\ Altug Alkan, Mar 21 2016

def a(n): return n//2
print([a(n) for n in range(74)]) # Michael S. Branicky, Apr 30 2022

def a(n) : return( dimension_cusp_forms( Gamma0(2), 2*n+4) ); # Michael Somos, Jul 03 2014

def a(n) : return( dimension_modular_forms( Gamma1(n+1), 1) ); # Michael Somos, Jul 03 2014

G.f.: x^2/((1+x)*(x-1)^2).
a(n) = floor(n/2).
a(n) = ceiling((n+1)/2). - Eric W. Weisstein, Jan 11 2024
a(n) = 1 + a(n-2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(2*n) = a(2*n+1) = n.
a(n+1) = n - a(n). - Henry Bottomley, Jul 25 2001
For n > 0, a(n) = Sum_{i=1..n} (1/2)/cos(Pi*(2*i-(1-(-1)^n)/2)/(2*n+1)). - Benoit Cloitre, Oct 11 2002
a(n) = (2*n-1)/4 + (-1)^n/4; a(n+1) = Sum_{k=0..n} k*(-1)^(n+k). - Paul Barry, May 20 2003
E.g.f.: ((2*x-1)*exp(x) + exp(-x))/4. - Paul Barry, Sep 03 2003
G.f.: (1/(1-x)) * Sum_{k >= 0} t^2/(1-t^4) where t = x^2^k. - Ralf Stephan, Feb 24 2004
a(n+1) = A000120(A001045(n)). - Paul Barry, Jan 13 2005
a(n) = (n-(1-(-1)^n)/2)/2 = (1/2)*(n-|sin(n*Pi/2)|). Likewise: a(n) = (n-A000035(n))/2. Also: a(n) = Sum_{k=0..n} A000035(k). - Hieronymus Fischer, Jun 01 2007
The expression floor((x^2-1)/(2*x)) (x >= 1) produces this sequence. - Mohammad K. Azarian, Nov 08 2007; corrected by M. F. Hasler, Nov 17 2008
a(n+1) = A002378(n) - A035608(n). - Reinhard Zumkeller, Jan 27 2010
a(n+1) = A002620(n+1) - A002620(n) = floor((n+1)/2)*ceiling((n+1)/2) - floor(n^2/4). - Jonathan Vos Post, May 20 2010
For n >= 2, a(n) = floor(log_2(2^a(n-1) + 2^a(n-2))). - Vladimir Shevelev, Jun 22 2010
a(n) = A180969(2,n). - Adriano Caroli, Nov 24 2010
A001057(n-1) = (-1)^n*a(n), n > 0. - M. F. Hasler, Jul 19 2012
a(n) = A008615(n) + A002264(n). - Reinhard Zumkeller, Apr 28 2014
Euler transform of length 2 sequence [1, 1]. - Michael Somos, Jul 03 2014

Partially edited by Joerg Arndt, Mar 11 2010, and M. F. Hasler, Jul 19 2012

A002264 Nonnegative integers repeated 3 times.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25

Offset: 0

N. J. A. Sloane

Complement of A010872, since A010872(n) + 3*a(n) = n. - Hieronymus Fischer, Jun 01 2007
Chvátal proved that, given an arbitrary n-gon, there exist a(n) points such that all points in the interior are visible from at least one of those points; further, for all n >= 3, there exists an n-gon which cannot be covered in this fashion with fewer than a(n) points. This is known as the "art gallery problem". - Charles R Greathouse IV, Aug 29 2012
The inverse binomial transform is 0, 0, 0, 1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729,.. (see A000748). - R. J. Mathar, Feb 25 2023

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Václav Chvátal, A combinatorial theorem in plane geometry, Journal of Combinatorial Theory, Series B 18 (1975), pp. 39-41, doi:10.1016/0095-8956(75)90061-1.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001477, A002265, A002266, A004526, A008615, A008620, A010761, A010762, A010872, A010873, A010874, A022003, A110532, A110533, A137221 (binomial transform).
Partial sums give A130518.
Cf. A004523 interlaced with A004396.
Apart from the zeros, this is column 3 of A235791.

a002264 n = a002264_list !! n
a002264_list = 0 : 0 : 0 : map (+ 1) a002264_list
-- Reinhard Zumkeller, Nov 06 2012, Apr 16 2012

[Floor(n/3): n in [0..100]]; // Vincenzo Librandi, Apr 29 2015

&cat [[n,n,n]: n in [0..30]]; // Bruno Berselli, Apr 29 2015

seq(i$3,i=0..100); # Robert Israel, Aug 04 2014

Flatten[Table[{n, n, n}, {n, 0, 25}]] (* Harvey P. Dale, Jun 09 2013 *)
Floor[Range[0, 20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Floor[n/3], {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - Cos[2 (n - 2) Pi/3] + Sin[2 (n - 2) Pi/3]/Sqrt[3] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n - 2, -1/2] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 0, 0, 1}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[x^3/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

a(n)=n\3  /* Jaume Oliver Lafont, Mar 25 2009 */

v=[0,0];for(n=2,50,v=concat(v,n-2-v[#v]-v[#v-1]));v \\ Derek Orr, Apr 28 2015

[floor(n/3) for n in range(0,79)] # Zerinvary Lajos, Dec 01 2009

a(n) = floor(n/3).
a(n) = (3*n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. - Hieronymus Fischer, Sep 18 2007
a(n) = (n - A010872(n))/3. - Hieronymus Fischer, Sep 18 2007
Complex representation: a(n) = (n - (1 - r^n)*(1 + r^n/(1 - r)))/3 where r = exp(2*Pi/3*i) = (-1 + sqrt(3)*i)/2 and i = sqrt(-1). - Hieronymus Fischer, Sep 18 2007; - corrected by Guenther Schrack, Sep 26 2019
a(n) = Sum_{k=0..n-1} A022003(k). - Hieronymus Fischer, Sep 18 2007
G.f.: x^3/((1-x)*(1-x^3)). - Hieronymus Fischer, Sep 18 2007
a(n) = (n - 1 + 2*sin(4*(n+2)*Pi/3)/sqrt(3))/3. - Jaume Oliver Lafont, Dec 05 2008
For n >= 3, a(n) = floor(log_3(3^a(n-1) + 3^a(n-2) + 3^a(n-3))). - Vladimir Shevelev, Jun 22 2010
a(n) = (n - 3 + A010872(n-1) + A010872(n-2))/3 using Zumkeller's 2008 formula in A010872. - Adriano Caroli, Nov 23 2010
a(n) = A004526(n) - A008615(n). - Reinhard Zumkeller, Apr 28 2014
a(2*n) = A004523(n) and a(2*n+1) = A004396(n). - L. Edson Jeffery, Jul 30 2014
a(n) = n - 2 - a(n-1) - a(n-2) for n > 1 with a(0) = a(1) = 0. - Derek Orr, Apr 28 2015
From Wesley Ivan Hurt, May 27 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4), n > 4.
a(n) = (n - 1 + 0^((-1)^(n/3) - (-1)^n) - 0^((-1)^(n/3)*(-1)^(1/3) + (-1)^n))/3. (End)
a(n) = (3*n - 3 + r^n*(1 - r) + r^(2*n)*(r + 2))/9 where r = (-1 + sqrt(-3))/2. - Guenther Schrack, Sep 26 2019
E.g.f.: exp(x)*(x - 1)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022

A002266 Integers repeated 5 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

N. J. A. Sloane

For n > 3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000032 (see example). E.g., the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ...] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre, Jan 08 2006
Complement of A010874, since A010874(n) + 5*a(n) = n. - Hieronymus Fischer, Jun 01 2007
From Paul Curtz, May 13 2020: (Start)
Main N-S vertical of the pentagonal spiral built with this sequence is A001105:
                              21
                          20  15  15
                      20  14  10  10  15
                  20  14   9   6   6  10  15
              20  14   9   5   3   3   6  10  15
          20  14   9   5   2   1   1   3   6  10  16
      19  14   9   5   2   0   0   0   1   3   6  11  16
        19   13   9   5  2   0   0    1   3   7 11  16
          19  13    8  5   2   2   1   4   7  11  16
            19   13   8  4   4   4   4   7  11  16
              19  13   8   8   8   7   7  11  17
                 18  13  12 12  12  12  12  17
                   18  18 18  18  17  17  17
The main S-N vertical and the next one are A000217. (End)

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

a(n) = A010766(n, 5).
Cf. A002162, A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A010872, A010873, A010874.
Partial sums: A130520.

a002266 = (`div` 5)
a002266_list = [0,0,0,0,0] ++ map (+ 1) a002266_list
-- Reinhard Zumkeller, Nov 27 2012

A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/5], {n,0,20}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Jun 17 2022 *)

a(n)=n\5 \\ Charles R Greathouse IV, Dec 10 2013

def A002266(n): return n//5 # Chai Wah Wu, Nov 08 2022

[floor(n/5) - 1 for n in range(5,88)] # Zerinvary Lajos, Dec 01 2009

a(n) = floor(n/5), n >= 0.
G.f.: x^5/((1-x)(1-x^5)).
a(n) = (n - A010874(n))/5. - Hieronymus Fischer, May 29 2007
For n >= 5, a(n) = floor(log_5(5^a(n-1) + 5^a(n-2) + 5^a(n-3) + 5^a(n-4) + 5^a(n-5))). - Vladimir Shevelev, Jun 22 2010
Sum_{n>=5} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Incorrect formula removed by Ridouane Oudra, Oct 16 2021

A010761 a(n) = floor(n/2) + floor(n/3).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 6, 7, 8, 8, 10, 10, 11, 12, 13, 13, 15, 15, 16, 17, 18, 18, 20, 20, 21, 22, 23, 23, 25, 25, 26, 27, 28, 28, 30, 30, 31, 32, 33, 33, 35, 35, 36, 37, 38, 38, 40, 40, 41, 42, 43, 43, 45, 45, 46, 47, 48, 48, 50, 50, 51, 52, 53, 53, 55, 55, 56, 57, 58, 58, 60, 60

Offset: 1

Simon Plouffe

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A001622, A010762, A110532.

Cf. A002264, A004526.

[Floor (n/2)+Floor(n/3): n in [0..100]]; // Vincenzo Librandi, Feb 22 2014

Maple
```
seq(floor(n/2) + floor(n/3), n=1..64);
```

Table[Floor[n/2] + Floor[n/3], {n, 100}] (* Wesley Ivan Hurt, Feb 21 2014 *)
LinearRecurrence[{0,1,1,0,-1},{0,1,2,3,3},80] (* Harvey P. Dale, May 05 2018 *)

PARI
```
a(n)=n\2+n\3
```

a(n) = A004526(n) + A002264(n). - Reinhard Zumkeller, Jul 25 2005
From Michael Somos, Aug 07 2005: (Start)
G.f.: x^2*(1+2*x+2*x^2)/((1-x^2)*(1-x^3)).
a(-n) = -2-a(n-1). (End)
a(n) = a(n-2)+a(n-3)-a(n-5). - Wesley Ivan Hurt, Jan 27 2022
Sum_{n>=2} (-1)^n/a(n) = sqrt(2*(1-1/sqrt(5)))*Pi/10 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 30 2023

A110533 a(n) = floor(n/2) * floor(n/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 3, 4, 4, 10, 10, 12, 12, 14, 21, 24, 24, 27, 27, 40, 40, 44, 44, 48, 60, 65, 65, 70, 70, 90, 90, 96, 96, 102, 119, 126, 126, 133, 133, 160, 160, 168, 168, 176, 198, 207, 207, 216, 216, 250, 250, 260, 260, 270, 297, 308, 308, 319, 319, 360, 360, 372

Offset: 0

Reinhard Zumkeller, Jul 25 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A001622, A002266, A004526, A010762, A110532.

Table[Floor[n/2]*Floor[n/5], {n, 0, 50}] (* G. C. Greubel, Aug 30 2017 *)

for(n=0,50, print1(floor(n/2)*floor(n/5), ", ")) \\ G. C. Greubel, Aug 30 2017

a(n) = A004526(n)*A002266(n).
From R. J. Mathar, Feb 20 2011: (Start)
a(n) = +a(n-2) +a(n-5) -a(n-7) +a(n-10) -a(n-12) -a(n-15) +a(n-17).
G.f.: -x^5*(2+3*x+x^2+x^3+x^4+4*x^5+3*x^6+x^7+x^8+x^9+x^10+x^11) / ( (x^4-x^3+x^2-x+1) *(1+x)^2 *(x^4+x^3+x^2+x+1)^2 *(x-1)^3 ). (End)
Sum_{n>=5} (-1)^(n+1)/a(n) = sqrt(5*(5-2*sqrt(5)))*Pi/8 - (5/8)*(1 + sqrt(5)*log(phi)) + (25/16)*log(5) - 2*log(2), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 30 2023

A242669 a(n) = n*floor(n/3).

Original entry on oeis.org

0, 0, 0, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 108, 114, 120, 147, 154, 161, 192, 200, 208, 243, 252, 261, 300, 310, 320, 363, 374, 385, 432, 444, 456, 507, 520, 533, 588, 602, 616, 675, 690, 705, 768, 784, 800, 867, 884, 901, 972, 990

Offset: 0

Bruno Berselli, Jul 01 2014

For n = 0, 1, 2, 4, 8, 49, 98, 676, 1352, 9409, 18818, 131044, 262088, 1825201, 3650402, ... a(n) is a square.

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

Cf. A000290 (n^2), A010762 (floor(n/2)*floor(n/3)), A093353 (n*floor(n/2)), A213033 (n*floor(n/2)*floor(n/3)), A233035 (n*floor(n/4)).
Cf. A033428, A045944, A049451.
Cf. A002264 (floor(n/3)).

Magma
```
[n*Floor(n/3): n in [0..60]];
```
Mathematica
```
Table[n Floor[n/3], {n, 0, 60}]
```

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

Sage
```
[n*floor(n/3) for n in (0..60)];
```

G.f.: x^3*(3 + x + x^2 + x^3)/((1 - x)^3*(1 + x + x^2)^2).
a(3m) = A033428(m), a(3m+1) = A049451(m), a(3m+2) = A045944(m).
Sum_{n>=3} (-1)^(n+1)/a(n) = 9/4 + Pi^2/36 - Pi/(2*sqrt(3)) - 2*log(2). - Amiram Eldar, Mar 30 2023

A268251 Expansion of x(1 + 2x - 96x^2 - 148x^3 + 45x^4 + 50x^5 + 2x^6)/(1 - 99x^2 + 99*x^4 - x^6).

Original entry on oeis.org

0, 1, 2, 3, 50, 243, 4802, 23763, 470450, 2328483, 46099202, 228167523, 4517251250, 22358088723, 442644523202, 2190864527283, 43374646022450, 214682365584963, 4250272665676802, 21036680962799043, 416483346590304050, 2061380051988721203, 40811117693184120002

Offset: 0

Bruno Berselli, Jan 29 2016

Conjecture: The sequence lists all nonnegative m, in increasing order, such that floor(m/2)*floor(m/3) is a square.

This conjecture has been proved by Robert Israel (see paper in Links section).

Robert Israel, Table of n, a(n) for n = 0..1000
Robert Israel, Proof of a conjecture.
Index entries for linear recurrences with constant coefficients, signature (0,99,0,-99,0,1).

Cf. A010762.

Cf. A268742: m for which floor(m/2) + floor(m/3) is a square.

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6)));

gf:= x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6):
S:= series(gf,x,51):
seq(coeff(S,x,j),j=0..50); # Robert Israel, Feb 11 2016

CoefficientList[x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6) + O[x]^30, x] (* Jean-François Alcover, Feb 12 2016 *)

makelist(coeff(taylor(x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6), x, 0, n), x, n), n, 0, 30);

concat(0, Vec((1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6) + O(x^30)))

gf = x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6); taylor(gf, x, 0, 30).list()

G.f.: x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/((1 - x)*(1 + x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)). (For e.g.f see Israel's paper.)
a(n) = 99*a(n-2) - 99*a(n-4) + a(n-6) for n>7.
a(n) = -a(n-1) + 98*a(n-2) + 98*a(n-3) - a(n-4) - a(n-5) - 144 for n>6.

cp's OEIS Frontend

A004526 Nonnegative integers repeated, floor(n/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Sage

Formula

Extensions

A002264 Nonnegative integers repeated 3 times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A002266 Integers repeated 5 times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

A010761 a(n) = floor(n/2) + floor(n/3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A110533 a(n) = floor(n/2) * floor(n/5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242669 a(n) = n*floor(n/3).

Views

A268251 Expansion of x(1 + 2x - 96x^2 - 148x^3 + 45x^4 + 50x^5 + 2x^6)/(1 - 99x^2 + 99*x^4 - x^6).