A004524 -id:A004524 - cp's OEIS frontend

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590

Offset: 0

N. J. A. Sloane

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
Number of triangular partitions (see Almkvist).
Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
  .....{*}..................{*}*.*{*}{*}
  .....*.*....................*.*.*.{*}
  ....*.*.*....---------\......*.*.*
  ..{*}*.*.*...---------/.......*.*
  {*}{*}*.*{*}..................{*}
  - Lekraj Beedassy, Oct 13 2003
Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004
Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004
Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005
Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008
In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008
Column sums of:
1 2 3 4 5 6 7  8  9.....
      1 2 3 4  5  6.....
            1  2  3.....
........................
----------------------
1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010
a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011
a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012
a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013
Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
1  2  2  3  4  4  5  6  6...
      1  2  2  3  4  4  5...
            1  2  2  3  4...
                  1  2  2...
                        1...
------------------------------
1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014
a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020
Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...
1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
[n] a(n)
--------
[1] 1
[2] 2
[3] 3
[4] 1 + 4
[5] 2 + 5
[6] 3 + 6
[7] 1 + 4 + 7
[8] 2 + 5 + 8
[9] 3 + 6 + 9
a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - _Bruno Berselli_, Aug 29 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
Hansraj Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
Richard K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
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=3]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
Michael Somos, Somos Polynomials.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for sequences related to Lyndon words.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000031, A001037, A008748, A051168.
Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
Cf. A058937.
A column of triangle A011847.
Cf. A000217, A130205.
Cf. A258708.
A001399 counts 3-part partitions, ranked by A014612.
A337483 counts either weakly increasing or weakly decreasing triples.
A337484 counts neither strictly increasing nor strictly decreasing triples.
A014311 ranks 3-part compositions, with strict case A337453.
Cf. A007052, A007304, A011782, A069905, A156040, A218004.

a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012

[ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

A001840 := n->floor((n+1)*(n+2)/6);
A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
seq(floor(binomial(n-1,2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009
A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n,0,100}]
f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)
a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]]; (* Michael Somos, Jul 11 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* Harvey P. Dale, Jul 25 2011 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */

[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009

a(n) = (A000217(n+1) - A022003(n-1))/3;
a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
Euler transform of length 3 sequence [2, 0, 1].
a(3*k-1) = k*(3*k + 1)/2;
a(3*k)   = 3*k*(k + 1)/2;
a(3*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
a(n+1) = a(n) + A008620(n) = A002264(n+3). - Reinhard Zumkeller, Aug 01 2002
From Michael Somos, Feb 11 2004: (Start)
G.f.: x / ((1-x)^2 * (1-x^3)).
a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
a(-3-n) = a(n). (End)
a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - Paul Barry, Jul 14 2004
a(n) = binomial(n+3, 3)/(n+3) + cos(2*Pi*(n-1)/3)/9 + sqrt(3)sin(2*Pi*(n-1)/3)/9 - 1/9. - Paul Barry, Jan 01 2005
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} k*(cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3).
a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011
a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012
G.f.: x*G(0), where G(k) = 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013
Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013
a(n) = Sum_{i=0..n} floor((i+2)/3). - Bruno Berselli, Aug 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 05 2023

A032528 Concentric hexagonal numbers: floor(3*n^2/2).

Original entry on oeis.org

0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601, 3750

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Aug 20 2011: (Start)
Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.
A033581 and A003154 interleaved.
Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example).  (End)
Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - Omar E. Pol, Sep 08 2011

			From _Omar E. Pol_, Aug 20 2011: (Start)
Using the numbers A008458 we can write:
  0, 1, 6, 12, 18, 24, 30, 36, 42,  48,  54, ...
  0, 0, 0,  1,  6, 12, 18, 24, 30,  36,  42, ...
  0, 0, 0,  0,  0,  1,  6, 12, 18,  24,  30, ...
  0, 0, 0,  0,  0,  0,  0,  1,  6,  12,  18, ...
  0, 0, 0,  0,  0,  0,  0,  0,  0,   1,   6, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ...
...
Illustration of initial terms as concentric hexagons:
.
.                                         o o o o o
.                         o o o o        o         o
.             o o o      o       o      o   o o o   o
.     o o    o     o    o   o o   o    o   o     o   o
. o  o   o  o   o   o  o   o   o   o  o   o   o   o   o
.     o o    o     o    o   o o   o    o   o     o   o
.             o o o      o       o      o   o o o   o
.                         o o o o        o         o
.                                         o o o o o
.
. 1    6        13           24               37
.
(End)

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

Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040.
Cf. A004524, A004525, A033436, A212959, A238410, A227017, A282513.
Cf. A033581, A003154, A011934, A184533.

a032528 n = a032528_list !! n
a032528_list = scanl (+) 0 a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[Floor(3*n^2/2): n in [0..50]]; // Vincenzo Librandi, Aug 21 2011

f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* Clark Kimberling, Apr 20 2012 *)

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

From Joerg Arndt, Aug 22 2011: (Start)
G.f.: (x+4*x^2+x^3)/(1-2*x+2*x^3-x^4) = x*(1+4*x+x^2)/((1+x)*(1-x)^3).
a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End)
a(n) = (6*n^2+(-1)^n-1)/4. - Bruno Berselli, Aug 22 2011
a(n) = A184533(n), n >= 2. - Clark Kimberling, Apr 20 2012
First differences of A011934: a(n) = A011934(n) - A011934(n-1) for n>0. - Franz Vrabec, Feb 17 2013
From Paul Curtz, Mar 31 2019: (Start)
a(-n) = a(n).
a(n) = a(n-2) + 6*(n-1) for n > 1.
a(2*n) = A033581(n).
a(2*n+1) = A003154(n+1). (End)
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Aug 19 2022
Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Jan 16 2023

New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011

A011848 a(n) = floor(binomial(n, 2)/2).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588, 612, 637, 663, 689, 715, 742, 770, 798

Offset: 0

N. J. A. Sloane, Dec 11 1996

Column sums of an array of the odd numbers repeatedly shifted 4 places to the right:
1 3 5 7  9 11 13 15 17...
         1  3  5  7  9...
                     1...
.........................
-------------------------
1 3 5 7 10 14 18 22 27...
Floor of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Beginning with a(4)=3, the sequence might be called the "off-axis" Ulam-Spiral numbers because they are the numbers in ascending order on the horizontal and vertical spokes (heading outward) starting with the first turning points on the spiral (i.e., 3, 5, 7 and 10). That is, starting with: 3 (upward); 5 (leftward); 7 (downward) and 10 (rightward). These are A033991 (starting at a(1)), A007742 (starting at a(1)), A033954 (starting at a(1)) and A001107 (starting at a(2)), respectively. These quadri-sections are summarized in the formulas of Sep 26 2015. - Bob Selcoe, Oct 05 2015
Conjecture: For n = 2, a(n) is the greatest k such that A123663(k) < A000217(n - 2). - Peter Kagey, Nov 18 2016
a(n) is also the matching number of the n-triangular graph, (n-1)-triangular honeycomb queen graph, (n-1)-triangular honeycomb bishop graphs, and (for n > 7) (n-1)-triangular honeycomb obtuse knight graphs. - Eric W. Weisstein, Jun 02 2017 and Apr 03 2018
After 0, 0, 0, add 1, then add 2 three times, then add 3, then add 4 three times, then add 5, etc.; i.e., first differences are A004524 = (0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, ...). - M. F. Hasler, May 09 2018
Let s(0) = s(1) = 1, s(-1) = s(2) = x, and s(n+2)*s(n-2) = s(n+1)*s(n-1) + s(n)^2 for all n in Z. Then s(n) = p(n) / x^e(n) is a Laurent polynomial in x with p(n) a polynomial with nonnegative integer coefficients of degree a(n) for all n in Z. If x = 1, then s(n) = p(n) = A006720(n+1). - Michael Somos, Mar 22 2023

			G.f. = x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 14*x^8 + 18*x^9 + 22*x^10 + ...
p(0) = p(1) = 1, p(2) = 1 + x, p(3) = 1 + x + x^3, p(4) = 1 + 2*x + 2*x^2 + x^3 + x^5. - _Michael Somos_, Mar 22 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

A column of triangle A011857.
First differences are in A004524.
Cf. A007318, A033991, A007742, A033954, A001107, A006720, A035608 (bisection), A156859 (bisection).

List([0..60],n->Int(Binomial(n,2)/2)); # Muniru A Asiru, Apr 05 2018

a011848 n = if n < 2 then 0 else flip div 2 $ a007318 n 2
-- Reinhard Zumkeller, Mar 04 2015

[ Floor(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

seq(floor(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Floor[n (n - 1)/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
CoefficientList[Series[x^3/((1 + x^2) (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 21 2013 *)
LinearRecurrence[{3, -4, 4, -4, 1}, {0, 0, 1, 3, 5}, {0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[Floor[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Table[1/4 (-1 + (-1 + n) n + Cos[n Pi/2] + Sin[n Pi/2]), {n, 0, 20}] (* Eric W. Weisstein, Jun 02 2017 *)
Floor[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Apr 03 2018 *)

PARI
```
a(n) = binomial(n, 2)\2;
```

vector(100, n, n--; floor(n*(n-1)/4)) \\ Altug Alkan, Sep 30 2015

def a(n): return n*(n-1)//4 # Christoph B. Kassir, Oct 07 2022

[floor(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

G.f.: x^3*(1-x^2)/((1-x)^3*(1-x^4)).
G.f.: x^3/((1+x^2)*(1-x)^3). - Jon Perry, Mar 31 2004
a(n) = +3*a(n-1) -4*a(n-2) +4*a(n-3) -3*a(n-4) +a(n-5). - R. J. Mathar, Apr 15 2010
a(n) = floor((n/(1+e^(1/n)))^2). - Richard R. Forberg, Jun 19 2013
a(n) = floor(n*(n-1)/4). - T. D. Noe, Jun 20 2013
a(n) = (1/4) * ( n^2 - n - 1 + (-1)^floor(n/2) ). - Ralf Stephan, Aug 11 2013
a(n) = A054925(n) - A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
a(4*n) = A033991(n). a(4*n+1) = A007742(n). a(4*n+2) = A033954(n). a(4*n+3) = A001107(n+1). - Bob Selcoe, Sep 26 2015
E.g.f.: (sin(x) + cos(x) + (x^2 - 1)*exp(x))/4. - Ilya Gutkovskiy, Nov 18 2016
A054925(n) = a(-n). A035608(n) = a(2*n+1). Wesley Ivan Hurt, Jun 09 2014
A156859(n) = a(2*n+2). - Michael Somos, Nov 18 2016
Euler transform of length 4 sequence [ 3, -1, 0, 1]. - Michael Somos, Nov 18 2016
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=3} 1/a(n) = 40/9 - 2*Pi/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = 32/9 - 4*log(2). (End)
0 = a(n+2)*(a(n)*(a(n) -6*a(n+1) +4*a(n+2)) +a(n+1)*(8*a(n+1) -10*a(n+2)) + 3*a(n+2)^2) +a(n+3)*(a(n)*(+a(n) -2*a(n+1)) +a(n+2)*(2*a(n+1) -a(n+2))) for all n in Z. - Michael Somos, Mar 22 2023
2*a(n) + 2*a(n-2) = (n-1)*(n-2). - R. J. Mathar, Feb 12 2024

A004525 One even followed by three odd.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 27, 28, 29, 29, 29, 30, 31, 31, 31, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 37, 37

Offset: 0

N. J. A. Sloane

a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_6 (binary tetrahedral group). - Paul Boddington, Oct 23 2003
(1 + x + x^2 + x^3 + x^4 + x^5) / ( (1-x^3)*(1- x^4)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(GL_2(F_3)). - N. J. A. Sloane, Jun 12 2004
The Fi1 and Fi2 sums, see A180662 for the definition of these sums, of triangle A101950 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 06 2011
Also the domination number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
Also the domination number of the (n-1)-Moebius laddder. - Eric W. Weisstein, Jun 30 2017
Also the rook domination number of the hexagonal hexagon board B_n [Harborth and Nienborg] - N. J. A. Sloane, Aug 31 2021
Two players play a game, the object of which is to determine a score. Player 1 prefers larger scores, while player 2 prefers smaller scores. The game begins with a set of potential scores {1,2,3, ... n}. Player 1 divides this set into two nonempty sets, one of which player 2 chooses. Player 2 the divides their chosen set into two nonempty sets, one of which player 1 chooses, and so on, until the final score is arrived at. a(n+1) is the final score when both players play optimally. - Thomas Anton, Jul 14 2023

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

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 247.
Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Heiko Harborth and Hauke Nienborg, Rook domination on hexagonal hexagon boards, INTEGERS 21A (2021), #A14.
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, Moebius Ladder
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A002620, A004396, A004524.

a004525 n = a004525_list !! n
a004525_list = 0 : 1 : 1 : zipWith3 (\x y z -> x - y + z + 1)
               a004525_list (tail a004525_list) (drop 2 a004525_list)
-- Reinhard Zumkeller, Jul 14 2012

[Floor(n/4) + Ceiling(n/4): n in [0..70]]; // Vincenzo Librandi, Aug 07 2011

A004525 := proc(n): floor(n/4) + ceil(n/4) end: seq(A004525(n), n=0..75); # Johannes W. Meijer, Aug 06 2011

Table[Floor[n/4] + Ceiling[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 22 2013 *)
Table[(n + Sin[n Pi/2])/2, {n, 0, 30}] (* Eric W. Weisstein, Jun 30 2017 *)
LinearRecurrence[{2, -2, 2, -1}, {1, 1, 1, 2}, {0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
Table[{n - 1, n, n, n}, {n, 1, 41, 2}] // Flatten (* Harvey P. Dale, Oct 18 2019 *)

makelist((1/4)*(2*n-(1-(-1)^n)*(-1)^(n*(n+1)/2)), n, 0, 75); /* Bruno Berselli, Mar 13 2012 */

{a(n) = n\4 + (n+3)\4}; /* Michael Somos, Jul 19 2003 */

def A004525(n): return ((n>>1)&-2)+bool(n&3) # Chai Wah Wu, Jan 27 2023

a(n) = a(n-1) - a(n-2) + a(n-3) + 1 = n - A004524(n+1). - Henry Bottomley, Mar 08 2000
G.f.: x*(1-x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^6)/((1-x)*(1-x^3)*(1-x^4)). - Michael Somos, Jul 19 2003
a(n) = -a(-n) for all n in Z. - Michael Somos, Jul 19 2003
a(n) = floor(n/4) + ceiling(n/4). See also A004396, one even followed by two odd and A002620, quarter-squares: floor(n/2)*ceiling(n/2). - Jonathan Vos Post, Mar 19 2006
a(n) = Sum_{k=0..n-1} (1 + (-1)^binomial(k+1, 2))/2. - Paul Barry, Mar 31 2008
E.g.f: A(x) = (x*exp(x) + sin(x))/2. - Vladimir Kruchinin, Feb 20 2011
a(n) = (1/4)*(2*n - (1 - (-1)^n)*(-1)^(n*(n+1)/2)). - Bruno Berselli, Mar 13 2012
a(n) = (n - floor(cos(Pi*(n+1)/2)))/2. - Wesley Ivan Hurt, Oct 22 2013
Euler transform of length 6 sequence [1, 0, 1, 1, 0, -1]. - Michael Somos, Apr 03 2017
a(n) = (n + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Oct 02 2017
a(n) = n-1-a(n-2) for n >= 2. - Kritsada Moomuang, Oct 29 2019

A008621 Expansion of 1/((1-x)*(1-x^4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: 1/((1-x^2)*(1-x^8)) is the Molien series for the real 2-dimensional Clifford group (a dihedral group of order 16) of genus 1.
Thickness of the hypercube graph Q_n. - Eric W. Weisstein, Sep 09 2008
Count of odd numbers between consecutive quarter-squares, A002620. Oppermann's conjecture states that for each count there will be at least one prime. - Fred Daniel Kline, Sep 10 2011
Number of partitions into parts 1 and 4. - Joerg Arndt, Jun 01 2013
a(n-1) is the minimum independence number over all planar graphs with n vertices. The bound follows from the Four Color Theorem. It is attained by a union of 4-cliques. Other extremal graphs are examined in the Bickle link. - Allan Bickle, Feb 04 2022

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.

T. D. Noe, Table of n, a(n) for n = 0..1000
Allan Bickle, Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 211
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Eric Weisstein's World of Mathematics, Graph Thickness
Wikipedia, Oppermann's conjecture
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265 (equals this - 1).

Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880, A008620.

Table[Floor[n/4]+1, {n, 0, 80}] (* Stefan Steinerberger, Apr 03 2006 *)
CoefficientList[Series[1/((1-x)(1-x^4)),{x,0,80}],x] (* Harvey P. Dale, Feb 19 2012 *)
Flatten[ Table[ PadRight[{},4,n],{n,19}]] (* Harvey P. Dale, Feb 19 2012 *)

a(n)=n\4+1 \\ Charles R Greathouse IV, Feb 06 2017

[n//4+1 for n in range(85)] # Gennady Eremin, Mar 01 2022

a(n) = floor(n/4) + 1.
a(n) = A010766(n+4, 4).
Also, a(n) = ceiling((n+1)/4), n >= 0. - Mohammad K. Azarian, May 22 2007
a(n) = Sum_{i=0..n} A121262(i) = n/4 + 5/8 + (-1)^n/8 + A057077(n)/4. - R. J. Mathar, Mar 14 2011
a(x,y) := floor(x/2) + floor(y/2) - x where x = A002620(n) and y = A002620(n+1), n > 2. - Fred Daniel Kline, Sep 10 2011
a(n) = a(n-1) + a(n-4) - a(n-5); a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=2. - Harvey P. Dale, Feb 19 2012
From R. J. Mathar, Jun 04 2021: (Start)
G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).
a(n) + a(n-1) = A004524(n+3).
a(n) + a(n-2) = A008619(n). (End)
a(n) = A002265(n) + 1. - M. F. Hasler, Oct 17 2022

More terms from Stefan Steinerberger, Apr 03 2006

A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

Original entry on oeis.org

0, 0, 2, 4, 4, 4, 6, 8, 8, 8, 10, 12, 12, 12, 14, 16, 16, 16, 18, 20, 20, 20, 22, 24, 24, 24, 26, 28, 28, 28, 30, 32, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 44, 44, 44, 46, 48, 48, 48, 50, 52, 52, 52, 54, 56, 56, 56, 58, 60, 60, 60, 62, 64, 64, 64, 66, 68, 68

Offset: 0

Vaclav Kotesovec, Dec 03 2011

For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.
Compare this with the binary triangle construction of A240828.
Minimal number of straight segments in a rook circuit of an (n-1) X n board (see example). - Ruediger Jehn, Feb 26 2021

			G.f. = 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + 8*x^9 + ...
From _Ruediger Jehn_, Feb 26 2021: (Start)
a(5) = 4:
   +----+----+----+----+----+
   |  __|____|_   |   _|__  |
   | /  |    | \  |  / |  \ |
   +----+----+----+----+----+
   | \__|__  | |  |  | |  | |
   |    |  \ | \__|__/ |  | |
   +----+----+----+----+----+
   |  __|__/ |  __|__  |  | |
   | /  |    | /  |  \ |  | |
   +----+----+----+----+----+
   | \  |    | |  |  | |  | |
   |  \_|____|_/  |  \_|__/ |
   +----+----+----+----+----+
There are at least 4 squares on the 4 X 5 board with straight lines (here in squares a_12, a_25, a_35 and a_42).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Rüdiger Jehn, Properties of Hamiltonian Circuits in Rectangular Grids, arXiv:2103.15778 [math.GM], 2021.
Craig Knecht, Row sums of superimposed and added binary filled triangles.
Vaclav Kotesovec, Non-attacking chess pieces

Cf. A004524, A085801, A189889, A190394.

a201629 = (* 2) . a004524 . (+ 1) -- Reinhard Zumkeller, Aug 05 2014

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^2/((1-x)^2*(1+x^2)))); // G. C. Greubel, Aug 13 2018

seq(n-sin(Pi*n/2), n=0..30); # Robert Israel, Jul 14 2015

Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* Alonso del Arte, Jan 27 2012 *)
CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 06 2014 *)
a[ n_] := n - KroneckerSymbol[ -4, n]; (* Michael Somos, Jul 18 2015 *)

a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ Charles R Greathouse IV, Jan 27 2012

{a(n) = n - kronecker( -4, n)}; /* Michael Somos, Jul 18 2015 */

a(n) = n - sin(n*Pi/2).
G.f.: 2*x^2/((1-x)^2*(1+x^2)).
a(n) = 2*A004524(n+1). - R. J. Mathar, Feb 02 2012
a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. - Bruno Berselli, Aug 06 2014
E.g.f.: x*exp(x) - sin(x). - G. C. Greubel, Aug 13 2018

Formula corrected by Robert Israel, Jul 14 2015

A122196 Fractal sequence: count down by 2's from successive integers.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2, 13, 11, 9, 7, 5, 3, 1, 14, 12, 10, 8, 6, 4, 2, 15, 13, 11, 9, 7, 5, 3, 1, 16, 14, 12, 10, 8, 6, 4, 2, 17, 15, 13, 11, 9, 7, 5, 3, 1, 18, 16, 14, 12, 10, 8, 6, 4, 2, 19, 17

Offset: 1

Franklin T. Adams-Watters, Aug 25 2006

First differences of A076644. Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, original sequence plus 1. 1's occur at square indices. New values occur at indices m^2+1 and m^2+m+1.
Ordinal transform of A122197.
Row sums give A002620. - Gary W. Adamson, Nov 29 2008
From Gary W. Adamson, Dec 05 2009: (Start)
A122196 considered as an infinite lower triangular matrix * [1,2,3,...] =
A006918 starting (1, 2, 5, 8, 14, 20, 30, 40, ...).
Let A122196 = an infinite lower triangular matrix M; then lim_{n->infinity} M^n = A171238, a left-shifted vector considered as a matrix. (End)
A122196 is the fractal sequence associated with the dispersion A082156; that is, A122196(n) is the number of the row of A082156 that contains n. - Clark Kimberling, Aug 12 2011
From Johannes W. Meijer, Sep 09 2013: (Start)
The alternating row sums lead to A004524(n+2).
The antidiagonal sums equal A001840(n). (End)

			The first few rows of the sequence a(n) as a triangle T(n, k):
n/k  1   2   3
1    1
2    2
3    3,  1
4    4,  2
5    5,  3,  1
6    6,  4,  2

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A076644, A122197, A000290, A033638, A002620, A006918, A171238, A082156, A000267.

a122196 n = a122196_list !! (n-1)
a122196_list = concatMap (\x -> enumFromThenTo x (x - 2) 1) [1..]
-- Reinhard Zumkeller, Jul 19 2012

From Johannes W. Meijer, Sep 09 2013: (Start)
a := proc(n) local t: t:=floor((sqrt(4*n-3)-1)/2): floor(sqrt(4*n-1))-2*((n-1) mod (t+1)) end: seq(a(n), n=1..92); # End first program.
T := (n, k) -> n-2*k+2: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..18); # End second program. (End)

Flatten@Range[Range[10], 1, -2] (* Birkas Gyorgy, Apr 07 2011 *)

From Boris Putievskiy, Sep 09 2013: (Start)
a(n) = 2*(1-A122197(n)) + A000267(n-1).
a(n) = floor(sqrt(4*n-1)) - 2*((n-1) mod (t+1)), where t = floor((sqrt(4*n-3)-1)/2). (End)
From Johannes W. Meijer, Sep 09 2013: (Start)
T(n, k) = n - 2*k + 2, for n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A002260(n, n-2*k+2). (End)

A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

N. J. A. Sloane

Complement of A047235. - Reinhard Zumkeller, Oct 01 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004524, A007310, A047228, A047235, A047241, A047261, A093719, A096981.
First differences of A281026.
See A301729 for an essentially identical sequence.

a047273 n = a047273_list !! (n-1)
a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list
-- Reinhard Zumkeller, Feb 19 2013

[(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 20 2016

seq(2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, n = 0..69); # Gary Detlefs, Mar 19 2010

LinearRecurrence[{2,-2,2,-1},{0,1,3,5},80] (* Harvey P. Dale, Jan 04 2015 *)

PARI
```
a(n)=n+(n+1)\4+(n+2)\4
```

[(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # Zerinvary Lajos, Mar 09 2009

G.f.: x*(1+x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^2)*(1-x^3)/((1-x)^3*(1-x^4)).
a(n) = n + A004524(n+1) = -a(-n) for all n in Z.
Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
A093719(a(n)) = 1. - Reinhard Zumkeller, Oct 01 2008
a(n) = 2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, with offset 0..a(0)=0. - Gary Detlefs, Mar 19 2010
a(n) = (3*n-3+cos(Pi*n/2))/2. - R. J. Mathar, Oct 08 2010
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.
a(n) = (6*n-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)
Euler transform of length 4 sequence [3, -1, -1, 1]. - Michael Somos, Jun 24 2017
Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/2. - Amiram Eldar, Dec 16 2021
E.g.f.: (2 + 3*exp(x)*(x - 1) + cos(x))/2. - Stefano Spezia, Jul 26 2024

cp's OEIS Frontend

A378413 Irregular triangle read by rows: T(n,k) is the coefficient of x^k in the domination polynomial of the n-prism graph (n>=1, A004524(n+2)<=k<=2*n).

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A132402 Binomial transform of A004524 starting at 1.

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A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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A032528 Concentric hexagonal numbers: floor(3*n^2/2).

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A011848 a(n) = floor(binomial(n, 2)/2).

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A004525 One even followed by three odd.

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