A070893 -id:A070893 - cp's OEIS frontend

A032766 Numbers that are congruent to 0 or 1 (mod 3).

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 0

Patrick De Geest, May 15 1998

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Partial sums of A000034. - Richard Choulet, Jan 28 2010
Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
The number of integer rectangles with a side of length n+1 and the property: the bisectors of the angles form a square within its limits. - Alexander M. Domashenko, Oct 17 2024
The maximum possible number of 5-cycles in an outerplanar graph on n+4 vertices. - Stephen Bartell, Jul 10 2025

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Erich Friedman, Problem of the month November 2009
Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]
Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.
Kival Ngaokrajang, Illustration of initial terms (U).
Eric Weisstein's World of Mathematics, Andrásfai Graph
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Hadwiger Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).
Essentially the same: A049624.
Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Row 1 of A254051.
Row sums of A171370.
Cf. A066272 for anti-divisors.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.
Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.
Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.
Cf. A118111, A132463, A133083, A196592, A249745.

a032766 n = div n 2 + n  -- Reinhard Zumkeller, Dec 13 2014
(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015

&cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010
select(n->member(n mod 3,{0,1}), [$0..103]); # Peter Luschny, Apr 06 2014

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Floor[3 Range[0, 69]/2] (* L. Edson Jeffery, Jan 14 2017 *)
Drop[Range[0,110],{3,-1,3}] (* Harvey P. Dale, Sep 02 2023 *)

PARI
```
{a(n) = n + n\2}
```

concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

[int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(-n) = -A007494(n).
a(n) = A049615(n, 2), for n > 2.
From Paul Barry, Sep 04 2003: (Start)
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010
a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011
a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011
a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011
a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013
a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014
a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = floor(3*n/2). - L. Edson Jeffery, Jan 18 2015
a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015
E.g.f.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021

Better description from N. J. A. Sloane, Aug 01 1998

A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

Original entry on oeis.org

0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485

Offset: 0

N. J. A. Sloane

4 times the variance of the area under an n-step random walk: e.g., with three steps, the area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 each with probability 1/8, giving a variance of 35/4 or a(3)/4. - Henry Bottomley, Jul 14 2003
Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004
Also a(n) = (1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers (vertex structure 9). Cf. A059722 = alternate vertex; A000447 = structured diamonds; and structured tetragonal anti-diamond numbers (vertex structure 9). Cf. A096000 = alternate vertex; A100188 = structured anti-diamonds. Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - Valentin Bakoev, Mar 03 2009
Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - J. M. Bergot, Aug 24 2011
This sequence is related to A070893 by A070893(2*n-1) = n*a(n)-sum(i=0..n-1, a(i)). - Bruno Berselli, Aug 26 2011
Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011
Partial sums of A016754. - Reinhard Zumkeller, Apr 02 2012
Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017

			G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.

G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Valentin Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
Milan Janjic, Two Enumerative Functions.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
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.
Eric Weisstein's World of Mathematics, Haüy Construction.
Eric Weisstein's World of Mathematics, Square Pyramid.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Column 1 in triangles A008956 and A008958.
Cf. A035328, A069072, A190152.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of A002577. - Valentin Bakoev, Mar 03 2009
Cf. A002412, A016061.

a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
-- Reinhard Zumkeller, Apr 02 2012

[n*(4*n^2-1)/3: n in [0..50]]; // Vincenzo Librandi, Jan 12 2016

A000447:=z*(1+6*z+z**2)/(z-1)**4; # Simon Plouffe, 1992 dissertation.
A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n * (4*n^2 - 1) / 3};
```

concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016

def A000447(n): return n*((n**2<<2)-1)//3 # Chai Wah Wu, Feb 12 2023

a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
G.f.: x*(1+6*x+x^2)/(1-x)^4.
a(n) = -a(-n) for all n in Z.
a(n) = A000330(2*n)-4*A000330(n) = A000466(n)*n/3 = A000578(n)+A007290(n-2) = A000583(n)-2*A024196(n-1) = A035328(n)/3. - Henry Bottomley, Jul 14 2003
a(n+1) = (2*n+1)*(2*n+2)(2*n+3)/6. - Valentin Bakoev, Mar 03 2009
a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, May 25 2012
a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - Mircea Merca, Jan 25 2014
a(n) = A005917(n+1) - A100157(n+1), where A005917 are the rhombic dodecahedral numbers and A100157 are the structured rhombic dodecahedral numbers (vertex structure 9). - Peter M. Chema, Jan 09 2016
For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - Ivan N. Ianakiev, Mar 04 2017
E.g.f.: exp(x)*x*(1 + 4*x  + (4/3)*x^2). - Wolfdieter Lang, Mar 11 2017
a(n) = A002412(n) + A016061(n-1), for n>0. - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)

Chrystal and Durell references from R. K. Guy, Apr 02 2004

A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

Original entry on oeis.org

0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054

Offset: 0

N. J. A. Sloane

Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009
We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010
Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010
Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012
Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

			G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, Hexagonal spiral containing the initial terms.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.
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.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A001859, A077043, A002620, A002378.
Row sums of A104567.
Cf. A000034, A032766, A002717, A070893. - Richard Choulet, Jan 28 2010
Cf. A051125.

[(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // Vincenzo Librandi, Aug 20 2011

with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
A006578 := proc(n)
    (6*n^2 + 4*n - 1 + (-1)^n)/8 ;
end proc: # R. J. Mathar, Apr 28 2017

Accumulate[LinearRecurrence[{1,1,-1}, {0,1,3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1},{0, 1, 4, 8},53] (* Ray Chandler, Aug 03 2015 *)

{a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */

Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) + A002620(n) = A002378(n) = n*(n+1).
Partial sums of A032766. - Paul Barry, May 30 2003
a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002620(n) + A004526(n) = A001859(n) - A004526(n+1). - Henry Bottomley, Mar 08 2000
a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003
a(n) = A001859(-1-n) for all n in Z. - Michael Somos, May 10 2006
a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = (3*n^2 + 2*n - (n mod 2))/4. - Ctibor O. Zizka, Mar 11 2012
a(n) = Sum_{i=1..n} floor(3*i/2) = Sum_{i=0..n} (i + floor(i/2)). - Enrique Pérez Herrero, Apr 21 2012
a(n) = 3*n*(n+1)/2 - A001859(n). - Clark Kimberling, Jul 02 2012
a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014
a(n) = A002717(n) - A002717(n-1). - Michael Somos, Jun 09 2014
a(n) = Sum_{k=1..n} floor((n+k+1)/2). - Wesley Ivan Hurt, Mar 31 2017
a(n) = A002620(n+1)+2*A002620(n). - R. J. Mathar, Apr 28 2017
Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Offset and description changed by N. J. A. Sloane, Nov 30 2006

A002717 a(n) = floor(n(n+2)(2n+1)/8).

Original entry on oeis.org

0, 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, 1945, 2255, 2596, 2970, 3378, 3822, 4303, 4823, 5383, 5985, 6630, 7320, 8056, 8840, 9673, 10557, 11493, 12483, 13528, 14630, 15790, 17010, 18291, 19635, 21043, 22517, 24058

Offset: 0

N. J. A. Sloane

Number of triangles in triangular matchstick arrangement of side n, for n >= 1. Row sums of A085691.
We observe that the sequence is the transform of A006578 by the following transform T: T(u_0,u_1,u_2,u_3,...)=(u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In another terms v_p=sum(u_k,k=0..p) and the G.f phi_v of v is given by: phi_v=phi_u/(1-z). - Richard Choulet, Jan 28 2010
Row sums of A220053, for n > 0. - Reinhard Zumkeller, Dec 03 2012
a(n) has the expansion (1*0)+(1*1)+(4*1)+(4*2)+(7*2)+(7*3)+..., where the expansion stops when a(n) has n+1 number of terms. The expansion starts at (1*0), and progresses by alternating addition of 1 to the second number and 3 to the first number. - Arlu Genesis A. Padilla, Jun 04 2014
Taking the absolute values of each n-th difference and excluding the first n terms of each mentioned sequence, A002717 has the first difference A006578 (see formula of Michael Somos dated Jun 09 2014), the second difference A032766 (see 'partial sum' crossref), the third difference A000034, the fourth difference A000012, and the fifth to n-th difference A000004. - Arlu Genesis A. Padilla, Jun 12 2014

			f(3)=13 because the following figure contains 13 triangles if horizontal bars are added:
....... /\
...... /\/\
..... /\/\/\
G.f. = x + 5*x^2 + 13*x^3 + 27*x^4 + 48*x^5 + 78*x^6 + 118*x^7 + 170*x^8 + ...

J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
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
Ralph E. Edwards et al., Problem 889: A well-known problem, Math. Mag., 47 (1974), 289-292.
F. Gerrish, How many triangles, Math. Gaz., 54 (1970), 241-246.
J. Halsall, An interesting series, Math. Gaz., 46 (1962), 55-56.
J. Halsall, An interesting series, Math. Gaz., 46 (1962), 55-56. [Annotated scanned copy]
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
B. D. Mastrantone, How Many Triangles?, Math. Gaz., 55 (1971), 438-440.
Hugo Pfoertner, Illustration of A002717(5) and A002717(6)
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
L. Smiley, A Quick Solution of Triangle Counting, Mathematics Magazine, 66, #1, Feb '93, p. 40.
Eric Weisstein's World of Mathematics, Triangle Tiling.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000292 number of triangles with same orientation as largest triangle, A002623 number of triangles pointing in opposite direction to largest triangle, A085691 number of triangles of side k in arrangement of side n.
Bisections: A135712 (odd part), A135713 (even part).
Cf. A006578, A032766, A000034, A070893. - Richard Choulet, Jan 28 2010
Cf. A045947, A016777.
Cf. A000004, A000012, A000217, A220053.

[Floor(n*(n+2)*(2*n+1)/8): n in [0..50]]; // Wesley Ivan Hurt, Jun 04 2014

A002717:=n->floor(n*(n+2)*(2*n+1)/8); seq(A002717(n), n=0..100);

Table[Floor[n(n+2)(2n+1)/8],{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,5,13,27},50] (* Harvey P. Dale, Jan 20 2013 *)

PARI
```
{a(n) = n * (n+2) * (2*n+1) \ 8};
```

a(n) = (1/16)*[2n(2n+1)(n+2)+cos(Pi*n)-1]. - Justin C. Bozonier (justinb67(AT)excite.com), Dec 05 2000
a(m+1)-2a(m)+2a(m-2)-a(m-3) = 3. - Len Smiley, Oct 08 2001
a(n) = (2n(2n+1)(n+2)+(-1)^n-1)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Oct 25 2003
a(n) = A000292(n-1) + A002623(n-2). - Hugo Pfoertner, Mar 06 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*k*binomial(k+1,2).
G.f.: x(1+2x)/((1+x)(1-x)^4). - Simon Plouffe in his 1992 dissertation (with a different offset).
a(0)=0, a(1)=1, a(2)=5, a(3)=13, a(4)=27, a(n)=3*a(n-1)-2*a(n-2)-2*a(n-3)+ 3*a(n-4)- a(n-5). - Harvey P. Dale, Jan 20 2013
a(n) = a(n-1) + A016777(floor(0.5*n))*floor(0.5+0.5*n). - Arlu Genesis A. Padilla, Jun 04 2014
a(-n) = - A045947(n). a(n) = a(n-1) + A006578(n). - Michael Somos, Jun 09 2014
a(n) = Sum_{i=1..n} T(n-i+1)+T(n-2*i+1), where T(n)=n*(n+1)/2=A000217(n) if n>0 and 0 if n<=0. So we have a(n+2)-a(n)=(n+2)^2+(n+1)*(n+2)/2. - Maurice Mischler, Sep 08 2014
E.g.f.: (x*(2*x^2 + 11*x + 9)*cosh(x) + (2*x^3 + 11*x^2 + 9*x - 1)*sinh(x))/8. - Stefano Spezia, Jul 19 2022

A194498 T(n,k)=Number of ways to arrange k nonattacking queens on the lower triangle of an n X n board.

Original entry on oeis.org

1, 0, 3, 0, 0, 6, 0, 0, 2, 10, 0, 0, 0, 12, 15, 0, 0, 0, 0, 38, 21, 0, 0, 0, 0, 12, 92, 28, 0, 0, 0, 0, 0, 82, 188, 36, 0, 0, 0, 0, 0, 8, 330, 344, 45, 0, 0, 0, 0, 0, 0, 118, 1008, 580, 55, 0, 0, 0, 0, 0, 0, 4, 802, 2566, 920, 66, 0, 0, 0, 0, 0, 0, 0, 114, 3708, 5742, 1390, 78, 0, 0, 0, 0, 0, 0, 0

Offset: 1

R. H. Hardin Aug 26 2011

Table starts
...1....0......0.......0........0........0........0........0........0......0..0
...3....0......0.......0........0........0........0........0........0......0..0
...6....2......0.......0........0........0........0........0........0......0..0
..10...12......0.......0........0........0........0........0........0......0..0
..15...38.....12.......0........0........0........0........0........0......0..0
..21...92.....82.......8........0........0........0........0........0......0..0
..28..188....330.....118........4........0........0........0........0......0..0
..36..344...1008.....802......114........0........0........0........0......0..0
..45..580...2566....3708.....1384.......64........0........0........0......0..0
..55..920...5742...13280.....9890.....1644.......10........0........0......0..0
..66.1390..11652...39734....50662....19306.....1210........0........0......0..0
..78.2020..21926..104000...205512...146718....27198......484........0......0..0
..91.2842..38802..244948...698688...820218...322782....26084.......38......0..0
.105.3892..65322..530632..2074530..3670288..2564988...531344....14870......0..0
.120.5208.105428.1072776..5525902.13846830.15372702..6285270...629414...3656..0
.136.6832.164214.2048056.13476246.45661556.74615814.51834064.11949712.501700.56

			Some solutions for n=6 k=4
..0............0............0............0............0............0
..0.0..........0.0..........0.1..........0.0..........1.0..........0.1
..1.0.0........1.0.0........0.0.0........0.0.1........0.0.1........0.0.0
..0.0.1.0......0.0.0.1......0.0.1.0......1.0.0.0......0.0.0.0......1.0.0.0
..0.0.0.0.1....0.1.0.0.0....1.0.0.0.0....0.0.0.1.0....0.1.0.0.0....0.0.1.0.0
..0.1.0.0.0.0..0.0.0.0.1.0..0.0.0.1.0.0..0.1.0.0.0.0..0.0.0.1.0.0..0.0.0.0.1.0

R. H. Hardin, Table of n, a(n) for n = 1..518

Column 1 is A000217

Column 2 is 2*A070893(n-2)

A082289 Expansion of x^4(2+x)/((1+x)(1-x)^5).

Original entry on oeis.org

2, 9, 26, 59, 116, 206, 340, 530, 790, 1135, 1582, 2149, 2856, 3724, 4776, 6036, 7530, 9285, 11330, 13695, 16412, 19514, 23036, 27014, 31486, 36491, 42070, 48265, 55120, 62680, 70992, 80104, 90066, 100929, 112746, 125571, 139460, 154470

Offset: 4

Michael Somos, Apr 07 2003

Vincenzo Librandi, Table of n, a(n) for n = 4..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A070893, A082290.

Cf. A045947 (which contains the first differences). - Bruno Berselli, Aug 26 2011

[(1/96)*(2*(n-2)*n*(3*n^2-10*n+4)+3*(-1)^n-3): n in [4..50]]; // Vincenzo Librandi, Aug 29 2011

Drop[CoefficientList[Series[x^4(2+x)/((1+x)(1-x)^5),{x,0,50}],x],4] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{2,9,26,59,116,206},50] (* Harvey P. Dale, Aug 26 2013 *)

a(n)=polcoeff(if(n>0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

G.f.: x^4*(2+x)/((1+x)*(1-x)^5).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 3 by this equation, then a(n)=0 for 0 <= n <= 3 and a(n) = A070893(-n) for n < 0.
a(n) = A082290(2*n-7).
a(n) = (1/96)*(2*(n-2)*n*(3*n^2 - 10*n + 4) + 3*(-1)^n - 3). a(n) - a(n-2) = A006002(n-3) for n > 5. - Bruno Berselli, Aug 26 2011
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6); a(4)=2, a(5)=9, a(6)=26, a(7)=59, a(8)=116, a(9)=206. - Harvey P. Dale, Aug 26 2013

A166278 Square array A(n,k), n,k>=0, read by antidiagonals: A(n,k) is the total element sum of the k-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 6, 10, 10, 5, 0, 1, 8, 19, 20, 15, 6, 0, 1, 12, 33, 46, 35, 21, 7, 0, 1, 16, 63, 100, 94, 56, 28, 8, 0, 1, 24, 111, 220, 242, 172, 84, 36, 9, 0, 1, 32, 201, 488, 633, 514, 290, 120, 45, 10, 0, 1, 48, 369, 1104, 1643, 1518, 984, 460, 165, 55, 11

Offset: 0

Alois P. Heinz, Oct 10 2009

			A(3,4) = 33, because f([1,1,1]) = [1,2,3], (f^2)([1,1,1]) = [3,3,4], (f^3)([1,1,1]) = [4,6,9], (f^4)([1,1,1]) = [9,12,12], and 9+12+12 = 33.
Square array A(n,k) begins:
  0,  0,  0,  0,   0,   0, ...
  1,  1,  1,  1,   1,   1, ...
  2,  3,  4,  6,   8,  12, ...
  3,  6, 10, 19,  33,  63, ...
  4, 10, 20, 46, 100, 220, ...
  5, 15, 35, 94, 242, 633, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-3 give: A001477, A000217, A000292, A070893.
Rows n=0-2 give: A000004, A000012, A029744(k+2).
Main diagonal gives A261490.

f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
A:= (n, k)-> add(i, i=(f@@k)([1$n])):
seq(seq(A(n, d-n), n=0..d), d=0..15);

f[L_List] := f[L] = Sort[Reverse[Sort[L]]*Range[Length[L]]];
A[0, ] = 0; A[n, 0] := n; A[n_, k_] := Total[Nest[f, Range[n], k-1]];
Table[A[n, k-n], {k, 0, 15}, {n, 0, k}] // Flatten (* Jean-François Alcover, Jun 07 2016 *)

A082290 Expansion of (1+x+x^2)/((1+x^2)(1+x)^4(1-x)^5).

Original entry on oeis.org

1, 2, 6, 9, 19, 26, 46, 59, 94, 116, 172, 206, 290, 340, 460, 530, 695, 790, 1010, 1135, 1421, 1582, 1946, 2149, 2604, 2856, 3416, 3724, 4404, 4776, 5592, 6036, 7005, 7530, 8670, 9285, 10615, 11330, 12870, 13695, 15466, 16412, 18436, 19514, 21814, 23036

Offset: 0

Michael Somos, Apr 07 2003

			G.f. = 1 + 2*x + 6*x^2 + 9*x^3 + 19*x^4 + 26*x^5 + 46*x^6 + 59*x^7 + ...

Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (1, 3, -3, -2, 2, -2, 2, 3, -3, -1, 1).

Cf. A070893, A082289.

[(6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^Floor((6*n -1 +(-1)^n)/4))/1536: n in [0..50]]; // Vincenzo Librandi, Oct 23 2014

Table[(6 n^4 + 108 n^3 + 666 n^2 + 1620 n + 1251 + (4 n^3 + 54 n^2 + 236 n + 333) (-1)^n - 48 (-1)^((6 n - 1 + (-1)^n)/4))/1536, {n, 0, 50}] (* after Luce ETIENNE; or, by definition: *) CoefficientList[Series[(1 + x + x^2)/((1 + x^2)*(1 + x)^4*(1 - x)^5), {x, 0, 50}], x] (* Bruno Berselli, Oct 26 2014 *)

{a(n) = if( n<-8, a(-9-n), polcoeff( (1 + x + x^2) / ((1 + x^2) *(1 + x)^4 * (1 - x)^5) + x * O(x^n), n))};

Euler transform of length 4 sequence [ 2, 3, -1, 1]. - Michael Somos, Feb 15 2006
G.f.: (1 + x + x^2) / ((1 + x^2) * (1 + x)^4 * (1 - x)^5).
a(n) = 3*a(n-2) - 2*a(n-4) - 2*a(n-6) + 3*a(n-8) - a(n-10) + 3.
a(n) = a(-9-n) for all n in Z.
a(2*n) = A070893(n+1). a(2*n + 1) = A082289(n+4).
a(n) = (6*n^4+108*n^3+666*n^2+1620*n+1251+(4*n^3+54*n^2+236*n+333)*(-1)^n-48*(-1)^((6*n-1+(-1)^n)/4))/1536. - Luce ETIENNE, Oct 23 2014

A185958 Accumulation array of the array max{n,k}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 7, 6, 10, 13, 13, 10, 15, 21, 22, 21, 15, 21, 31, 34, 34, 31, 21, 28, 43, 49, 50, 49, 43, 28, 36, 57, 67, 70, 70, 67, 57, 36, 45, 73, 88, 94, 95, 94, 88, 73, 45, 55, 91, 112, 122, 125, 125, 122, 112, 91, 55, 66, 111, 139, 154, 160, 161, 160, 154, 139, 111, 66, 78, 133, 169, 190, 200, 203, 203, 200, 190, 169, 133, 78, 91, 157, 202, 230, 245, 251, 252, 251, 245, 230, 202, 157, 91, 105, 183, 238, 274, 295, 305, 308, 308, 305, 295, 274, 238, 183, 105

Offset: 1

Clark Kimberling, Feb 07 2011

A member of the accumulation chain
... < A185917 < A051125 < A185958 < ...,
where A051125, written as a rectangular array M, is given by M(n,k)=max{n,k}.  See A144112 for the definition of accumulation array.
row 1: A000217
row 2: A002061
diag (1,7,...): A002412
diag (3,13,..): A016061
antidiagonal sums: A070893

			Northwest corner:
1....3....6....10....15
3....7....13...21....31
6....13...22...34....49
10...21...34...50....70

Cf. A144112, A051125.

A := proc(n,k) option remember; ## n >= 0 and k = 0 .. n
    if k < 0 or k > n then
        0
    elif n = 0 then
        1
    else
        A(n-1,k) + A(n-1,k-1) - A(n-2,k-1) + max(n-k+1,k+1)
    end if
end proc: # Yu-Sheng Chang, Jun 05 2020

From Yu-Sheng Chang, Jun 05 2020: (Start)
O.g.f.: F(z,v) = -(v^2*z^3+v*z^3-3*v*z^2+1)/((v*z^2-v*z-z+1)^2*(v*z^2-1)*(z-1)*(v*z-1)).
T(n,k) = [v^k] 1/2*n^2*(v^(n+2)+1)/(1-v)^2+1/2*n*(3*v^(n+3)-7*v^(n+2)+7*v-3)/(-1+v)^3-1/2*v*((1-v^(1/2))^4*(-1)^n+(1+v^(1/2))^4)*v^(1/2*n)/(1-v)^4+(6*v^2+6*v^(n+2)+v^(n+4)-3*v^(n+3)-3*v+1)/(1-v)^4.
(End)

cp's OEIS Frontend

A032766 Numbers that are congruent to 0 or 1 (mod 3).

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A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.

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A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

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A002717 a(n) = floor(n(n+2)(2n+1)/8).

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A194498 T(n,k)=Number of ways to arrange k nonattacking queens on the lower triangle of an n X n board.

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A082289 Expansion of x^4*(2+x)/((1+x)*(1-x)^5).

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A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

A082289 Expansion of x^4(2+x)/((1+x)(1-x)^5).

A082290 Expansion of (1+x+x^2)/((1+x^2)(1+x)^4(1-x)^5).