A001105 -id:A001105 - cp's OEIS frontend

A033991 a(n) = n(4n-1).

0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335, 3570, 3813, 4064, 4323, 4590, 4865, 5148, 5439, 5738, 6045, 6360, 6683, 7014, 7353, 7700, 8055, 8418

Offset: 0

N. J. A. Sloane

Write 0,1,2,... in a clockwise spiral; sequence gives numbers on negative x axis. (See illustration in Example.)
This sequence is the number of expressions x generated for a given modulus n in finite arithmetic. For example, n=1 (modulus 1) generates 3 expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting n from 4n^2, we eliminate the counting of those expressions that would include division by zero, which would be, of course, undefined. - David Quentin Dauthier, Nov 04 2007
From Emeric Deutsch, Sep 21 2010: (Start)
a(n) is also the Wiener index of the windmill graph D(3,n).
The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
Example: a(2)=14; indeed if the triangles are OAB and OCD, then, denoting distance by d, we have d(O,A)=d(O,B)=d(A,B)=d(O,C)=d(O,D)=d(C,D)=1 and d(A,C)=d(A,D)=d(B,C)=d(B,D)=2. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(4,n), D(5,n), and D(6,n) see A152743, A028994, and A180577, respectively. (End)
Even hexagonal numbers divided by 2. - Omar E. Pol, Aug 18 2011
For n > 0, a(n) equals the number of length 3*n binary words having exactly two 0's with the n first bits having at most one 0. For example a(2) = 14. Words are 010111, 011011, 011101, 011110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Franck Maminirina Ramaharo, Mar 09 2018
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n-1; {1, 2, 1, 4n-2}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 06 2022

			Clockwise spiral (with sequence terms parenthesized) begins
   16--17--18--19
    |
   15   4---5---6
    |   |       |
  (14) (3) (0)  7
    |   |   |   |
   13   2---1   8
    |           |
   12--11--10---9

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Eric W. Weisstein, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences from spirals: A001107, A002939, A007742, A033951-A033954, A033989-A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A074378, A014848, A152743, A028994, A000326, A001105, A005476, A014635, A016742, A049452, A118729.

[seq(binomial(4*n, 2)/2, n=0..45)]; # Zerinvary Lajos, Jan 16 2007

Table[n*(4*n - 1), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
LinearRecurrence[{3,-3,1},{0,3,14},50] (* Harvey P. Dale, Oct 10 2011 *)

PARI
```
a(n)=4*n^2-n;
```

a(n) = A007742(-n) = A074378(2n-1) = A014848(2n).
G.f.: x*(3+5*x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = A014635(n)/2. - Zerinvary Lajos, Jan 16 2007
From Zerinvary Lajos, Jun 12 2007: (Start)
a(n) = A000326(n) + A005476(n).
a(n) = A049452(n) - A001105(n). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Harvey P. Dale, Oct 10 2011
a(n) = A118729(8n+2). - Philippe Deléham, Mar 26 2013
From Ilya Gutkovskiy, Dec 04 2016: (Start)
E.g.f.: x*(3 + 4*x)*exp(x).
Sum_{n>=1} 1/a(n) = 3*log(2) - Pi/2 = 0.50864521488... (End)
a(n) = Sum_{i=n..3n-1} i. - Wesley Ivan Hurt, Dec 04 2016
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n, 2) + 2*n^2.
a(n) = A054556(n+1) - 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3-2*sqrt(2)))/sqrt(2) - log(2). - Amiram Eldar, Mar 20 2022

Two remarks combined into one by Emeric Deutsch, Oct 03 2010

A055998 a(n) = n*(n+5)/2.

Original entry on oeis.org

0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272

Offset: 0

Barry E. Williams, Jun 14 2000

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Milan Janjic, Two Enumerative Functions.
Kival Ngaokrajang, Illustration from A000027 (contains errors).
Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
Leo Tavares, Illustration: Truncated Point Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Cf. A000096, A000217, A001477, A002522.
Row n=2 of A255961.
Cf. other rows, columns and diagonals of A000027 written as a table: A034856, A046691, A052905, A055999, A155212, A051936, A056000, A183897, A056115, A051938; A000124, A022856, A152950, A145018, A077169, A166136, A167487, A173036; A059993, A090288, A054000, A142463, A056220, A001105, A001844, A058331, A051890, A097080, A093328, A137882.

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)

A033581 a(n) = 6*n^2.

Original entry on oeis.org

0, 6, 24, 54, 96, 150, 216, 294, 384, 486, 600, 726, 864, 1014, 1176, 1350, 1536, 1734, 1944, 2166, 2400, 2646, 2904, 3174, 3456, 3750, 4056, 4374, 4704, 5046, 5400, 5766, 6144, 6534, 6936, 7350, 7776, 8214, 8664, 9126, 9600, 10086, 10584, 11094, 11616

Offset: 0

N. J. A. Sloane

Number of edges of a complete 4-partite graph of order 4n, K_n,n,n,n. - Roberto E. Martinez II, Oct 18 2001
Number of edges of the complete bipartite graph of order 7n, K_n, 6n. - Roberto E. Martinez II, Jan 07 2002
Number of edges in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n). - Roberto E. Martinez II, Jan 07 2002
Total surface area of a cube of edge length n. See A000578 for cube volume. See A070169 and A071399 for surface area and volume of a regular tetrahedron and links for the other Platonic solids. - Rick L. Shepherd, Apr 24 2002
a(n) can represented as n concentric hexagons (see example). - Omar E. Pol, Aug 21 2011
Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A003154 in the same spiral. - Omar E. Pol, Sep 08 2011
Together with 1, numbers m such that floor(2*m/3) and floor(3*m/2) are both squares. Example: floor(2*150/3) = 100 and floor(3*150/2) = 225 are both squares, so 150 is in the sequence. - Bruno Berselli, Sep 15 2014
a(n+1) gives the number of vertices in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter has 18 vertices. The core hexagon has 6 vertices. a(2) = 18 + 6 = 24 is the total number of vertices. - Ivan N. Ianakiev, Mar 11 2015
a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n). - Sergey Pavlov, Mar 31 2017
More generally, if k >= 5 we have that the sequence whose formula is a(n) = (2*k - 4)*n^2 is also the sequence found by reading the line from 0, in the direction 0, (2*k - 4), ..., in the square spiral whose vertices are the generalized k-gonal numbers. In this case k = 5. - Omar E. Pol, May 13 2018
The sequence also gives the number of size=1 triangles within a match-made hexagon of size n. - John King, Mar 31 2019
For hexagons, the number of matches required is A045945; thus number of size=1 triangles is A033581; number of larger triangles is A307253 and total number of triangles is A045949. See A045943 for analogs for Triangles; see A045946 for analogs for Stars. - John King, Apr 04 2019

			From _Omar E. Pol_, Aug 21 2011: (Start)
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 o o o
.
.    6            24                   54
.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Bruno Berselli, An interpretation of initial terms.
Ivan N. Ianakiev, Hexagon-like honeycomb built form regular congruent hexagons.
Leo Tavares, Illustration: Diamond Star Rays
Eric Weisstein's World of Mathematics, Platonic Solid.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A032528. Central column of triangle A001283.
Cf. A000217, A000290, A001105, A009111, A033428, A033583, A085250, A086729, A152734, A152751.
Cf. A000578, A001318, A003215, A005897, A045943, A045945, A045949, A070169, A071399.
Cf. A017593 (first differences).

a033581 = (* 6) . (^ 2)  -- Reinhard Zumkeller, Apr 27 2014

seq(6*n^2,n=0..44); # Nathaniel Johnston, Jun 26 2011

6 Range[44]^2 (* Michael De Vlieger, Apr 02 2017 *)
LinearRecurrence[{3,-3,1},{0,6,24},50] (* Harvey P. Dale, Jul 03 2017 *)

vector(100,n,6*(n-1)^2) \\ Derek Orr, Mar 11 2015

a(n) = A000290(n)*6. - Omar E. Pol, Dec 11 2008
a(n) = A001105(n)*3 = A033428(n)*2. - Omar E. Pol, Dec 13 2008
a(n) = 12*n + a(n-1) - 6, with a(0)=0. - Vincenzo Librandi, Aug 05 2010
G.f.: 6*x*(1+x)/(1-x)^3. - Colin Barker, Feb 14 2012
For n > 0: a(n) = A005897(n) - 2. - Reinhard Zumkeller, Apr 27 2014
a(n) = 3*floor(1/(1-cos(1/n))) = floor(1/(1-n*sin(1/n))) for n > 0. - Clark Kimberling, Oct 08 2014
a(n) = t(4*n) - 4*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(4*n) - 4*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/36.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/72 (A086729).
Product_{n>=1} (1 + 1/a(n)) = sqrt(6)*sinh(Pi/sqrt(6))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(6)*sin(Pi/sqrt(6))/Pi. (End)
E.g.f.: 6*exp(x)*x*(1 + x). - Stefano Spezia, Aug 19 2022

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001

A045991 a(n) = n^3 - n^2.

Original entry on oeis.org

0, 0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400, 67240, 72324

Offset: 0

N. J. A. Sloane

Number of edges in the line graph of the complete bipartite graph of order 2n, L(K_n,n). - Roberto E. Martinez II, Jan 07 2002
Number of edges of the Cartesian product of two complete graphs K_n X K_n. - Roberto E. Martinez II, Jan 07 2002
That is, number of edges in the n X n rook graph. - Eric W. Weisstein, Jun 20 2017
n such that x^3 + x^2 + n factors over the integers. - James R. Buddenhagen, Apr 19 2005
Also the number of triangles in a 2 X n grid of points and therefore also (n choose 2) * (n choose 1) * 2, or (2n choose 3) - 2*(n choose 3). - Joshua Zucker, Jan 11 2006
Nonnegative X values of solutions to the equation (X-Y)^3-XY=0. To find Y values: b(n)=(n+1)*n^2 (see A011379). I proved that, if(X,Y) is different from (0,0) and m=2, 4, 6, 8, 10, 12,..., then the equation (X-Y)^m-XY=0,... has no solution. - Mohamed Bouhamida, May 10 2006
For n>=1, a(n) is equal to the number of functions f:{1,2,3}->{1,2,...,n} such that for a fixed x in {1,2,3} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - John M. Campbell, Jul 16 2011
Define the infinite square array m(n,k) = (n-k)^2 for 1<=k<=n below the diagonal and m(n,k) = (k+n)(k-n) for 1<=n<=k above the diagonal. Then a(n) = Sum_{k=1..n} m(n,k) + Sum_{r=1..n} m(r,n), the "hook sum" of the terms left from m(n,n) and above m(n,n). - J. M. Bergot, Aug 16 2013
Partial sums of A049451. - Bruno Berselli, Feb 10 2014
Volume of an extruded rectangular brick with side lengths n, n and n-1. - Luciano Ancora, Jun 24 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
R. J. Mathar, On the Diophantine equation (X-Y)^m-XY=0.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
J.S. Seneschal, Oblong cuboid illustration.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Rook Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011379, A047929, A114364 (essentially the same).
Cf. A000217, A000290, A000578, A006002, A049451, A146485.
Cf. A002378, A001105, A011379.
Cf. A028724, A052289.

[n^3-n^2: n in [0..40]]; // Vincenzo Librandi, May 02 2011

A045991:=n->n^3 - n^2: seq(A045991(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[n^3 - n^2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
Table[4 Binomial[n, 2] + 6 Binomial[n, 3], {n, 0, 50}] (* Robert G. Wilson v, Mar 25 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 18, 48}, 20] (* Eric W. Weisstein, Jun 20 2017 *)

a(n)=n^2*(n-1) \\ Charles R Greathouse IV, Jul 17 2011

[n^2*(n-1) for n in range(0, 40)] # Zerinvary Lajos, Dec 03 2009

G.f.: 2*x^2*(x+2)/(-1+x)^4 = 6/(-1+x)^4+10/(-1+x)^2+14/(-1+x)^3+2/(-1+x). - R. J. Mathar, Nov 19 2007
a(n) = floor(n^5/(n^2+n+1)). - Gary Detlefs, Feb 10 2010
a(n) = 4*binomial(n,2) + 6*binomial(n,3). - Gary Detlefs, Mar 25 2012
a(n+1) = 2*A006002(n). - R. J. Mathar, Oct 31 2012
a(n) = (A000217(n-1)+A000217(n))*(A000217(n-1)-A000217(n-2)). - J. M. Bergot, Oct 31 2012
From Wesley Ivan Hurt, May 19 2015: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n) = Sum_{k=0..n-1} Sum_{i=n-k-1..n+k-1} i. (End)
Sum_{n>=2} 1/a(n) = 2 - Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) - 2. - Amiram Eldar, Jul 05 2020
E.g.f.: exp(x)*x^2*(2 + x). - Stefano Spezia, May 20 2021
Product_{n>=2} (1 - 1/a(n)) = A146485. - Amiram Eldar, Nov 22 2022
From J.S. Seneschal, Jun 21 2024: (Start)
a(n) = (n-1)*A000290(n).
a(n) = n*A002378(n-1).
a(n) = A011379(n) - A001105(n). (End)

A028983 Numbers whose sum of divisors is even.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Patrick De Geest

The even terms of this sequence are the even terms appearing in A178910. [Edited by M. F. Hasler, Oct 02 2014]
A071324(a(n)) is even. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152678(n). - Jaroslav Krizek, Oct 06 2009
A083207 is a subsequence. - Reinhard Zumkeller, Jul 19 2010
Numbers k such that the number of odd divisors of k (A001227) is even. - Omar E. Pol, Apr 04 2016
Numbers k such that the sum of odd divisors of k (A000593) is even. - Omar E. Pol, Jul 05 2016
Numbers with a squarefree part greater than 2. - Peter Munn, Apr 26 2020
Equivalently, numbers whose odd part is nonsquare. Compare with the numbers whose square part is even (i.e., nonodd): these are the positive multiples of 4, A008586\{0}, and A225546 provides a self-inverse bijection between the two sets. - Peter Munn, Jul 19 2020
Also numbers whose reversed prime indices have alternating product > 1, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). Also Heinz numbers of the partitions counted by A347448. - Gus Wiseman, Oct 29 2021
Numbers whose number of middle divisors is not odd (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

The complement is A028982 = A000290 U A001105.
Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.
Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.
Cf. A334748 (a permutation).
Related to A008586 via A225546.
Ranks the partitions counted by A347448, complement A119620.
Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.

Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* Jayanta Basu, Jun 05 2013 *)

is(n)=!issquare(n)&&!issquare(n/2) \\ Charles R Greathouse IV, Jan 11 2013

from math import isqrt
def A028983(n):
    def f(x): return n-1+isqrt(x)+isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

a(n) ~ n. - Charles R Greathouse IV, Jan 11 2013
a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - Charles R Greathouse IV, Sep 01 2015
A007913(a(n)) > 2. - Peter Munn, May 05 2020

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

A006331 a(n) = n(n+1)(2*n+1)/3.

Original entry on oeis.org

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

N. J. A. Sloane

Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020
a(n) is the irregularity of the n-th power of a path of length at least 3*n.  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023
a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012

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
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
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]
Rowan Beckworth, Basic atomic information.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
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
Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A132339.
Cf. A002378, A048147, A098077, A163102.
Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI
```
a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
```

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
From Bruno Berselli, May 17 2018: (Start)
a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018
E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022

A014132 Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals.

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

N. J. A. Sloane

Numbers that are not triangular (nontriangular numbers).
Also definable as follows: a(1)=2; for n>1, a(n) is smallest integer greater than a(n-1) such that the condition "n and a(a(n)) have opposite parities" can always be satisfied. - Benoit Cloitre and Matthew Vandermast, Mar 10 2003
Record values in A256188 that are greater than 1. - Reinhard Zumkeller, Mar 26 2015
From Daniel Forgues, Apr 10 2015: (Start)
With n >= 1, k >= 1:
  t(n+k) - k, 1 <= k <= n+k-1, n >= 1;
  t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
where t(n+k) = t(n+k-1) + (n+k) is the (n+k)-th triangular number, while the number of compositions of n+k into 2 parts is C(n+k-1, 2-1) = n+k-1, the number of nontriangular numbers between t(n+k-1) and t(n+k), just right!
Related to Hilbert's Infinite Hotel:
0) All rooms, numbered through the positive integers, are full;
1) An infinite number of trains, each containing an infinite number of passengers, arrives: i.e., a 2-D lattice of pairs of positive integers;
2) Move occupant of room m, m >= 1, to room t(m) = m*(m+1)/2, where t(m) is the m-th triangular number;
3) Assign n-th passenger from k-th train to room t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
4) Everybody has his or her own room, no room is empty, for m >= 1.
If situation 1 happens again, repeat steps 2 and 3, you're back to 4.
(End)
1711 + 2*a(n)*(58 + a(n)) is prime for n<=21. The terms that do not have this property start 29,32,34,43,47,58,59,60,62,63,65,68,70,73,...  - Benedict W. J. Irwin, Nov 22 2016
Also numbers k with the property that in the symmetric representation of sigma(k) both Dyck paths have a central peak or both Dyck paths have a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

			From _Boris Putievskiy_, Jan 14 2013: (Start)
Start of the sequence as a table (read by antidiagonals, right to left), where the k-th row corresponds to the k-th column of the triangle (shown thereafter):
   2,  4,  7, 11, 16, 22, 29, ...
   5,  8, 12, 17, 23, 30, 38, ...
   9, 13, 18, 24, 31, 39, 48, ...
  14, 19, 25, 32, 40, 49, 59, ...
  20, 26, 33, 41, 50, 60, 71, ...
  27, 34, 42, 51, 61, 72, 84, ...
  35, 43, 52, 62, 73, 85, 98, ...
  (...)
Start of the sequence as a triangle (read by rows), where the i elements of the i-th row are t(i) + 1 up to t(i+1) - 1, i >= 1:
   2;
   4,  5;
   7,  8,  9;
  11, 12, 13, 14;
  16, 17, 18, 19, 20;
  22, 23, 24, 25, 26, 27;
  29, 30, 31, 32, 33, 34, 35;
  (...)
Row number i contains i numbers, where t(i) = i*(i+1)/2:
  t(i) + 1, t(i) + 2, ..., t(i) + i = t(i+1) - 1
(End) [Edited by _Daniel Forgues_, Apr 11 2015]

T. D. Noe, Table of n, a(n) for n = 1..1000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence arXiv:math/0305308 [math.NT], 2003.
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011. See Example 5 p. 456.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000217, A006002, A035214, A080036, A002024, A007401, A003057, A114327, A002260, A004736, A118011, A237593.
Cf. A000124 (left edge: quasi-triangular numbers), A000096 (right edge: almost-triangular numbers), A006002 (row sums), A001105 (central terms).
Cf. A242401 (subsequence).
Cf. A004202, A256188.
Cf. A145397 (the non-tetrahedral numbers).

a014132 n = n + round (sqrt $ 2 * fromInteger n)
a014132_list = filter ((== 0) . a010054) [0..]
-- Reinhard Zumkeller, Dec 12 2012

IsTriangular:=func< n | exists{ k: k in [1..Isqrt(2*n)] | n eq (k*(k+1) div 2)} >; [ n: n in [1..90] | not IsTriangular(n) ]; // Klaus Brockhaus, Jan 04 2011

f[n_] := n + Round[Sqrt[2n]]; Array[f, 71] (* or *)
Complement[ Range[83], Array[ #(# + 1)/2 &, 13]] (* Robert G. Wilson v, Oct 21 2005 *)
DeleteCases[Range[80],?(OddQ[Sqrt[8#+1]]&)] (* _Harvey P. Dale, Jul 24 2021 *)

PARI
```
a(n)=if(n<1,0,n+(sqrtint(8*n-7)+1)\2)
```

isok(n) = !ispolygonal(n,3); \\ Michel Marcus, Mar 01 2016

from math import isqrt
def A014132(n): return n+(isqrt((n<<3)-7)+1>>1) # Chai Wah Wu, Jun 17 2024

a(n) = n + round(sqrt(2*n)).
a(a(n)) = n + 2*floor(1/2 + sqrt(2n)) + 1.
a(n) = a(n-1) + A035214(n), a(1)=2.
a(n) = A080036(n) - 1.
a(n) = n + A002024(n). - Vincenzo Librandi, Jul 08 2010
A010054(a(n)) = 0. - Reinhard Zumkeller, Dec 10 2012
From Boris Putievskiy, Jan 14 2013: (Start)
a(n) = A007401(n)+1.
a(n) = A003057(n)^2 - A114327(n).
a(n) = ((t+2)^2 + i - j)/2, where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)
A248952(a(n)) < 0. - Reinhard Zumkeller, Oct 20 2014
a(n) = A256188(A004202(n)). - Reinhard Zumkeller, Mar 26 2015
From Robert Israel, Apr 20 2015 (Start):
a(n) = A118011(n) - n.
G.f.: x/(1-x)^2 + x/(1-x) * Sum(j>=0, x^(j*(j+1)/2)) = x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
G.f. as array: x*y*(2 - 2*y + x^2*y + y^2 - x*(1 + y))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Apr 22 2024

Following Alford Arnold's comment: keyword tabl and correspondent crossrefs added by Reinhard Zumkeller, Dec 12 2012

I restored the original definition. - N. J. A. Sloane, Jan 27 2019

A033429 a(n) = 5*n^2.

Original entry on oeis.org

0, 5, 20, 45, 80, 125, 180, 245, 320, 405, 500, 605, 720, 845, 980, 1125, 1280, 1445, 1620, 1805, 2000, 2205, 2420, 2645, 2880, 3125, 3380, 3645, 3920, 4205, 4500, 4805, 5120, 5445, 5780, 6125, 6480, 6845, 7220, 7605, 8000, 8405, 8820, 9245, 9680, 10125, 10580, 11045, 11520, 12005, 12500

Offset: 0

Jeff Burch

Number of edges of the complete bipartite graph of order 6n, K_n,5n. - Roberto E. Martinez II, Jan 07 2002
Number of edges of the complete tripartite graph of order 4n, K_n,n,2n. - Roberto E. Martinez II, Jan 07 2002
a(n+1)-a(n) : 5, 15, 25, 35, 45, ... (see A017329). - Philippe Deléham, Dec 08 2011
From Larry J Zimmermann, Feb 21 2013: (Start)
The sum of the areas of 2 squares that equals the area of a rectangle with whole number sides using the formula x^2 + y^2 = (x+y+sqrt(2*x*y))(x+y-sqrt(2*x*y)), where the substitution y=2*x obtains the whole number sides of the rectangle. So x^2+(2*x)^2=5x(x).
  x     squares sum      rectangle (l,w)   area
  1     1,4     5                   5,1    5
  2     4,16    20                  10,2   20  (End)

G. C. Greubel, Table of n, a(n) for n = 0..5000
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Central column of A055096.
Cf. A000290.
Cf. numbers of the form  n*(d*n+10-d)/2:  A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. A185019.
Cf. A001105, A033428.
Similar sequences are listed in A316466.

5*Range[50]^2 (* Alonso del Arte, May 23 2012 *)

PARI
```
a(n)=5*n^2
```

a(n) = 5*A000290(n). - Omar E. Pol, Dec 11 2008
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: 5*x*(1+x)/(1-x)^3.
a(n) = 4*A000217(n) + A000567(n). (End)
a(n) = a(n-1)+5*(2*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
a(n) = A131242(10*n+4). - Philippe Deléham, Mar 27 2013
a(n) = a(n-1) + 10*n - 5, with a(0)=0. - Jean-Bernard François, Oct 04 2013
a(n) = A001105(n) + A033428(n). - Altug Alkan, Sep 28 2015
E.g.f.: 5*x*(x+1)*exp(x). - G. C. Greubel, Jul 17 2017
a(n) = Sum_{i = 2..6} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/30.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/60.
Product_{n>=1} (1 + 1/a(n)) = sqrt(5)*sinh(Pi/sqrt(5))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(5)*sin(Pi/sqrt(5))/Pi. (End)

Better description from N. J. A. Sloane, May 15 1998

A054556 a(n) = 4n^2 - 9n + 6.

Original entry on oeis.org

1, 4, 15, 34, 61, 96, 139, 190, 249, 316, 391, 474, 565, 664, 771, 886, 1009, 1140, 1279, 1426, 1581, 1744, 1915, 2094, 2281, 2476, 2679, 2890, 3109, 3336, 3571, 3814, 4065, 4324, 4591, 4866, 5149, 5440, 5739, 6046, 6361, 6684, 7015, 7354, 7701, 8056, 8419, 8790

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Move in 1-4 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 3, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
Ulam's spiral (N spoke). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonpositive m. - Bruno Berselli, Jan 06 2016

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054555, A068225, A054552, A054554, A054567, A054569, A033951.
Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A001105, A058331, A130883.

[4*n^2-9*n+6 : n in [1..50]]; // Vincenzo Librandi, Mar 10 2018

a:=n->4*n^2-9*n+6: seq(a(n),n=1..50); # Muniru A Asiru, Mar 09 2018

a[n_] := 4*n^2 - 9*n + 6; Array[a, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,4,15},50] (* Harvey P. Dale, Sep 06 2015 *)
CoefficientList[Series[-(6x^2 + x + 1)/(x - 1)^3, {x, 0, 49}], x] (* Robert G. Wilson v, Mar 12 2018 *)

a(n)=4*n^2-9*n+6 \\ Charles R Greathouse IV, Sep 24 2015

a(n)^2 = Sum_{i = 0..2*(4*n-5)} (4*n^2-13*n+9+i)^2*(-1)^i = ((n-1)*(4*n-5)+1)^2. - Bruno Berselli, Apr 29 2010
From Harvey P. Dale, Aug 21 2011: (Start)
a(0)=1, a(1)=4, a(2)=15; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(6*x^2+x+1)/(x-1)^3. (End)
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n - 2, 2) + 2*(n - 1)^2 + 1.
a(n) = A000384(n-1) + A058331(n-1).
a(n) = A130883(n-1) + A001105(n-1). (End)
E.g.f.: exp(x)*(6 - 5*x + 4*x^2) - 6. - Stefano Spezia, Apr 24 2024

Edited by Frank Ellermann, Feb 24 2002

Incorrect formula deleted by N. J. A. Sloane, Aug 02 2009

cp's OEIS Frontend

A033991 a(n) = n*(4*n-1).

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A045991 a(n) = n^3 - n^2.

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A028983 Numbers whose sum of divisors is even.

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A002266 Integers repeated 5 times.

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A033991 a(n) = n(4n-1).

A006331 a(n) = n(n+1)(2*n+1)/3.

A054556 a(n) = 4n^2 - 9n + 6.