A035608 -id:A035608 - cp's OEIS frontend

A002943 a(n) = 2n(2*n+1).

0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190, 8556, 8930

Offset: 0

N. J. A. Sloane

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel, Jan 12 2000
In other words, the edge count of the (n+1) X (n+1) king graph. - Eric W. Weisstein, Jun 20 2017
Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2 - a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012
Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 27 2010
The hyper-Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1 <= i,j <= n, i != j} (= the complete bipartite graph K(n,n) with horizontal edges removed). The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
Sum of the numbers from n to 3n. - Wesley Ivan Hurt, Oct 27 2014

			64--65--66--67--68--69--70--71--72
|
63  36--37--38--39--40--41--42
|   |                       |
62  35  16--17--18--19--20  43
|   |   |               |   |
61  34  15   4---5---6  21  44
|   |   |    |       |  |   |
60  33  14   3   0   7  22  45
|   |   |    |   |   |  |   |
59  32  13   2---1   8  23  46
|   |   |            |  |   |
58  31  12--11--10---9  24  47
|   |                   |   |
57  30--29--28--27--26--25  48
|                           |
56--55--54--53--52--51--50--49

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

T. D. Noe, Table of n, a(n) for n = 0..1000
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Leo Tavares, Illustration: Twin Diamond Stars.
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001477, A007395, A007494, A007742, A014105, A016813, A033954, A045896, A046092, A054000, A118729, A173511.
Same as A033951 except start at 0.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, this sequence, 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, this sequence = 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. A003154, A056220, A002939.

a002943 n = 2 * n * (2 * n + 1)  -- Reinhard Zumkeller, Jan 12 2014

[ 4*n^2+2*n: n in [0..50]]; // Vincenzo Librandi, Nov 25 2012

A002943 := proc(n)
    2*n*(2*n+1) ;
end proc: # R. J. Mathar, Jun 28 2013

LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* Harvey P. Dale, Aug 11 2011 *)
Table[2 n (2 n + 1), {n, 0, 40}] (* Harvey P. Dale, Aug 11 2011 *)

a(n)=2*n*(2*n+1) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = 4*n^2 + 2*n.
a(n) = 2*A014105(n). - Omar E. Pol, May 21 2008
a(n) = floor((2*n + 1/2)^2). - Reinhard Zumkeller, Feb 20 2010
a(n) = A007494(n) + A173511(n) = A007742(n) + n. - Reinhard Zumkeller, Feb 20 2010
a(n) = 8*n+a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 11 2011
a(n+1) = A045896(2*n+1). - Reinhard Zumkeller, Dec 12 2011
G.f.: 2*x*(3+x)/(1-x)^3. - Colin Barker, Jan 14 2012
From R. J. Mathar, Jan 15 2013: (Start)
Sum_{n>=1} 1/a(n) = 1 - log(2).
Sum_{n>=1} 1/a(n)^2 = 2*log(2) + Pi^2/6 - 3. (End)
a(n) = A118729(8*n+5). - Philippe Deléham, Mar 26 2013
a(n) = 1*A001477(n) + 2*A000217(n) + 3*A000290(n). - J. M. Bergot, Apr 23 2014
a(n) = 2 * A000217(2*n) = 2 * A014105(n). - Jon Perry, Oct 27 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - Amiram Eldar, Feb 22 2022
a(n) = A003154(n+1) - A056220(n+1). - Leo Tavares, Mar 31 2022
E.g.f.: 2*exp(x)*x*(3 + 2*x). - Stefano Spezia, Apr 24 2024
a(n) = A002939(-n) for all n in Z. - Charles Kusniec, Aug 12 2025

Formula fixed by Reinhard Zumkeller, Apr 09 2010

A007742 a(n) = n(4n+1).

Original entry on oeis.org

0, 5, 18, 39, 68, 105, 150, 203, 264, 333, 410, 495, 588, 689, 798, 915, 1040, 1173, 1314, 1463, 1620, 1785, 1958, 2139, 2328, 2525, 2730, 2943, 3164, 3393, 3630, 3875, 4128, 4389, 4658, 4935, 5220, 5513, 5814, 6123, 6440, 6765, 7098, 7439, 7788, 8145

Offset: 0

N. J. A. Sloane

Write 0,1,2,... in a clockwise spiral; sequence gives the numbers that fall on the positive y-axis. (See Example section.)
Central terms of the triangle in A126890. - Reinhard Zumkeller, Dec 30 2006
a(n)*Pi is the total length of 4 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A004770. The spiral length ratio rounded down [floor(L(n)/L(1))] is A047497. See illustration in links. - Kival Ngaokrajang, Dec 27 2013
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n; {4, 4n}]. For n=1, this collapses to [2, {4}]. - Magus K. Chu, Sep 15 2022

			Part of the spiral:
.
  64--65--66--67--68
   |
  63  36--37--38--39--40--41--42
   |   |                       |
  62  35  16--17--18--19--20  43
   |   |   |               |   |
  61  34  15   4---5---6  21  44
   |   |   |   |       |   |   |
  60  33  14   3   0   7  22  45
   |   |   |   |   |   |   |   |
  59  32  13   2---1   8  23  46
   |   |   |           |   |   |
  58  31  12--11--10---9  24  47
   |   |                   |   |
  57  30--29--28--27--26--25  48
   |                           |
  56--55--54--53--52--51--50--49

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Emilio Apricena, A version of the Ulam spiral
Robert FERREOL, Illustration by pentagons
Kival Ngaokrajang, Illustration of 4 points circle center spiral
Leo Tavares, Illustration: Triangular Layers
G. Thimm, Emails to N. J. A. Sloane, Sep. 1994
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033991, A074378.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, 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. index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.
Cf. A081266.

I:=[0, 5, 18]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012

LinearRecurrence[{3,-3,1},{0,5,18},50] (* Vincenzo Librandi, Jan 29 2012 *)
Table[n(4n+1),{n,0,50}] (* Harvey P. Dale, Aug 10 2017 *)

PARI
```
a(n)=4*n^2+n
```

G.f.: x*(5+3*x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = A033991(-n) = A074378(2*n).
a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Feb 20 2010
a(n) = A110654(n) + A173511(n) = A002943(n) - n. - Reinhard Zumkeller, Feb 20 2010
a(n) = 8*n + a(n-1) - 3. - Vincenzo Librandi, Nov 21 2010
Sum_{n>=1} 1/a(n) = Sum_{k>=0} (-1)^k*zeta(2+k)/4^(k+1) = 0.349762131... . - R. J. Mathar, Jul 10 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=18. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+4). - Philippe Deléham, Mar 26 2013
a(n) = A000217(3*n) - A000217(n). - Bruno Berselli, Sep 21 2016
E.g.f.: (4*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = 4 - Pi/2 - 3*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(2) + log(2) + sqrt(2)*log(1 + sqrt(2)) - 4. (End)
a(n) = A081266(n) - A000217(n). - Leo Tavares, Mar 25 2022

A053755 a(n) = 4*n^2 + 1.

Original entry on oeis.org

1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, 1025, 1157, 1297, 1445, 1601, 1765, 1937, 2117, 2305, 2501, 2705, 2917, 3137, 3365, 3601, 3845, 4097, 4357, 4625, 4901, 5185, 5477, 5777, 6085, 6401, 6725, 7057

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 06 2000

Subsequence of A004613: all numbers in this sequence have all prime factors of the form 4k+1. E.g., 40001 = 13*17*181, 13 = 4*3 + 1, 17 = 4*4 + 1, 181 = 4*45 + 1. - Cino Hilliard, Aug 26 2006, corrected by Franklin T. Adams-Watters, Mar 22 2011
A000466(n), A008586(n) and a(n) are Pythagorean triples. - Zak Seidov, Jan 16 2007
Solutions x of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0, 1, 2, ... - Michel Lagneau, Feb 12 2010
Ulam's spiral (NW spoke). - Robert G. Wilson v, Oct 31 2011
For n >= 1, a(n) is numerator of radius r(n) of circle with sagitta = n and cord length = 1. The denominator is A008590(n). - Kival Ngaokrajang, Jun 13 2014
a(n)+6 is prime for n = 0..6 and for n = 15..20. - Altug Alkan, Sep 28 2015

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1 Nr. 11, p. 19.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Kival Ngaokrajang, Illustration of initial terms.
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).

Column 2 of array A188647.
Cf. A016742, A256970 (smallest prime factors), A214345.
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. A000466, A001844, A004613, A008586, A008590, A017113, A046092, A087475, A156701, A176271.

List([0..45],n->4*n^2+1); # Muniru A Asiru, Nov 01 2018

a053755 = (+ 1) . (* 4) . (^ 2)  -- Reinhard Zumkeller, Apr 20 2015

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+5*x^2)/((1-x)^3))); /* or */ I:=[1,5]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+8: n in [1..50]]; // Vincenzo Librandi, Jun 26 2013

with (combinat):seq(fibonacci(3,2*n), n=0..42); # Zerinvary Lajos, Apr 21 2008

f[n_] := 4n^2 +1; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[(1 + 2 x + 5 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2013 *)
LinearRecurrence[{3,-3,1},{1,5,17},50] (* Harvey P. Dale, Dec 28 2021 *)

for(x=0,100,print1(4*x^2+1",")) \\ Cino Hilliard, Aug 26 2006

for n in range(0,50): print(4*n**2+1, end=', ') # Stefano Spezia, Nov 01 2018

a(n) = A000466(n) + 2. - Zak Seidov, Jan 16 2007
From R. J. Mathar, Apr 28 2008: (Start)
O.g.f.: (1 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 3a(n-1) - 3a(n-2) + a(n-3). (End)
Equals binomial transform of [1, 4, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = A156701(n)/A087475(n). - Reinhard Zumkeller, Feb 13 2009
For n>0: a(n) = A176271(2*n,n+1); cf. A016754, A000466. - Reinhard Zumkeller, Apr 13 2010
a(n+1) = denominator of Sum_{k=0..n} (-1)^n*(2*n + 1)^3/((2*n + 1)^4 + 4), see Knuth reference. - Reinhard Zumkeller, Apr 11 2010
a(n) = 8*n + a(n-1) - 4. with a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n) = ((2*n - 1)^2 + (2*n + 1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = 2*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=5. - Vincenzo Librandi, Jun 26 2013
a(n+1) = a(n) + A017113(n), a(0) = 1. - Altug Alkan, Sep 26 2015
a(n) = A001844(n) + A046092(n-1) = A001844(n-1) + A046092(n). - Bruce J. Nicholson, Aug 07 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/2)*coth(Pi/2))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/2)*csch(Pi/2))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/2)*sinh(Pi/sqrt(2)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/2)*csch(Pi/2). (End)
E.g.f.: exp(x)*(1 + 2*x)^2. - Stefano Spezia, Jun 10 2021

Equation corrected, and examples that were based on a different offset removed, by R. J. Mathar, Mar 18 2010

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

Original entry on oeis.org

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

A033951 Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.

Original entry on oeis.org

1, 8, 23, 46, 77, 116, 163, 218, 281, 352, 431, 518, 613, 716, 827, 946, 1073, 1208, 1351, 1502, 1661, 1828, 2003, 2186, 2377, 2576, 2783, 2998, 3221, 3452, 3691, 3938, 4193, 4456, 4727, 5006, 5293, 5588, 5891, 6202, 6521, 6848, 7183, 7526, 7877, 8236, 8603, 8978

Offset: 0

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Ulam's spiral (S spoke of A054552). - Robert G. Wilson v, Oct 31 2011

a(n) is the first term in a sum of 2*n + 1 consecutive integers that equals (2*n + 1)^3. - Patrick J. McNab, Dec 24 2016

			Spiral begins:
.
  65--66--67--68--69--70--71--72--73
   |                               |
  64  37--38--39--40--41--42--43  74
   |   |                       |   |
  63  36  17--18--19--20--21  44  75
   |   |   |               |   |   |
  62  35  16   5---6---7  22  45  76
   |   |   |   |       |   |   |   |
  61  34  15   4   1   8  23  46  77
   |   |   |   |   |   |   |   |
  60  33  14   3---2   9  24  47
   |   |   |           |   |   |
  59  32  13--12--11--10  25  48
   |   |                   |   |
  58  31--30--29--28--27--26  49
   |                           |
  57--56--55--54--53--52--51--50
From _Aaron David Fairbanks_, Mar 06 2025: (Start)
Illustration of initial terms:
                                            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        8              23                   46
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
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).

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, 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. A014848, A132774.

A033951:=n->4*n^2 + 3*n + 1: seq(A033951(n), n=0..100); # Wesley Ivan Hurt, Feb 11 2017

lst={};Do[p=4*n^2+3*n+1;AppendTo[lst, p], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,8,23},60] (* Harvey P. Dale, Feb 07 2015 *)
CoefficientList[Series[(1 + 5 x + 2 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Feb 12 2017 *)

PARI
```
a(n)=4*n^2+3*n+1
```

[4*n**2 + 3*n + 1 for n in range(46)] # Michael S. Branicky, Jan 08 2021

a(n) = 4*n^2 + 3*n + 1.
G.f.: (1 + 5*x + 2*x^2)/(1-x)^3.
A014848(2n+1) = a(n).
Equals A132774 * [1, 2, 3, ...]; = binomial transform of [1, 7, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 28 2007
a(n) = A016754(n) - n. - Reinhard Zumkeller, May 17 2009
a(n) = a(n-1) + 8*n-1 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
a(0)=1, a(1)=8, a(2)=23, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 07 2015
E.g.f.: exp(x)*(1 + 7*x + 4*x^2). - Stefano Spezia, Apr 24 2024

Extended (with formula) by Erich Friedman

A054552 a(n) = 4n^2 - 3n + 1.

Original entry on oeis.org

1, 2, 11, 28, 53, 86, 127, 176, 233, 298, 371, 452, 541, 638, 743, 856, 977, 1106, 1243, 1388, 1541, 1702, 1871, 2048, 2233, 2426, 2627, 2836, 3053, 3278, 3511, 3752, 4001, 4258, 4523, 4796, 5077, 5366, 5663, 5968, 6281, 6602, 6931, 7268, 7613, 7966, 8327

Offset: 0

Enoch Haga and G. L. Honaker, Jr., Apr 09 2000

Also indices in any square spiral organized like A054551.
Equals binomial transform of [1, 1, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 11 2008
Ulam's spiral (E spoke). - Robert G. Wilson v, Oct 31 2011
For n > 0: left edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012

			The spiral begins:
.
197-196-195-194-193-192-191-190-189-188-187-186-185-184-183
  |                                                       |
198 145-144-143-142-141-140-139-138-137-136-135-134-133 182
  |   |                                               |   |
199 146 101-100--99--98--97--96--95--94--93--92--91 132 181
  |   |   |                                       |   |   |
200 147 102  65--64--63--62--61--60--59--58--57  90 131 180
  |   |   |   |                               |   |   |   |
201 148 103  66  37--36--35--34--33--32--31  56  89 130 179
  |   |   |   |   |                       |   |   |   |   |
202 149 104  67  38  17--16--15--14--13  30  55  88 129 178
  |   |   |   |   |   |               |   |   |   |   |   |
203 150 105  68  39  18   5---4---3  12  29  54  87 128 177
  |   |   |   |   |   |   |       |   |   |   |   |   |   |
204 151 106  69  40  19   6   1---2  11  28  53  86 127 176
  |   |   |   |   |   |   |           |   |   |   |   |   |
205 152 107  70  41  20   7---8---9--10  27  52  85 126 175
  |   |   |   |   |   |                   |   |   |   |   |
206 153 108  71  42  21--22--23--24--25--26  51  84 125 174
  |   |   |   |   |                           |   |   |   |
207 154 109  72  43--44--45--46--47--48--49--50  83 124 173
  |   |   |   |                                   |   |   |
208 155 110  73--74--75--76--77--78--79--80--81--82 123 172
  |   |   |                                           |   |
209 156 111-112-113-114-115-116-117-118-119-120-121-122 171
  |   |                                                   |
210 157-158-159-160-161-162-163-164-165-166-167-168-169-170
  |
211-212-213-214-215-216-217-218-219-220-221-222-223-224-225
.
- _Robert G. Wilson v_, Jul 04 2014

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Scientific American, Cover of the March 1964 issue
Leo Tavares, Illustration: Hexagon/Square Pairs
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033293, A054551, A108781.
Spokes of square spiral: A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.
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. A003215.

[4*n^2-3*n+1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 11 2014

A054552:=n->4*n^2-3*n+1: seq(A054552(n), n=0..50); # Wesley Ivan Hurt, Jul 11 2014

f[n_] := 4*n^2 - 3*n + 1; Array[f, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,2,11},50] (* Harvey P. Dale, Jun 01 2024 *)

a(n)= 4*n^2-3*n+1 \\ Charles R Greathouse IV, Jan 15 2012

G.f.: (1 - x + 8*x^2)/(1-x)^3.
a(n) = 8*n + a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 07 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=11. - Harvey P. Dale, Oct 10 2011
E.g.f.: exp(x)*(1 + x + 4*x^2). - Stefano Spezia, May 14 2021
a(n) = A003215(n-1) + A000290(n). - Leo Tavares, Jul 21 2022

A033954 Second 10-gonal (or decagonal) numbers: n(4n+3).

Original entry on oeis.org

0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451, 3690, 3937, 4192, 4455, 4726, 5005, 5292, 5587, 5890, 6201, 6520, 6847, 7182, 7525, 7876, 8235

Offset: 0

N. J. A. Sloane

Same as A033951 except start at 0. See example section.

Bisection of A074377. Also sequence found by reading the line from 0, in the direction 0, 22, ... and the line from 7, in the direction 7, 45, ..., in the square spiral whose vertices are the generalized 10-gonal numbers A074377. - Omar E. Pol, Jul 24 2012

			  36--37--38--39--40--41--42
   |                       |
  35  16--17--18--19--20  43
   |   |               |   |
  34  15   4---5---6  21  44
   |   |   |       |   |   |
  33  14   3   0===7==22==45==76=>
   |   |   |   |   |   |
  32  13   2---1   8  23
   |   |           |   |
  31  12--11--10---9  24
   |                   |
  30--29--28--27--26--25

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.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Leo Tavares, Illustration: V numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002620, A033951.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, 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.
Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705.
Cf. A060544.

List([0..50], n-> n*(4*n+3)) # G. C. Greubel, May 24 2019

[n*(4*n+3): n in [0..50]]; // G. C. Greubel, May 24 2019

Table[n(4n+3),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,22},50] (* Harvey P. Dale, May 06 2018 *)

PARI
```
a(n)=4*n^2+3*n
```

[n*(4*n+3) for n in (0..50)] # G. C. Greubel, May 24 2019

a(n) = A001107(-n) = A074377(2*n).
G.f.: x*(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = a(n-1) + 8*n - 1 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
For n>0, Sum_{j=0..n} (a(n) + j)^4 + (4*A000217(n))^3 = Sum_{j=n+1..2n} (a(n) + j)^4; see also A045944. - Charlie Marion, Dec 08 2007, edited by Michel Marcus, Mar 14 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 22. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+6). - Philippe Deléham, Mar 26 2013
a(n) = A002943(n) + n = A007742(n) + 2n = A016742(n) + 3n = A033991(n) + 4n = A002939(n) + 5n = A001107(n) + 6n = A033996(n) - n. - Philippe Deléham, Mar 26 2013
Sum_{n>=1} 1/a(n) = 4/9 + Pi/6 - log(2) = 0.2748960394827980081... . - Vaclav Kotesovec, Apr 27 2016
E.g.f.: exp(x)*x*(7 + 4*x). - Stefano Spezia, Jun 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 - 4/9 - sqrt(2)*arcsinh(1)/3. - Amiram Eldar, Nov 28 2021
For n>0, (a(n)^2 + n)/(a(n) + n) = (4*n + 1)^2/4, a ratio of two squares. - Rick L. Shepherd, Feb 23 2022
a(n) = A060544(n+1) - A000217(n+1). - Leo Tavares, Mar 31 2022

A054569 a(n) = 4n^2 - 6n + 3.

Original entry on oeis.org

1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163, 2353, 2551, 2757, 2971, 3193, 3423, 3661, 3907, 4161, 4423, 4693, 4971, 5257, 5551, 5853, 6163, 6481, 6807, 7141, 7483, 7833, 8191

Offset: 1

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

Move in 1-7 direction in a spiral organized like A068225 etc.
Third row of A082039. - Paul Barry, Apr 02 2003
Inverse binomial transform of A036826. - Paul Barry, Jun 11 2003
Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - Johannes W. Meijer, May 20 2011
Ulam's spiral (SW spoke). - Robert G. Wilson v, Oct 31 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000
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. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.
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.

List([1..50], n-> 4*n^2-6*n+3); # G. C. Greubel, Jul 04 2019

[4*n^2-6*n+3: n in [1..50]]; // G. C. Greubel, Jul 04 2019

f[n_]:= 4*n^2-6*n+3; Array[f, 50] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,7,21},50] (* Harvey P. Dale, Nov 17 2012 *)

a(n)=4*n^2-6*n+3 \\ Charles R Greathouse IV, Sep 24 2015

[4*n^2-6*n+3 for n in (1..50)] # G. C. Greubel, Jul 04 2019

a(n+1) = 4*n^2 + 2*n + 1. - Paul Barry, Apr 02 2003
a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - Paul Barry, Jun 11 2003
Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Mar 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1+x)*(1+3*x)/(1-x)^3. (End)
a(n) = A000384(n) + A000384(n-1). - Bruce J. Nicholson, May 07 2017
E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - G. C. Greubel, Jul 04 2019
Sum_{n>=1} 1/a(n) = A339237. - R. J. Mathar, Jan 22 2021

Edited by Frank Ellermann, Feb 24 2002

A054554 a(n) = 4n^2 - 10n + 7.

Original entry on oeis.org

1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071, 2257, 2451, 2653, 2863, 3081, 3307, 3541, 3783, 4033, 4291, 4557, 4831, 5113, 5403, 5701, 6007, 6321, 6643, 6973, 7311, 7657, 8011, 8373, 8743

Offset: 1

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

Move in 1-3 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008
Ulam's spiral (NE spoke). - Robert G. Wilson v, Oct 31 2011
Number of ternary strings of length 2*(n-1) that have one or no 0's, one or no 1's, and an even number of 2's. For n=2, the 3 strings of length 2 are 01, 10 and 22. For n=3, the 13 strings of length 4 are the 12 permutations of 0122 and 2222. - Enrique Navarrete, Jul 25 2025

Ivan Panchenko, Table of n, a(n) for n = 1..1000
James Grime and Brady Haran, Prime Spirals, Numberphile video (2013).
Scientific American, Cover of the March 1964 issue
Leo Tavares, Illustration: Hexagonal Dual Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054553, A068225, A054552, A054556, A054567, A054569, A033951.
Cf. A014105.
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. A003215, A227776.

A054554:=n->4*n^2 -10*n + 7; seq(A054554(k),k=1..100); # Wesley Ivan Hurt, Nov 05 2013

f[n_] := 4n^2 -10n + 7; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,3,13},60] (* Harvey P. Dale, Jan 20 2025 *)

a(n)=4*n^2-10*n+7 \\ Charles R Greathouse IV, Nov 05 2013

a(n) = 8*n + a(n-1) - 14 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: -x*(7*x^2+1)/(x-1)^3. - Colin Barker, Sep 21 2012
For n > 2, a(n) = A014105(n) + A014105(n-1). - Bruce J. Nicholson, May 07 2017
From Leo Tavares, Feb 21 2022: (Start)
a(n) = A003215(n-2) + 2*A000217(n-1). See Hexagonal Dual Rays illustration in links.
a(n) = A227776(n-1) - 4*A000217(n-1). (End)
a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - Frank Ellermann, Mar 11 2020
E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - Stefano Spezia, Apr 24 2024

Edited by Frank Ellermann, Feb 24 2002

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

A002943 a(n) = 2*n*(2*n+1).

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A007742 a(n) = n*(4*n+1).

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A053755 a(n) = 4*n^2 + 1.

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

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A033951 Write 1,2,... in a clockwise spiral; sequence gives numbers on positive x axis.

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A002943 a(n) = 2n(2*n+1).

A007742 a(n) = n(4n+1).

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

A054552 a(n) = 4n^2 - 3n + 1.

A033954 Second 10-gonal (or decagonal) numbers: n(4n+3).

A054569 a(n) = 4n^2 - 6n + 3.

A054554 a(n) = 4n^2 - 10n + 7.

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