A214295 -id:A214295 - cp's OEIS frontend

A033428 a(n) = 3*n^2.

0, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, 507, 588, 675, 768, 867, 972, 1083, 1200, 1323, 1452, 1587, 1728, 1875, 2028, 2187, 2352, 2523, 2700, 2883, 3072, 3267, 3468, 3675, 3888, 4107, 4332, 4563, 4800, 5043, 5292, 5547, 5808, 6075, 6348

Offset: 0

Jeff Burch

The number of edges of a complete tripartite graph of order 3n, K_n,n,n. - Roberto E. Martinez II, Oct 18 2001
From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,3,.... The spiral begins:
.
            33--32--31--30
            /             \
          34  16--15--14  29
          /   /         \   \
        35  17   5---4  13  28
        /   /   /     \   \   \
      36  18   6   0---3--12--27--48-->
      /   /   /   /   /   /   /   /
    37  19   7   1---2  11  26  47
      \   \   \         /   /   /
      38  20   8---9--10  25  46
        \   \             /   /
        39  21--22--23--24  45
          \                 /
          40--41--42--43--44
(End)
Number of edges of the complete bipartite graph of order 4n, K_n,3n. - Roberto E. Martinez II, Jan 07 2002
Also the number of partitions of 6n + 3 into at most 3 parts. - R. K. Guy, Oct 23 2003
Also the number of partitions of 6n into exactly 3 parts. - Colin Barker, Mar 23 2015
Numbers n such that the imaginary quadratic field Q[sqrt(-n)] has six units. - Marc LeBrun, Apr 12 2006
The denominators of Hoehn's sequence (recalled by G. L. Honaker, Jr.) and the numerators of that sequence reversed. The sequence is 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11), (1+3+5+7)/(9+11+13+15), ...; reduced to 1/3, 4/12, 9/27, 16/48, ... . For the reversal, the reduction is 3/1, 12/4, 27/9, 48/16, ... . - Enoch Haga, Oct 05 2007
Right edge of tables in A200737 and A200741: A200737(n, A000292(n)) = A200741(n, A100440(n)) = a(n). - Reinhard Zumkeller, Nov 21 2011
The 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). Example: a(3)=27 because G(3) is the cycle C(6) and 6*1 + 6*2 + 3*3 = 27. The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
From Michel Lagneau, May 04 2015: (Start)
Integer area A of equilateral triangles whose side lengths are in the commutative ring Z[3^(1/4)] = {a + b*3^(1/4) + c*3^(1/2) + d*3^(3/4), a,b,c and d in Z}.
The area of an equilateral triangle of side length k is given by A = k^2*sqrt(3)/4. In the ring Z[3^(1/4)], if k = q*3^(1/4), then A = 3*q^2/4 is an integer if q is even. Example: 27 is in the sequence because the area of the triangle (6*3^(1/4), 6*3^(1/4), 6*3^(1/4)) is 27. (End)
a(n) is 2*sqrt(3) times the area of a 30-60-90 triangle with short side n. Also, 3 times the area of an n X n square. - Wesley Ivan Hurt, Apr 06 2016
Consider the hexagonal tiling of the plane. Extract any four hexagons adjacent by edge. This can be done in three ways. Fold the four hexagons so that all opposite faces occupy parallel planes. For all parallel projections of the resulting object, at least two correspond to area a(n) for side length of n of the original hexagons. - Torlach Rush, Aug 17 2022
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(3*n))/(1 + q^(3*n)) = ( Sum_{n in Z} q^(n*(3*n+1)/2) ) / ( Product_{n >= 1} 1 + q^n ) = 1 - 2*q^3 + 2*q^12 - 2*q^27 + 2*q^48 - 2*q^75 + - .... - Peter Bala, Dec 30 2024

			From _Ilya Gutkovskiy_, Apr 13 2016: (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  o o  o  o  o  o  o  o o  o  o  o
. n=1      n=2            n=3                 n=4
(End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21. [Incomplete annotated scanned copy]
Frank Ellermann, Illustration of binomial transforms.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Leo Tavares, Hexagonal illustration
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Eric Weisstein's World of Mathematics, Unit.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A000217, A000290, A033581, A033583, A092205, A092206.
3 times n-gonal numbers: A045943, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Cf. A219056.
Cf. A000326, A005449, A086463.
Cf. A003215, A016777.

a033428 = (* 3) . (^ 2)
a033428_list = 0 : 3 : 12 : zipWith (+) a033428_list
   (map (* 3) $ tail $ zipWith (-) (tail a033428_list) a033428_list)
-- Reinhard Zumkeller, Jul 11 2013

[3*n^2: n in [0..50]]; // Vincenzo Librandi, May 18 2015

seq(3*n^2, n=0..46); # Nathaniel Johnston, Jun 26 2011

3 Range[0, 50]^2
LinearRecurrence[{3, -3, 1}, {0, 3, 12}, 50] (* Harvey P. Dale, Feb 16 2013 *)

makelist(3*n^2,n,0,30); /* Martin Ettl, Nov 12 2012 */

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

def a(n): return 3 * (n**2) # Torlach Rush, Aug 25 2022

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
G.f.: 3*x*(1+x)/(1-x)^3. - R. J. Mathar, Sep 09 2008
Main diagonal of triangle in A132111: a(n) = A132111(n,n). - Reinhard Zumkeller, Aug 10 2007
A214295(a(n)) = -1. - Reinhard Zumkeller, Jul 12 2012
a(n) = A215631(n,n) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) = A174709(6n+2). - Philippe Deléham, Mar 26 2013
a(n) = a(n-1) + 6*n - 3, with a(0)=0. - Jean-Bernard François, Oct 04 2013
E.g.f.: 3*x*(1 + x)*exp(x). - Ilya Gutkovskiy, Apr 13 2016
a(n) = t(3*n) - 3*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): A000217(3*n) - 3*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = A000326(n) + A005449(n). - Bruce J. Nicholson, Jan 10 2020
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36. (End)
From Amiram Eldar, Feb 03 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sqrt(3)*sinh(Pi/sqrt(3))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(3)*sin(Pi/sqrt(3))/Pi. (End)
a(n) = A003215(n) - A016777(n). - Leo Tavares, Apr 29 2023

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

A092206 Positive integers that are not of the form n^2 or 3n^2.

Original entry on oeis.org

2, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Eric W. Weisstein, Feb 24 2004

nonn

Values of n such that Q(sqrt(-n)) has 2 units.

A214295(a(n)) = 0. - Reinhard Zumkeller, Jul 12 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unit

Cf. A000290, A033428, A092205.

a092206 n = a092206_list !! (n-1)
a092206_list = filter ((== 0) . a214295) [1..]
-- Reinhard Zumkeller, Jul 12 2012

Select[ Range[100], Not[ IntegerQ[ Sqrt[#] ] || IntegerQ[ Sqrt[#/3] ] ]&] (* Jean-François Alcover, Oct 30 2012 *)

is(n)=!issquare(n/3^valuation(n,3)) \\ Charles R Greathouse IV, Oct 30 2012

A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

Original entry on oeis.org

1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 10 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + q^4 - q^5 + q^9 + q^16 - q^20 + q^25 + q^36 - q^45 + q^49 + q^64 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A214284, A214295, A214960, A244612, A245485, A322159.

Basis( ModularForms( Gamma1(20), 1/2), 65) [2]; /* Michael Somos, Jul 01 2014 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^5]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [5 n]]]];

PARI
```
{a(n) = issquare(n) - issquare(5*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==5, 1 - X, 1) / (1 - X^2 ))[n])};

Expansion of (phi(q) - phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 24 2013
Expansion of q * f(q^3, q^7) * f(-q^4, -q^16) / f(-q^8, -q^12) in powers of q where f() is Ramanujan's two-variable theta function.
Expansion of q * f(x, x^9) * f(-q, -q^4) / f(-q^2, -q^3) in powers of q where f() is Ramanujan's two-variable theta function. - Michael Somos, Sep 24 2013
Euler transform of period 20 sequence [ 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, -1, ...].
Multiplicative with a(5^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^5)) / 2 = Sum_{k>0} x^(k^2) - x^(5*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 5^-s).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = a(n). a(5*n) = -a(n).
a(4*n) = A214293(n). a(4*n+1) = A214960(n). - Michael Somos, Sep 24 2013
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(5) = 0.5527864... (A322159). - Amiram Eldar, Oct 24 2023

A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

Original entry on oeis.org

1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 01 2014

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + q^4 - q^6 + q^9 + q^16 - q^24 + q^25 + q^36 + q^49 - q^54 + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A089801, A214293, A214295, A245485.

Basis( ModularForms( Gamma1(24), 1/2), 64) [2];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^6]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [6 n]]]];

PARI
```
{a(n) = issquare(n) - issquare(6*n)};
```

Expansion of (phi(q) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
G.f.: (theta_3(q) - theta_3(q^6)) / 2 = Sum_{k>0} x^(k^2) - x^(6*k^2).
a(3*n) = A089801(n). a(3*n + 2) = 0.
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(6) = 0.5917517... . - Amiram Eldar, Oct 24 2023

More terms from Antti Karttunen, Dec 15 2017

A245485 a(n) = 1 if n is a square, -1 if n is seven times a square, 0 otherwise.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Michael Somos, Jul 23 2014

			G.f. = q + q^4 - q^7 + q^9 + q^16 + q^25 - q^28 + q^36 + q^49 - q^63 + ...

Antti Karttunen, Table of n, a(n) for n = 1..100489

Cf. A214293, A214295, A244612.

Basis( ModularForms( Gamma1(28), 1/2), 86) [2] ;

a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 7 n]]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^7]) / 2, {q, 0, n}];
Table[Which[IntegerQ[Sqrt[n]],1,IntegerQ[Sqrt[n/7]],-1,True,0],{n,120}] (* Harvey P. Dale, May 19 2019 *)

PARI
```
{a(n) = issquare(n) - issquare(7*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==7, 1 - X, 1) / (1 - X^2 ))[n])};

a(n) is multiplicative with a(p^e) = (-1)^e if p = 7, a(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f.: (theta_3(q) - theta_3(q^7)) / 2 = Sum_{k>0} x^(k^2) - x^(7*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 7^(-*s)). [corrected by Amiram Eldar, Oct 24 2023]
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(7) = 0.6220355... . - Amiram Eldar, Oct 24 2023

Data section extended up to a(112) by Antti Karttunen, Jul 02 2024

A214505 a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.

Original entry on oeis.org

1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Michael Somos, Jul 19 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x + x^4 + x^12 - x^13 + x^24 - x^37 + x^40 + x^60 - x^73 + x^84 + ...
G.f. = q - q^3 + q^9 + q^25 - q^27 + q^49 - q^75 + q^81 + q^121 - q^147 + q^169 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
S. Cooper and M. Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 5.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010052, A214295.

{a(n) = n = 2*n + 1; issquare(n) - issquare(3*n)};

Expansion of psi(x^4) - x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.
Expansion of f(-x, x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
Euler transform of period 24 sequence [ -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, ...].
G.f.: (Sum_{k} x^(2*k*(k + 1)) - x^(6*k*(k + 1) + 1)) / 2.
a(n) = A214295(2*n + 1).

cp's OEIS Frontend

A033428 a(n) = 3*n^2.

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A092206 Positive integers that are not of the form n^2 or 3n^2.

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A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

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A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

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A245485 a(n) = 1 if n is a square, -1 if n is seven times a square, 0 otherwise.

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Magma

Mathematica

PARI

PARI

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A214505 a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.

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