



1, 1, 7, 17, 31, 49, 71, 97, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 647, 721, 799, 881, 967, 1057, 1151, 1249, 1351, 1457, 1567, 1681, 1799, 1921, 2047, 2177, 2311, 2449, 2591, 2737, 2887, 3041, 3199, 3361, 3527, 3697, 3871, 4049, 4231, 4417
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OFFSET

0,3


COMMENTS

Image of squares (A000290) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2  c_{n+1}*c_{n1}.
Also surround numbers of an n X n square.  Jason Earls (zevi_35711(AT)yahoo.com), Apr 16 2001
Also 8n + 8 is a square.  Cino Hilliard, Dec 18 2003
The sums of the consecutive integer sequences 2n^2 to 2(n+1)^21 are cubes, as 2n^2+...+2(n+1)^21 = (1/2)(2(n+1)^212n^2+1)(2(n+1)^21+2n^2)=(2n+1)^3. E.g., 2+3+4+5+6+7 = 27 =3^3, then 8+9+10+..+17 = 125 = 5^3.  Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 29 2005
Sequence allows us to find X values of the equation: 2*X^3 + 2*X^2 = Y^2. To find Y values: b(n)=2n(2*n^2  1).  Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
Average of the squares of two consecutive terms is also a square. In fact: (2*n^2  1)^2 + (2*(n+1)^2  1)^2 = 2*(2*n^2 + 2*n + 1)^2.  Matias Saucedo (solomatias(AT)yahoo.com.ar), Aug 18 2008
Equals row sums of triangle A143593 and binomial transform of [1, 6, 4, 0, 0, 0,...] with n>1.  Gary W. Adamson, Aug 26 2008
Sqrt(a(n) + a(n+1) + 1) = 2n+1.  Doug Bell (bell.doug(AT)gmail.com), Mar 09 2009
Apart the first term which is 1 the number of units of a(n) belongs to a periodic sequence: 1, 7, 7, 1, 9. We conclude that a(n) and a(n+5) have the same number of units.  Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009
Start a spiral of square tiles. Trivially the first tile fits in a 1 X 1 square. 7 tiles fit in a 3 X 3 square, 17 tiles fit in a 5 X 5 square and so on.  Juhani Heino, Dec 13 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2, A[i,i1]=1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=coeff(charpoly(A,x),x^(n2)).  Milan Janjic, Jan 26 2010
For each n>0, the recursive series, formula S(b) = 6*S(b1)  S(b2)  2*a(n) with S(0) = 4n^24n+1 and S(1) = 2n^2, has the property that every even term is a perfect square and every odd term is twice a perfect square.  Kenneth J Ramsey, Jul 18 2010
Also, fourth diagonal of A154685 for n>2.  Vincenzo Librandi, Aug 07 2010
Also first integer of (2*n)^2 consecutive integers, where the last integer is 3 times the first + 1. As example, n = 2: term = 7; (2*n)^2 = 16; 7, 8, 9,..., 20, 21, 22 : 7*3 + 1 = 22.  Denis Borris, Nov 18 2012
For n > 0: a(n) = A162610(2*n1,n).  Reinhard Zumkeller, Jan 19 2013
Chebyshev polynomial of the first kind T(2,n).  Vincenzo Librandi, May 30 2014
For n>3 a(n)=sum[(C(n+k,3)(C(n+k1,3))*(C(n+k,3)+C(n+k+1,3)) {0<=k<=2}](C(n+2,3)C(n1,3))*(C(n,3)+C(n+3,3)).  J. M. Bergot, Jun 16 2014


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Mitch Phillipson, Manda Riehl and Tristan Williams, Enumeration of Wilf classes in Sn ~ Cr for two patterns of length 3, PU. M. A. Vol. 21 (2010), No. 2, pp. 321338.
Index to sequences with linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

G.f.: (1+4x+x^2)/(1x)^3.
a(n) = A119258(n+1,2) for n>0.  Reinhard Zumkeller, May 11 2006
From Doug Bell (bell.doug(AT)gmail.com), Mar 08 2009: (Start)
a(0) = 1,
a(n) = sqrt(A001844(n)^2  A069074(n1)),
a(n+1) = sqrt(A001844(n)^2 + A069074(n1)) = sqrt(a(n)^2 + A069074(n1)*2). (End)
a(n) = a(n1)+4*n2 (with a(0)=1).  Vincenzo Librandi, Dec 25 2010
a(n) = A188653(2*n) for n>0.  Reinhard Zumkeller, Apr 13 2011


EXAMPLE

a(0) = 0^21*1 = 1, a(1) = 1^24*0 = 1, a(2) = 2^29*1 = 7, etc.
a(4) = 31 = (1, 3, 3, 1) dot (1, 6, 4, 0) = (1 + 18 + 12 + 0).  Gary W. Adamson, Aug 29 2008


MAPLE

A056220:=n>2*n^21; seq(A056220(n), n=0..50); # Wesley Ivan Hurt, Jun 16 2014


MATHEMATICA

Table[2*n^2 + 4*n + 1, {n, 1, 46}] (* Zerinvary Lajos, Jul 10 2009 *)
Table[ChebyshevT[2, n], {n, 0, 60}] (* Vincenzo Librandi, May 30 2014 *)


PROG

(PARI) a(n)=if(n<0, 0, 2*n^21)
(MAGMA) [2*n^21 : n in [0..50]]; // Vincenzo Librandi, May 30 2014


CROSSREFS

Cf. A047875, A000105, A077585, A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475, A000217.
Cf. A143593.  Gary W. Adamson, Aug 26 2008
Column 2 of array A188644 (starting at offset 1).
Cf. A001653.
Sequence in context: A046118 A120092 A130284 * A024840 A024835 A225251
Adjacent sequences: A056217 A056218 A056219 * A056221 A056222 A056223


KEYWORD

sign,easy


AUTHOR

N. J. A. Sloane, Aug 06 2000


EXTENSIONS

Formula and additional comments from Henry Bottomley, Dec 12 2000


STATUS

approved



