OFFSET
1,1
COMMENTS
The sum of the squares of two consecutive terms multiplied (or divided) by 2 is always a perfect square. In general, numbers represented by the quadratic form a(n) = (2*i*n + j)^2 - 2*i^2 for any i and j have 2(a(n)^2 + a(n+1)^2)) and (a(n)^2 + a(n+1)^2)/2 as perfect squares: in this case, i=j=1.
The terms of this sequence may be seen to be 2 less than the odd squares. As such they run parallel to those in the square spiral as well as the Ulam square spiral. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Oct 01 2002
The continued fraction expansion of sqrt(4*a(n)) is [4n+1; {1, n-1, 2, 2n, 2, n-1, 1, 8n+2}]. For n=1, this collapses to [5; {3, 2, 3, 10}]. - Magus K. Chu, Sep 12 2022
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, Expositiones Mathematicae, Vol. 38, No. 4 (2020), pp. 430-479; arXiv preprint, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.7, p. 36 (p. 35 in the preprint).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = FrobeniusNumber(2*n+1, 2*n+3). - Darrell Minor, Jul 29 2008
a(n) = 8*n + a(n-1) (with a(1)=7). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(7+2*x-x^2)/(1-x)^3. - Robert Israel, Jan 13 2015
E.g.f.: 1 - (1-8*x-4*x^2)*exp(x). - Robert Israel, Jan 13 2015
a(n+1) = a(n) + A008590(n+1), a(1) = 7. - Altug Alkan, Sep 28 2015
a(n) = (2*n+1)+(2*n-1) + (2*n+1)*(2*n-1). - J. M. Bergot, Apr 17 2016
a(n) = (2*n+1)^2 - 2. - Zhandos Mambetaliyev, Jun 13 2017
From Stefano Spezia, Nov 04 2018: (Start)
L.g.f.: 4*x*(2+x)/(1+x)^2-log(1+x).
L.h.g.f.: -4*(-2+x)*x/(-1+x)^2+log(1-x).
(End)
Sum_{n>=1} 1/a(n) = 1 + sqrt(2)*Pi*tan(Pi/sqrt(2))/8. - Amiram Eldar, Jan 03 2021
EXAMPLE
MAPLE
seq(4*n^2+4*n-1, n=1..100); # Robert Israel, Jan 13 2015
MATHEMATICA
Table[4*n^2+4*n-1, {n, 60}] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
LinearRecurrence[{3, -3, 1}, {7, 23, 47}, 50] (* Harvey P. Dale, Dec 04 2018 *)
PROG
(Maxima) A073577(n):=4*n^2+4*n-1$
makelist(A073577(n), n, 1, 30); /* Martin Ettl, Nov 01 2012 */
(PARI) vector(50, n, 4*n^2 + 4*n - 1) \\ Michel Marcus, Jan 14 2015
(Magma) [4*n^2 + 4*n - 1: n in [1..50]]; // Wesley Ivan Hurt, Apr 18 2016
(Python) for n in range(1, 50): print(4*n**2+4*n-1, end=', ') # Stefano Spezia, Nov 01 2018
(GAP) List([1..50], n->4*n^2+4*n-1); # Muniru A Asiru, Nov 01 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
M. N. Deshpande (dpratap(AT)nagpur.dot.net.in), Aug 27 2002
EXTENSIONS
Edited and extended by Henry Bottomley, Oct 10 2002
STATUS
approved