A073577 a(n) = 4*n^2 + 4*n - 1.

7, 23, 47, 79, 119, 167, 223, 287, 359, 439, 527, 623, 727, 839, 959, 1087, 1223, 1367, 1519, 1679, 1847, 2023, 2207, 2399, 2599, 2807, 3023, 3247, 3479, 3719, 3967, 4223, 4487, 4759, 5039, 5327, 5623, 5927, 6239, 6559, 6887, 7223, 7567, 7919, 8279, 8647

Offset: 1

Author

M. N. Deshpande (dpratap(AT)nagpur.dot.net.in), Aug 27 2002

Keywords

• nonn
• easy

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

Primes in the sequence are in A028871. - Russ Cox, Aug 26 2019

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

Examples

			a(2) = 8*2 + 7 = 23;
a(3) = 8*3 + 23 = 47;
a(4) = 8*4 + 47 = 79. - _Vincenzo Librandi_, Aug 08 2010
		

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).

Crossrefs

Cf. A008590, A028871, A214345.

Programs

GAP

List([1..50],n->4*n^2+4*n-1); # Muniru A Asiru, Nov 01 2018

Magma

[4*n^2 + 4*n - 1: n in [1..50]]; // Wesley Ivan Hurt, Apr 18 2016

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 *)

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

Python

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

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

Extensions

Edited and extended by Henry Bottomley, Oct 10 2002