The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A073577 a(n) = 4*n^2 + 4*n - 1. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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 a(2) = 8*2 + 7 = 23; a(3) = 8*3 + 23 = 47; a(4) = 8*4 + 47 = 79. - Vincenzo Librandi, Aug 08 2010 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 Cf. A008590, A028871, A214345. Sequence in context: A002146 A336092 A184882 * A348230 A139830 A153210 Adjacent sequences: A073574 A073575 A073576 * A073578 A073579 A073580 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 18 06:24 EDT 2024. Contains 371769 sequences. (Running on oeis4.)