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%I #137 Sep 08 2022 08:45:00
%S 1,5,17,37,65,101,145,197,257,325,401,485,577,677,785,901,1025,1157,
%T 1297,1445,1601,1765,1937,2117,2305,2501,2705,2917,3137,3365,3601,
%U 3845,4097,4357,4625,4901,5185,5477,5777,6085,6401,6725,7057
%N a(n) = 4*n^2 + 1.
%C Subsequence of A004613: all numbers in this sequence have all prime factors of the form 4k+1. E.g., 40001 = 13*17*181, 13 = 4*3 + 1, 17 = 4*4 + 1, 181 = 4*45 + 1. - _Cino Hilliard_, Aug 26 2006, corrected by _Franklin T. Adams-Watters_, Mar 22 2011
%C A000466(n), A008586(n) and a(n) are Pythagorean triples. - _Zak Seidov_, Jan 16 2007
%C Solutions x of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0, 1, 2, ... - _Michel Lagneau_, Feb 12 2010
%C Ulam's spiral (NW spoke). - _Robert G. Wilson v_, Oct 31 2011
%C For n >= 1, a(n) is numerator of radius r(n) of circle with sagitta = n and cord length = 1. The denominator is A008590(n). - _Kival Ngaokrajang_, Jun 13 2014
%C a(n)+6 is prime for n = 0..6 and for n = 15..20. - _Altug Alkan_, Sep 28 2015
%D Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1 Nr. 11, p. 19.
%H Vincenzo Librandi, <a href="/A053755/b053755.txt">Table of n, a(n) for n = 0..1000</a>
%H Tom M. Apostol, <a href="https://archive.org/details/introductiontoan00apos_0/page/2/mode/2up">Introduction to Analytic Number Theory</a>, Springer-Verlag, 1976, page 3.
%H Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%H Kival Ngaokrajang, <a href="/A053755/a053755.pdf">Illustration of initial terms</a>.
%H Robert G. Wilson v, <a href="/A244677/a244677.jpg">Cover of the March 1964 issue of Scientific American</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A000466(n) + 2. - _Zak Seidov_, Jan 16 2007
%F From _R. J. Mathar_, Apr 28 2008: (Start)
%F O.g.f.: (1 + 2*x + 5*x^2)/(1-x)^3.
%F a(n) = 3a(n-1) - 3a(n-2) + a(n-3). (End)
%F Equals binomial transform of [1, 4, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 30 2008
%F a(n) = A156701(n)/A087475(n). - _Reinhard Zumkeller_, Feb 13 2009
%F For n>0: a(n) = A176271(2*n,n+1); cf. A016754, A000466. - _Reinhard Zumkeller_, Apr 13 2010
%F a(n+1) = denominator of Sum_{k=0..n} (-1)^n*(2*n + 1)^3/((2*n + 1)^4 + 4), see Knuth reference. - _Reinhard Zumkeller_, Apr 11 2010
%F a(n) = 8*n + a(n-1) - 4. with a(0)=1. - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = ((2*n - 1)^2 + (2*n + 1)^2)/2. - _J. M. Bergot_, May 31 2012
%F a(n) = 2*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=5. - _Vincenzo Librandi_, Jun 26 2013
%F a(n+1) = a(n) + A017113(n), a(0) = 1. - _Altug Alkan_, Sep 26 2015
%F a(n) = A001844(n) + A046092(n-1) = A001844(n-1) + A046092(n). - _Bruce J. Nicholson_, Aug 07 2017
%F From _Amiram Eldar_, Jul 15 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + (Pi/2)*coth(Pi/2))/2.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/2)*csch(Pi/2))/2. (End)
%F From _Amiram Eldar_, Feb 05 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/2)*sinh(Pi/sqrt(2)).
%F Product_{n>=1} (1 - 1/a(n)) = (Pi/2)*csch(Pi/2). (End)
%F E.g.f.: exp(x)*(1 + 2*x)^2. - _Stefano Spezia_, Jun 10 2021
%p with (combinat):seq(fibonacci(3,2*n), n=0..42); # _Zerinvary Lajos_, Apr 21 2008
%t f[n_] := 4n^2 +1; Array[f, 40] (* _Vladimir Joseph Stephan Orlovsky_, Sep 02 2008 *)
%t CoefficientList[Series[(1 + 2 x + 5 x^2) / (1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 26 2013 *)
%t LinearRecurrence[{3,-3,1},{1,5,17},50] (* _Harvey P. Dale_, Dec 28 2021 *)
%o (PARI) for(x=0,100,print1(4*x^2+1",")) \\ _Cino Hilliard_, Aug 26 2006
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+5*x^2)/((1-x)^3))); /* or */ I:=[1,5]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+8: n in [1..50]]; // _Vincenzo Librandi_, Jun 26 2013
%o (Haskell)
%o a053755 = (+ 1) . (* 4) . (^ 2) -- _Reinhard Zumkeller_, Apr 20 2015
%o (Python) for n in range(0,50): print(4*n**2+1, end=', ') # _Stefano Spezia_, Nov 01 2018
%o (GAP) List([0..45],n->4*n^2+1); # _Muniru A Asiru_, Nov 01 2018
%Y Column 2 of array A188647.
%Y Cf. A016742, A256970 (smallest prime factors), A214345.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y Cf. A000466, A001844, A004613, A008586, A008590, A017113, A046092, A087475, A156701, A176271.
%K nonn,easy
%O 0,2
%A Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 06 2000
%E Equation corrected, and examples that were based on a different offset removed, by _R. J. Mathar_, Mar 18 2010
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