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A053755
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a(n) = 4*n^2 + 1.
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67
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1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, 1025, 1157, 1297, 1445, 1601, 1765, 1937, 2117, 2305, 2501, 2705, 2917, 3137, 3365, 3601, 3845, 4097, 4357, 4625, 4901, 5185, 5477, 5777, 6085, 6401, 6725, 7057
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OFFSET
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0,2
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COMMENTS
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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
A000466(n), A008586(n) and a(n) are Pythagorean triples. - Zak Seidov, Jan 16 2007
Solutions x of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0, 1, 2, ... - Michel Lagneau, Feb 12 2010
Ulam's spiral (NW spoke). - Robert G. Wilson v, Oct 31 2011
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
a(n)+6 is prime for n = 0..6 and for n = 15..20. - Altug Alkan, Sep 28 2015
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REFERENCES
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Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1 Nr. 11, p. 19.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013
Kival Ngaokrajang, Illustration of initial terms.
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = A000466(n) + 2. - Zak Seidov, Jan 16 2007
From R. J. Mathar, Apr 28 2008: (Start)
O.g.f.: (1 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 3a(n-1) - 3a(n-2) + a(n-3). (End)
Equals binomial transform of [1, 4, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = A156701(n)/A087475(n). - Reinhard Zumkeller, Feb 13 2009
For n>0: a(n) = A176271(2*n,n+1); cf. A016754, A000466. - Reinhard Zumkeller, Apr 13 2010
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
a(n) = 8*n + a(n-1) - 4. with a(0)=1. - Vincenzo Librandi, Aug 06 2010
a(n) = ((2*n - 1)^2 + (2*n + 1)^2)/2. - J. M. Bergot, May 31 2012
a(n) = 2*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=5. - Vincenzo Librandi, Jun 26 2013
a(n+1) = a(n) + A017113(n), a(0) = 1. - Altug Alkan, Sep 26 2015
a(n) = A001844(n) + A046092(n-1) = A001844(n-1) + A046092(n). - Bruce J. Nicholson, Aug 07 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/2)*coth(Pi/2))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/2)*csch(Pi/2))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/2)*sinh(Pi/sqrt(2)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/2)*csch(Pi/2). (End)
E.g.f.: exp(x)*(1 + 2*x)^2. - Stefano Spezia, Jun 10 2021
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MAPLE
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with (combinat):seq(fibonacci(3, 2*n), n=0..42); # Zerinvary Lajos, Apr 21 2008
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MATHEMATICA
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f[n_] := 4n^2 +1; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[(1 + 2 x + 5 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 5, 17}, 50] (* Harvey P. Dale, Dec 28 2021 *)
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PROG
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(PARI) for(x=0, 100, print1(4*x^2+1", ")) \\ Cino Hilliard, Aug 26 2006
(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
(Haskell)
a053755 = (+ 1) . (* 4) . (^ 2) -- Reinhard Zumkeller, Apr 20 2015
(Python) for n in range(0, 50): print(4*n**2+1, end=', ') # Stefano Spezia, Nov 01 2018
(GAP) List([0..45], n->4*n^2+1); # Muniru A Asiru, Nov 01 2018
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CROSSREFS
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Column 2 of array A188647.
Cf. A016742, A256970 (smallest prime factors), A214345.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
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.
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.
Cf. A000466, A001844, A004613, A008586, A008590, A017113, A046092, A087475, A156701, A176271.
Sequence in context: A273274 A357058 A273250 * A162373 A146781 A338277
Adjacent sequences: A053752 A053753 A053754 * A053756 A053757 A053758
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KEYWORD
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nonn,easy
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AUTHOR
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Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 06 2000
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EXTENSIONS
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Equation corrected, and examples that were based on a different offset removed, by R. J. Mathar, Mar 18 2010
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STATUS
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approved
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