A228564 Largest odd divisor of n^2 + 1.

1, 1, 5, 5, 17, 13, 37, 25, 65, 41, 101, 61, 145, 85, 197, 113, 257, 145, 325, 181, 401, 221, 485, 265, 577, 313, 677, 365, 785, 421, 901, 481, 1025, 545, 1157, 613, 1297, 685, 1445, 761, 1601, 841, 1765, 925, 1937, 1013, 2117, 1105, 2305, 1201, 2501, 1301, 2705

Offset

0,3

Comments

From Lamine Ngom, Jan 04 2023: (Start)

For n>2, a(n) = hypotenuse c of the primitive Pythagorean triple (a, b, c) such that n*a = b + c.

Terms that appear twice (1, 5, 145, 4901, ...) are the positive terms of A076218. Equivalently, the products of two consecutive terms of A001653, or one more than the squares of A001542.

These duplicated terms appear at indices i and j (i>j) such that (i^2-1)/2 = j^2 (A001541). In addition, they are hypotenuse in two primitive Pythagorean triples: (i, j^2, a(i)) and (2*j, j^2-1, a(i)). (End)

Links

Jeremy Gardiner, Table of n, a(n) for n = 0..1000

Wikipedia, Quasi-polynomial.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(n) = A000265(A002522(n)).

From Ralf Stephan, Aug 26 2013: (Start)

a(4n) = 4*a(2n) - 3.

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

a(4n+2) = 4*a(2n) + 16*n + 1.

a(4n+3) = 2*a(2n) + 12*n + 3. (End)

From Bruno Berselli, Aug 26 2013: (Start)

G.f.: (1 + x + 2*x^2 + 2*x^3 + 5*x^4 + x^5) / (1-x^2)^3.

a(n) = 3*a(n-2) -3*a(n-4) +a(n-6).

a(n) = (n^2+1)*(3+(-1)^n)/4. (End)

From Peter Bala, Feb 14 2019: (Start)

a(n) = numerator((n^2 + 1)/(n + 1)).

a(n) is a quasi-polynomial in n: a(2*n) = 4*n^2 + 1; a(2*n + 1) = 2*n^2 + 2*n + 1. (End)

From Amiram Eldar, Aug 09 2022: (Start)

a(n) = numerator((n^2+1)/2).

Sum_{n>=0} 1/a(n) = (1 + coth(Pi/2)*Pi/2 + tanh(Pi/2)*Pi)/2. (End)

Example

A002522(3) = 3^2+1 = 10 => a(3) = 10/2 = 5.

Maple

lod:= t -> t/2^padic:-ordp(t, 2):
seq(lod(n^2+1), n=0..60); # Robert Israel, Aug 19 2014

Mathematica

Table[(n^2 + 1) (3 + (-1)^n)/4, {n, 0, 60}] (* Bruno Berselli, Aug 26 2013 *)

Prog

(BASIC)
for n = 0 to 45 : t=n^2+1
x: if not t mod 2 then t=t/2 : goto x
print str$(t); ", "; : next n
print
end
(PARI) a(n)=if(n<2, n>0, m=n\4; [4*a(2*m)-3, 2*a(2*m)+4*m-1, 4*a(2*m)+16*m+1, 2*a(2*m)+12*m+3][(n%4)+1]) \\ Ralf Stephan, Aug 26 2013
(PARI) a(n)=(n^2+1)/2^valuation(n^2+1, 2) \\ Ralf Stephan, Aug 26 2013
(Magma) [(n^2+1)*(3+(-1)^n)/4: n in [0..60]]; // Bruno Berselli, Aug 26 2013
(Magma) [Denominator(2*n^2/(n^2+1)): n in [0..60]]; // Vincenzo Librandi, Aug 19 2014
(GAP) List([0..60], n->NumeratorRat((n^2+1)/(n+1))); # Muniru A Asiru, Feb 20 2019
(Sage) [(n^2+1)*(3+(-1)^n)/4 for n in (0..60)] # G. C. Greubel, Feb 20 2019

Crossrefs

Cf. A000265, A002522, A164900, A191871.

Sequence in context: A175614 A214022 A125256 * A075684 A146876 A147206

Adjacent sequences: A228561 A228562 A228563 * A228565 A228566 A228567

Keyword

nonn,easy

Author

Jeremy Gardiner, Aug 25 2013

Status

approved