A036666 Numbers k such that 5*k + 1 is a square.

0, 3, 7, 16, 24, 39, 51, 72, 88, 115, 135, 168, 192, 231, 259, 304, 336, 387, 423, 480, 520, 583, 627, 696, 744, 819, 871, 952, 1008, 1095, 1155, 1248, 1312, 1411, 1479, 1584, 1656, 1767, 1843, 1960, 2040, 2163, 2247, 2376, 2464, 2599, 2691

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

Third differences are 4, -6, 8, -10, 12, -14, 16, -18, 20, -22, 24, -26, 28, ...

X values of solutions to the equation 5*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007

Also, numbers 5*i^2 + 2*i for integer i. The characteristic function is A205633(n). - Jason Kimberley, Nov 15 2012

From Gary W. Adamson, Sep 22 2019: (Start)

Match the values a(n) with the squares 5k + 1 as follows:

3,....7,....16,....24,... .a, a, a, a,...

16,...36,....81,...121,... (base).

Then 1/5 in the matching base is equal to .a, a, a,...

Example: 1/5 in base 36 is equal to .7, 7, 7, 7...

Check: 7/36 + 7/36^2 = 259/1296 = .199845...; close to 1/5.

(End)

Links

• Jason Kimberley, Table of n, a(n) for n = 1..2000
• S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See D(q).
• Ralf Stephan, On the solutions to 'px+1 is square'.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A001082, A001622, A002378, A005563, A046092, A047209, A109043, A205633.

Programs

GAP

List([1..50],n->(10*n*(n-1)+(2*n-1)*(-1)^n+1)/8); # Muniru A Asiru, Nov 28 2018

Magma

[(n-1)^2+(n-1)+Ceiling((n-1)/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Jun 05 2014

Maple

seq(n^2+n+ceil(n/2)^2, n=0..46); # Gary Detlefs, Feb 23 2010

Mathematica

(Select[ Range[121], Mod[ #, 5] == 1 || Mod[ #, 5] == 4 &]^2 - 1)/5 (* Robert G. Wilson v, Jun 23 2004 *)
Flatten[Position[5*Range[0,3000]+1,?(IntegerQ[Sqrt[#]]&)]]-1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{0,3,7,16,24},50] (* _Harvey P. Dale, Feb 13 2018 *)
Accumulate[Table[n + LCM[n, 2], {n, 0, 121}]] (* Jon Maiga, Nov 28 2018 *)

PARI

a(n)=n^2+n+ceil(n/2)^2

Formula

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

a(n) has the form ((5*m + 1)^2 - 1)/5 if n is odd; a(n) has the form ((5*m + 4)^2 - 1)/5 if n is even.

a(2*k) = k*(5*k + 2), a(2*k + 1) = 5*k^2 + 8*k + 3. - Mohamed Bouhamida, Nov 06 2007

a(n+1) = n^2 + n + ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010

From Bruno Berselli, Nov 27 2010: (Start)

a(n) = (10*n*(n - 1)+(2*n - 1)*(-1)^n + 1)/8.

5*a(n) + 1 = A047209(n)^2. (End)

a(n) = Sum_{k=0..n} k + A109043(k). - Jon Maiga, Nov 28 2018

E.g.f.: (exp(x)*(1 + 10*x^2) - exp(-x)*(1 + 2*x))/8. - Franck Maminirina Ramaharo, Nov 29 2018

From Amiram Eldar, Mar 15 2022: (Start)

Sum_{n>=2} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/2.

Sum_{n>=2} (-1)^n/a(n) = 5*(log(5)-1)/4 - sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). (End)

Extensions

Better description and additional formula from Santi Spadaro, Jul 12 2001