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A036666 Numbers k such that 5*k+1 is a square. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Sequence allows us to find X values of the equation: 5*X^3 + X^2 = Y^2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

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

Equivalently, numbers of the form m*(5*m+2), where n = 0,-1,1,-2,2,-3,3,... - Bruno Berselli, Jan 05 2016

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. 274 (2004), no. 1-3, 9-24. See D(q).

R. Stephan, On the solutions to 'px+1 is square'

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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 (bhmd95(AT)yahoo.fr), Nov 06 2007

a(n) = n^2 + n + ceiling(n/2)^2, (with offset 0). - 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)

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 *)

PROG

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

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

CROSSREFS

Cf. A005563, A046092, A001082, A002378, A047209.

Sequence in context: A116040 A218276 A221025 * A218359 A117491 A211379

Adjacent sequences:  A036663 A036664 A036665 * A036667 A036668 A036669

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Dec 11 1999

EXTENSIONS

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

STATUS

approved

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Last modified February 19 14:46 EST 2018. Contains 299334 sequences. (Running on oeis4.)