A008845 Numbers k such that k+1 and k/2+1 are squares.

0, 48, 1680, 57120, 1940448, 65918160, 2239277040, 76069501248, 2584123765440, 87784138523760, 2982076586042448, 101302819786919520, 3441313796169221280, 116903366249966604048, 3971273138702695316400, 134906383349641674153600, 4582845760749114225906048

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			48+1 = 49 = 7^2 and 48/2+1 = 24+1 = 25 = 5^2.
		

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.

Links

• Colin Barker, Table of n, a(n) for n = 0..650
• Henry Ernest Dudeney, Amusements in Mathematics, 1917. See problem 114, "Curious numbers".
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Programs

GAP

a:=[0,48,1680];; for n in [4..20] do a[n]:=35*a[n-1]-35*a[n-2] +a[n-3]; od; a; # G. C. Greubel, Sep 13 2019

Magma

I:=[0,48]; [n le 2 select I[n] else  34*Self(n-1) - Self(n-2)+48: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016

Maple

seq(coeff(series(48*x/((1-x)*(1-34*x+x^2)), x, n+1), x, n), n = 0..20); # G. C. Greubel, Sep 13 2019

Mathematica

LinearRecurrence[{35,-35,1},{0,48,1680},20] (* Harvey P. Dale, May 24 2014 *)

PARI

concat(0, Vec(48*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 02 2016

Sage

def A008845_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(48*x/((1-x)*(1-34*x+x^2))).list()
A008845_list(20) # G. C. Greubel, Sep 13 2019

Formula

a(n) = 2*(A008844(n)-1) = 16*A075528(n) = 48*A029546(n). - corrected by Sean A. Irvine, Apr 07 2018

a(0)=0, a(1)=48, a(2)=1680, a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - Harvey P. Dale, May 24 2014

From Colin Barker, Mar 02 2016: (Start)

a(n) = (-6+(3-2*sqrt(2))*(17+12*sqrt(2))^(-n)+(3+2*sqrt(2))*(17+12*sqrt(2))^n)/4.

G.f.: 48*x / ((1-x)*(1-34*x+x^2)).

(End)

a(n) = 34*a(n-1) - a(n-2) + 48. - Vincenzo Librandi, Mar 03 2016