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

0,2

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).

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

Example

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

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 *)

Prog

(PARI) concat(0, Vec(48*x/((1-x)*(1-34*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 02 2016
(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
(Sage)
def A008845_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(48*x/((1-x)*(1-34*x+x^2))).list()
A008845_list(20) # G. C. Greubel, Sep 13 2019
(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

Crossrefs

Sequence in context: A266160 A004386 A076003 * A273627 A355998 A288455

Adjacent sequences: A008842 A008843 A008844 * A008846 A008847 A008848

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved