A057780 Multiples of 3 that are one less than a perfect square.

0, 3, 15, 24, 48, 63, 99, 120, 168, 195, 255, 288, 360, 399, 483, 528, 624, 675, 783, 840, 960, 1023, 1155, 1224, 1368, 1443, 1599, 1680, 1848, 1935, 2115, 2208, 2400, 2499, 2703, 2808, 3024, 3135, 3363, 3480, 3720, 3843, 4095, 4224, 4488, 4623, 4899, 5040

Offset

1,2

Comments

Also, numbers of the form 9*m^2+6*m, m an integer. - Jason Kimberley, Nov 08 2012

k is in this list iff k+1 is in the support of A033684. - Jason Kimberley, Nov 13 2012

Links

Jason Kimberley, Table of n, a(n) for n = 1..2001

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

Formula

a(n) = A001651(n)^2 - 1 = 3 * A001082(n).

G.f.: 3*x^2*(1+4*x+x^2) / ((1-x)^3*(1+x)^2). - Colin Barker, Nov 24 2012

From Colin Barker, Dec 26 2015: (Start)

a(n) = 3/8*(6*n^2-2*((-1)^n+3)*n+(-1)^n-1).

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.

(End)

Mathematica

Select[3*Range[0, 2000], IntegerQ[Sqrt[#+1]]&] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {0, 3, 15, 24, 48}, 50] (* Harvey P. Dale, Sep 10 2019 *)

Prog

(Magma) a:=func<n|9*n^2+6*n>; [0]cat[a(n*m):m in[-1, 1], n in[1..24]]; // Jason Kimberley, Nov 09 2012
(PARI) concat(0, Vec(3*x^2*(1+4*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 26 2015

Crossrefs

Numbers of the form 9n^2+kn, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), this sequence (k=6), A218864 (k=8). - Jason Kimberley, Nov 08 2012

Sequence in context: A061386 A366209 A365412 * A274697 A129024 A348770

Adjacent sequences: A057777 A057778 A057779 * A057781 A057782 A057783

Keyword

nonn,easy

Author

Benjamin Geiger (benjamin_geiger(AT)yahoo.com), Nov 02 2000

Extensions

Since this is a list, offset corrected to 1 by Jason Kimberley, Nov 09 2012

Status

approved