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

Author

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

Keywords

• nonn
• easy

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

Exponents in the expansion of Product_{n >= 1} (1 - q^(6*n))^2 * (1 - q ^(9*n)) * (1 - q^(36*n))/((1 - q^(3*n))*(1 - q^(12*n))*(1 - q^(18*n))) = 1 + q^3 + q^15 + q^24 + q^48 + q^63 + q^99 + ... (see Oliver, Theorem 1.1). - Peter Bala, Jan 06 2025

Links

• Jason Kimberley, Table of n, a(n) for n = 1..2001
• Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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

Programs

Magma

a:=func;[0]cat[a(n*m):m in[-1, 1],n in[1..24]]; // Jason Kimberley, Nov 09 2012

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 *)

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

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)

Extensions

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