A185039 Numbers of the form 9*m^2 + 4*m, m an integer.

0, 5, 13, 28, 44, 69, 93, 128, 160, 205, 245, 300, 348, 413, 469, 544, 608, 693, 765, 860, 940, 1045, 1133, 1248, 1344, 1469, 1573, 1708, 1820, 1965, 2085, 2240, 2368, 2533, 2669, 2844, 2988, 3173, 3325, 3520, 3680, 3885, 4053, 4268, 4444, 4669, 4853, 5088

Offset: 1

Author

N. J. A. Sloane, Feb 04 2012

Keywords

• nonn
• easy

Comments

Also, numbers m such that 9*m+4 is a square. After 0, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See B(q).
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Characteristic function is A205809.

Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), A132355 (k=2), this sequence (k=4), A057780 (k=6), A218864 (k=8). [Jason Kimberley, Nov 08 2012]

For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

Programs

Magma

[0] cat &cat[[9*n^2-4*n,9*n^2+4*n]: n in [1..32]]; // Bruno Berselli, Feb 04 2011

Mathematica

CoefficientList[Series[x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3), {x,0,50}], x] (* G. C. Greubel, Jun 20 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,5,13,28,44},50] (* Harvey P. Dale, Jan 23 2018 *)

PARI

x='x+O('x^50); Vec(x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3)) \\ G. C. Greubel, Jun 20 2017

Formula

From Bruno Berselli, Feb 04 2012: (Start)

G.f.: x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3).

a(n) = a(-n+1) = (18*n*(n-1)+(2*n-1)*(-1)^n+1)/8 = A004526(n)*A156638(n). (End).