A295867 - OEIS

A295867

Numbers of the form A000217(n)*A007494(n) that are divisible by 3.

0, 9, 30, 60, 120, 189, 432, 630, 825, 1122, 1404, 2205, 2760, 3264, 3978, 4617, 6300, 7392, 8349, 9660, 10800, 13689, 15498, 17052, 19140, 20925, 25344, 28050, 30345, 33390, 35964, 42237, 46020, 49200, 53382, 56889, 65340, 70380, 74589, 80088, 84672, 95625, 102102, 107484, 114480, 120285, 134064, 142158

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

L. Euler and J. Hewlett, Elements of Algebra, Chapter V, Of Figurate or Polygonal Numbers, Springer-Verlag. 1984. See pages 139-145.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,3,-3,0,0,0,-3,3,0,0,0,1,-1).

FORMULA

Let m = floor((6*n+4)/5), a(n) = (3*m/2)*binomial(m+1,2) if m is even, otherwise ((3*m+1)/2)*binomial(m+1,2).

Conjectures from Colin Barker, Feb 15 2018: (Start)

G.f.: 3*x*(3 + 7*x + 10*x^2 + 20*x^3 + 23*x^4 + 72*x^5 + 45*x^6 + 35*x^7 + 39*x^8 + 25*x^9 + 33*x^10 + 8*x^11 + 3*x^12 + x^13) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)^3).

a(n) = a(n-1) + 3*a(n-5) - 3*a(n-6) - 3*a(n-10) + 3*a(n-11) + a(n-15) - a(n-16) for n>15.

(End)

Colin Barker's conjecture is true. This is a cubic quasipolynomial of order 5: a(n) = 162/125*n^3 + 27/25*n^2 if n is 0 mod 5, 162/125*n^3 + 261/125*n^2 + 111/125*n + 12/125 if n is 4 mod 5, 162/125*n^3 + 297/125*n^2 + 144/125*n + 21/125 if n is 3 mod 5, 162/125*n^3 + 423/125*n^2 + 339/125*n + 84/125 if n is 2 mod 5, and 162/125*n^3 + 459/125*n^2 + 396/125*n + 108/125 if n is 1 mod 5. Generally a(n) = 162/125*n^3 + O(n^2). - Charles R Greathouse IV, Feb 20 2018

EXAMPLE

For n = 1, let m = floor((6*1+4)/5) = 2, a(1) = (3*2/2)*binomial(2+1,2) = 3*3 = 9.

For n = 2, let m = floor((6*2+4)/5) = 3, a(2) = ((3*3+1)/2)*binomial(3+1,2) = 5*6 = 30.

MATHEMATICA

Select[Array[Floor[(3 # + 1)/2] (# + 1) #/2 &, 58, 0], Divisible[#, 3] &] (* Michael De Vlieger, Feb 17 2018 *)

LinearRecurrence[{1, 0, 0, 0, 3, -3, 0, 0, 0, -3, 3, 0, 0, 0, 1, -1}, {0, 9, 30, 60, 120, 189, 432, 630, 825, 1122, 1404, 2205, 2760, 3264, 3978, 4617, 6300}, 48] (* Robert G. Wilson v, Feb 19 2018 *)

PROG

(PARI) a(n) = my(k=n%5); 162/125*n^3 + if(k==0, 27/25*n^2, k==1, 459/125*n^2 + 396/125*n + 108/125, k==2, 423/125*n^2 + 339/125*n + 84/125, k==3, 297/125*n^2 + 144/125*n + 21/125, 261/125*n^2 + 111/125*n + 12/125) \\ Charles R Greathouse IV, Feb 20 2018

(PARI) concat(0, Vec(3*x*(3 + 7*x + 10*x^2 + 20*x^3 + 23*x^4 + 72*x^5 + 45*x^6 + 35*x^7 + 39*x^8 + 25*x^9 + 33*x^10 + 8*x^11 + 3*x^12 + x^13) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)^3) + O(x^60))) \\ Colin Barker, Mar 01 2018

CROSSREFS

Cf. A000217, A007494, A047248.

Sequence in context: A167154 A063150 A063161 * A195319 A073399 A225275

Adjacent sequences: A295864 A295865 A295866 * A295868 A295869 A295870

KEYWORD

nonn,easy

AUTHOR

Justin Gaetano, Feb 13 2018

STATUS

approved