A299412 Pentagonal pyramidal numbers divisible by 3.

0, 6, 18, 75, 126, 288, 405, 726, 936, 1470, 1800, 2601, 3078, 4200, 4851, 6348, 7200, 9126, 10206, 12615, 13950, 16896, 18513, 22050, 23976, 28158, 30420, 35301, 37926, 43560, 46575, 53016, 56448, 63750, 67626, 75843, 80190, 89376, 94221, 104430, 109800, 121086, 127008, 139425, 145926, 159528, 166635

Offset: 0

Author

Justin Gaetano, Feb 20 2018

Keywords

• nonn

Examples

			The first 6 pentagonal pyramidal numbers are 0, 1, 6, 18, 40, 75; of these, 0, 6, 18, 75 are divisible by 3.
		

Links

• Justin Gaetano, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

Crossrefs

Cf. A001318, A002411, A007494, A117748, A269416.

Programs

Magma

[IsEven(n) select (3*n/2)^2*(3*n/2+1)/2 else ((3*n+1)/2)^2*((3*n+1)/2+1)/2: n in [0..50] ]; // Vincenzo Librandi, Mar 14 2018

Maple

f:= proc(n) if n::even then (3*n/2)^2*(3*n/2+1)/2 else
((3*n+1)/2)^2*((3*n+1)/2+1)/2 fi end proc:
map(f, [$0..100]); # Robert Israel, Feb 28 2018

Mathematica

Array[((3 #1 + #2)/2)^2*((3 #1 + #2)/2 + 1)/2 & @@ {#, Boole@ OddQ@ #} &, 47, 0] (* Michael De Vlieger, Feb 21 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,6,18,75,126,288,405},50] (* Harvey P. Dale, Jul 16 2021 *)

PARI

lista(nn) = {for (n=0, nn, if (!(n^2*(n+1)/2 % 3), print1(n^2*(n+1)/2, ", ")););} \\ Michel Marcus, Feb 21 2018

PARI

x='x+O('x^99); concat(0, Vec(3*x*(3*x^4+5*x^3+13*x^2+4*x+2)/((x-1)^4*(x+1)^3))) \\ Altug Alkan, Mar 14 2018

Formula

a(n) = A007494(n)*A117748(n).

a(n) = (3*n/2)^2*(3*n/2+1)/2 if n even.

a(n) = ((3*n+1)/2)^2*((3*n+1)/2+1)/2 if n odd.

From Omar E. Pol, Feb 21 2018: (Start)

a(n) = 3*A001318(n)*A007494(n).

a(n) = A001318(n)*abs(A269416(n-1)), n >= 1. (End)

G.f.: 3*x*(3*x^4 + 5*x^3 + 13*x^2 + 4*x + 2)/((x-1)^4*(x+1)^3). - Robert Israel, Feb 28 2018