A117748 -id:A117748 - cp's OEIS frontend

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

Justin Gaetano, Feb 20 2018

nonn

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

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)

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

[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

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

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 *)

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

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

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

A069499 Triangular numbers of the form 21*k.

Original entry on oeis.org

0, 21, 105, 210, 231, 378, 630, 861, 903, 1176, 1596, 1953, 2016, 2415, 3003, 3486, 3570, 4095, 4851, 5460, 5565, 6216, 7140, 7875, 8001, 8778, 9870, 10731, 10878, 11781, 13041, 14028, 14196, 15225, 16653, 17766, 17955, 19110, 20706, 21945, 22155, 23436, 25200

Offset: 1

Amarnath Murthy, Mar 30 2002

Intersection of A000217 and A008603. - Michel Marcus, Sep 17 2013

Let F(r) = Product_{n >= 0} 1 - q^(21*(14*n+r)). The sequence terms occur as the exponents in the expansion of (1 - q^21)*F(5)*F(6)*F(7)*F(8)*F(9)*F(13)*F(14)*F(15) = 1 - q^21 - q^105 + q^210 + q^231 - q^378 - q^630 + + - - ... (by the quintuple product identity). - Peter Bala, Dec 23 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000217, A008603, A069498, A117748.

a[0] := 0:a[1] := 6:a[2] := 14:a[3] := 20:a[4] := 21:a[5] := 27:a[6] := 35:a[7] := 41:seq((42*(floor(i/8))+a[i mod 8])*(42*(floor(i/8))+a[i mod 8]+1)/2,i=0..100);
# alternative program
A := proc (q) local n: for n from 0 to q do if type((1/21)*n*(n+1)/2, integer) then print(n*(n+1)/2) fi; od; end: A(250); # Peter Bala, Dec 24 2024

Select[21Range[1100],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Aug 16 2021 *)
Select[Accumulate[Range[0,300]],IntegerQ[#/21]&] (* Harvey P. Dale, Jun 12 2022 *)

G.f.: -21*x^2*(x^2-x+1)*(x^4+5*x^3+9*x^2+5*x+1) / ((x-1)^3*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 23 2013
From Peter Bala, Dec 24 2025: (Start)
a(n) is quasi-polynomial in n:
a(4*n) = 21 * n*(21*n - 1)/2; a(4*n+1) = 21 * n*(21*n + 1)/2;
a(4*n+2) = 21 * (3*n + 1)*(7*n + 2)/2; a(4*n+3) = 21 * (3*n + 2)*(7*n + 5)/2. (End)

More terms from Sascha Kurz, Apr 01 2002

a(1)=0 added and edited by Alois P. Heinz, Aug 19 2021

cp's OEIS Frontend

A299412 Pentagonal pyramidal numbers divisible by 3.

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