A069499 Triangular numbers of the form 21*k.

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

Author

Amarnath Murthy, Mar 30 2002

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A000217, A008603, A069498, A117748.

Programs

Maple

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

Mathematica

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 *)

Formula

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)

Extensions

More terms from Sascha Kurz, Apr 01 2002

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