cp's OEIS Frontend

A069499 Triangular numbers of the form 21*k.

%I A069499 #40 Dec 25 2024 04:57:58
%S A069499 0,21,105,210,231,378,630,861,903,1176,1596,1953,2016,2415,3003,3486,
%T A069499 3570,4095,4851,5460,5565,6216,7140,7875,8001,8778,9870,10731,10878,
%U A069499 11781,13041,14028,14196,15225,16653,17766,17955,19110,20706,21945,22155,23436,25200
%N A069499 Triangular numbers of the form 21*k.
%C A069499 Intersection of A000217 and A008603. - _Michel Marcus_, Sep 17 2013
%C A069499 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
%H A069499 Harvey P. Dale, <a href="/A069499/b069499.txt">Table of n, a(n) for n = 1..1000</a>
%H A069499 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).
%F A069499 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
%F A069499 From _Peter Bala_, Dec 24 2025: (Start)
%F A069499 a(n) is quasi-polynomial in n:
%F A069499 a(4*n) = 21 * n*(21*n - 1)/2; a(4*n+1) = 21 * n*(21*n + 1)/2;
%F A069499 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)
%p A069499 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);
%p A069499 # alternative program
%p A069499 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
%t A069499 Select[21Range[1100],OddQ[Sqrt[8#+1]]&] (* _Harvey P. Dale_, Aug 16 2021 *)
%t A069499 Select[Accumulate[Range[0,300]],IntegerQ[#/21]&] (* _Harvey P. Dale_, Jun 12 2022 *)
%Y A069499 Cf. A000217, A008603, A069498, A117748.
%K A069499 nonn,easy
%O A069499 1,2
%A A069499 _Amarnath Murthy_, Mar 30 2002
%E A069499 More terms from _Sascha Kurz_, Apr 01 2002
%E A069499 a(1)=0 added and edited by _Alois P. Heinz_, Aug 19 2021