A069498 -id:A069498 - cp's OEIS frontend

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

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

A180926 Numbers k such that 6k and 10k are triangular numbers.

Original entry on oeis.org

0, 1, 63, 3906, 242110, 15006915, 930186621, 57656563588, 3573776755836, 221516502298245, 13730449365735355, 851066344173293766, 52752382889378478138, 3269796672797292350791, 202674641330542747270905

Offset: 1

Vladimir Pletser, Sep 25 2010

From Klaus Purath, Jul 25 2024: (Start)
Numbers k such that 48k + 1 is a square as well as the sum of two consecutive terms.
a(n) = t(n-1)*t(n)/(8*t(1)^2) where (t) is any recurrence t(k) = 8*t(k-1) - t(k-2) with t(0) = 0 and arbitrary t(1) != 0. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Subsequence of A154293.

Cf. A069498, A154293.

a[1] = 0; a[n_] := a[n] = (62 a[n - 1] + 1 + Sqrt[(48 a[n - 1] + 1)*(80 a[n - 1] + 1)])/2; Array[a, 14] (* Robert G. Wilson v, Sep 27 2010 *)
Rest[CoefficientList[Series[-x^2/((x - 1) (x^2 - 62 x + 1)), {x, 0, 30}], x]] (* Vincenzo Librandi, Jun 26 2014 *)
LinearRecurrence[{63,-63,1},{0,1,63},20] (* Harvey P. Dale, Dec 25 2019 *)

isok(n) = ispolygonal(6*n, 3) && ispolygonal(10*n, 3); \\ Michel Marcus, Jun 25 2014

a(n) = (62*a(n-1) + 1 + ((48*a(n-1) + 1)*(80*a(n-1) + 1))^(1/2))/2 with a(1)=0.
G.f.: -x^2 / ((x-1)*(x^2-62*x+1)). - Colin Barker, Jun 25 2014
a(n) = (-8+(4+sqrt(15))*(31+8*sqrt(15))^(-n) - (-4+sqrt(15))*(31+8*sqrt(15))^n)/480. - Colin Barker, Mar 03 2016

a(8) onwards from Robert G. Wilson v, Sep 27 2010

cp's OEIS Frontend

A069499 Triangular numbers of the form 21*k.

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A180926 Numbers k such that 6k and 10k are triangular numbers.

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cp's OEIS Frontend

A069499 Triangular numbers of the form 21*k.

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Maple

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A180926 Numbers k such that 6*k and 10*k are triangular numbers.

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PARI

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A180926 Numbers k such that 6k and 10k are triangular numbers.