A053715 a(n) = n-th triangular number (the sum of the first n integers) in base n.

0, 1, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 110, 121, 120, 132, 130, 143, 140, 154, 150, 165, 160, 176, 170, 187, 180, 198, 190, 209, 200, 220, 210, 231, 220, 242, 230, 253, 240, 264, 250, 275, 260, 286, 270, 297, 280

Offset

0,3

Comments

Note that these numbers should be read in base n. For instance, a(19) = 100 is actually (10)(00) in base 19, which equals 190 (the 19th triangular number) in base 10. - T. D. Noe, Feb 01 2013

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

Apart from a(1), a(2n-1) = 10n and a(2n) = 11n.

From Colin Barker, Feb 01 2013: (Start)

a(n) = (-10*(-1+(-1)^n)+(21+(-1)^n)*n)/4 for n>1.

a(n) = 2*a(n-2)-a(n-4) for n>5.

G.f.: -x*(9*x^4-18*x^2-11*x-1) / ((x-1)^2*(x+1)^2). (End)

Example

a(3) = 1 + 2 + 3 = 6 = 20_3.

Mathematica

Join[{0, 1}, Table[FromDigits[IntegerDigits[n*(n+1)/2, n]], {n, 2, 50}]] (* T. D. Noe, Feb 01 2013 *)
Join[{0, 1}, LinearRecurrence[{0, 2, 0, -1}, {11, 20, 22, 30}, 60]] (* Vincenzo Librandi, Jan 04 2016 *)

Prog

(PARI) baseE(x, b)= { local(d, e=0, f=1); if (b<2, return(x)); while (x>0, d=x%b; x\=b; e+=d*f; f*=10); return(e) } { for (n=0, 1000, write("b053715.txt", n, " ", baseE(n*(n + 1)/2, n)) ) } \\ Harry J. Smith, Apr 27 2010
(Magma) I:=[0, 1, 11, 20, 22, 30]; [n le 6 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jan 04 2016

Crossrefs

Cf. A000217.

Sequence in context: A124251 A279431 A230541 * A343770 A250203 A339307

Adjacent sequences: A053712 A053713 A053714 * A053716 A053717 A053718

Keyword

base,easy,nonn

Author

Henry Bottomley, Feb 10 2000

Extensions

More terms from James A. Sellers, Feb 22 2000

Status

approved