A145389 Digital roots of triangular numbers.

0, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6

Offset

0,3

Comments

Decimal expansion of 45387733/3333333330. - Enrique Pérez Herrero, Nov 14 2021

Links

Table of n, a(n) for n=0..86.

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

Formula

a(n) = A010888(A000217(n)).

Periodic sequence for n>0: a(n+9) = a(n);

a(A016777(n)) = 1; a(A007494(n)) <> 1;

a(A090570(n)) = A010888(A090570(n)).

a(n) = 1 + ((n^2 + n - 2)/2) mod 9. - Ant King, Apr 25 2009

G.f.: x(1 + 3x + 6x^2 + x^3 + 6x^4 + 3x^5 + x^6 + 9x^7 + 9x^8)/((1-x)(1 + x + x^2)(1 + x^3 + x^6)). - Ant King, Nov 16 2010

Mathematica

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[If[n==0, 0, digitalRoot[n(n+1)/2]], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

Prog

(PARI) a(n)=if(n, n=n*(n+1)/2%9; if(n, n, 9), 0) \\ Charles R Greathouse IV, Dec 19 2016
(Python)
def A145389(n): return (9, 1, 3, 6, 1, 6, 3, 1, 9)[n%9] if n else 0 # Chai Wah Wu, Feb 09 2023

Crossrefs

Cf. A000217, A004157, A010888.

Sequence in context: A089078 A355072 A134804 * A055263 A004157 A331514

Adjacent sequences: A145386 A145387 A145388 * A145390 A145391 A145392

Keyword

nonn,base,easy

Author

Reinhard Zumkeller, Oct 10 2008

Status

approved