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

Author

Reinhard Zumkeller, Oct 10 2008

Keywords

• nonn
• base
• easy

Comments

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

Links

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

Crossrefs

Cf. A000217, A004157, A010888.

Programs

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 *)

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

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