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Digital roots of the stella octangula numbers.
3

%I #61 Mar 22 2017 03:39:42

%S 0,1,5,6,7,2,3,4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,2,3,

%T 4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,

%U 2,3,4,8,9,1,5,6,7,2,3,4,8,9,1,5,6,7,2

%N Digital roots of the stella octangula numbers.

%C This is the digital root sequence for A007588 and for A006003, two nice sequences relating to structured numbers (hexagonal anti-diamond numbers (vertex structure 13) and trigonal diamond numbers (vertex structure 4) respectively).

%C It is composed of all 9 of the nonzero digits, period 9. Root digits increase by 1 in sets of 3 [i.e., "5, 6, 7", "2, 3, 4" and "8, 9, 1"]. - _Peter M. Chema_, Aug 21 2016

%H Colin Barker, <a href="/A267017/b267017.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,1).

%F a(n) = A010888(A007588(n)).

%F From _Colin Barker_, Jan 10 2016: (Start)

%F a(n) = a(n-9) for n>9.

%F G.f.: x*(1+5*x+6*x^2+7*x^3+2*x^4+3*x^5+4*x^6+8*x^7+9*x^8) / ((1-x)*(1+x+x^2)*(1+x^3+x^6)).

%F (End)

%F a(n) = A010888(A006003(n)). - _Peter M. Chema_, Aug 17 2016

%t FixedPoint[Total@ IntegerDigits@ # &, #] & /@ Table[n (2 n^2 - 1), {n, 0, 108}] (* _Michael De Vlieger_, Jan 09 2016 *)

%o (PARI) A010888(n)=if(n, (n-1)%9+1);

%o a(n) = A010888(n*(2*n^2 - 1)); \\ _Michel Marcus_, Jan 10 2016

%o (PARI) concat(0, Vec(x*(1+5*x+6*x^2+7*x^3+2*x^4+3*x^5+4*x^6+8*x^7+9*x^8) / ((1-x)*(1+x+x^2)*(1+x^3+x^6)) + O(x^100))) \\ _Colin Barker_, Jan 10 2016

%Y Cf. A006003, A007588, A010888.

%K nonn,base,easy

%O 0,3

%A _Peter M. Chema_, Jan 08 2016