|
%I #7 Jul 26 2017 23:35:39
%S 0,1,4,1,2,8,2,3,3,3,4,7,4,5,2,5,6,6,6,7,1,7,8,5,8,9,9,9,1,4,1,2,8,2,
%T 3,3,3,4,7,4,5,2,5,6,6,6,7,1,7,8,5,8,9,9,9,1,4,1,2,8,2,3,3,3,4,7,4,5,
%U 2,5,6,6,6,7,1,7,8,5,8,9,9,9,1,4,1,2,8,2,3,3,3,4,7,4,5,2,5,6,6,6,7,1,7,8,5,8,9,9,9
%N Digital roots of tetrahedral numbers (A000292).
%C Period 27: repeat [1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9, 9, 9] for n >= 1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1,0,-1,1).
%F G.f.: x*(1 + 4*x + x^2 + 2*x^3 + 8*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 7*x^10 + 4*x^11 + 5*x^12 + 2*x^13 + 5*x^14 + 6*x^15 + 6*x^16 + 6*x^17 + 7*x^18 + x^19 + 7*x^20 + 8*x^21 + 5*x^22 + 8*x^23 + 9*x^24 + 9*x^25 + 9*x^26)/(1 - x^27).
%F a(n) = A010888(A000292(n)).
%e a(6) = 2 because the 6th tetrahedral number is 56, 5 + 6 = 11 -> 1 + 1 = 2.
%t Join[{0}, Table[n (n + 1) (n + 2)/6 - 9 Floor[(n - 1) (n^2 + 4 n + 6)/54], {n, 108}]]
%t Join[{0}, LinearRecurrence[{1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1}, {1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9}, 108]]
%Y Cf. A000292, A004160, A010888, A145389.
%K nonn,base,easy
%O 0,3
%A _Ilya Gutkovskiy_, Dec 19 2016
|
