A279816 Digital roots of tetrahedral numbers (A000292).

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, 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, 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

Offset: 0

Author

Ilya Gutkovskiy, Dec 19 2016

Keywords

• nonn
• base
• easy

Comments

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.

Examples

			a(6) = 2 because the 6th tetrahedral number is 56, 5 + 6 = 11 -> 1 + 1 = 2.
		

Links

• Eric Weisstein's World of Mathematics, Digital Root
• Eric Weisstein's World of Mathematics, Tetrahedral Number
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A000292, A004160, A010888, A145389.

Programs

Mathematica

Join[{0}, Table[n (n + 1) (n + 2)/6 - 9 Floor[(n - 1) (n^2 + 4 n + 6)/54], {n, 108}]]
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]]

Formula

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).

a(n) = A010888(A000292(n)).