A256496 a(n) = 15(n mod 2) + 10(n mod 3) + 6(n mod 5).

31, 32, 33, 34, 35, 6, 37, 38, 39, 10, 41, 12, 43, 44, 15, 16, 47, 18, 49, 20, 21, 22, 53, 24, 25, 26, 27, 28, 59, 0, 31, 32, 33, 34, 35, 6, 37, 38, 39, 10, 41, 12, 43, 44, 15, 16, 47, 18, 49, 20, 21, 22, 53, 24, 25, 26, 27, 28, 59, 0, 31, 32, 33, 34, 35, 6

Offset: 1

Author

Aaron Kastel, Mar 31 2015

Keywords

• nonn
• easy

Comments

After 0 it cycles again from 31.

This is the simplest example of a(n) = b*c*(n mod a) + a*c*(n mod b) + a*b*(n mod c) with a=2, b=3, c=5.

a(n) mod abc = n mod abc.

Other values for a,b,c require modification so that bc + ac + ab = abc + 1. For example, for a=3, b=5, c=7, a(n)= 2b*c*(n mod a) + a*c*(n mod b) + a*b*(n mod c) so that 2*5*7 + 3*7 + 3*5 = 3*5*7 = 70 + 21 + 15 = 106.

This expression (with a=3, b=5, c=7) (A255818) is found in the Kol Bo (Hebrew: כלבו), a book of religious Jewish law from the 13th to 14th centuries. It is given there as a method for calculating a person's age without anyone saying it explicitly.

a(n) = n for n = 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Aaron Hacohen from Lunil?, כלבו (in Hebrew).
• Menachem Mendel Schneerson, אגרות קודש (in Hebrew).
• Wikipedia, Kol Bo
• Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Crossrefs

Cf. A255818 for an example with 1 modification, A256643 for 2 modifications and A256668 for 3 modifications.

Programs

Mathematica

Table[15 Mod[n, 2] + 10 Mod[n, 3] + 6 Mod[n, 5], {n, 60}] (* Michael De Vlieger, Mar 31 2015 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{31,32,33,34,35,6,37,38},70] (* or *) PadRight[ {},70,{31,32,33,34,35,6,37,38,39,10,41,12,43,44,15,16,47,18,49,20,21,22,53,24,25,26,27,28,59,0}] (* Harvey P. Dale, Oct 31 2016 *)

PARI

vector(30, n, 15*(n%2) + 10*(n%3) + 6*(n%5)) \\ Michel Marcus, Mar 31 2015

Formula

a(n) = 15(n mod 2) + 10(n mod 3) + 6(n mod 5).

G.f.: -x*(59*x^6+146*x^5+201*x^4+195*x^3+159*x^2+94*x+31) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Apr 07 2015

a(n) = -2*a(n-1) - 2*a(n-2) - a(n-3) + a(n-5) + 2*a(n-6) + 2*a(n-7) + a(n-8), for n>=9. - Vaclav Kotesovec, Apr 07 2015