A076310 a(n) = floor(n/10) + 4*(n mod 10).

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 7, 11, 15, 19, 23

Offset

0,2

Comments

(n==0 modulo 13) iff (a(n)==0 modulo 13); applied recursively, this property provides a divisibility test for numbers given in base 10 notation.

References

Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Divisibility Tests.

Wikipedia, Divisibility rule

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

Formula

a(n) = +a(n-1) +a(n-10) -a(n-11). G.f.: -x*(-4-4*x-4*x^2-4*x^3-4*x^4-4*x^5-4*x^6-4*x^7-4*x^8+35*x^9) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x-1)^2 ). - R. J. Mathar, Feb 20 2011

Example

435598 is not a multiple of 13, as 435598 -> 43559+4*8=43591 -> 4359+4*1=4363 -> 436+4*3=448 -> 44+4*8=76 -> 7+4*6=29=13*2+3, therefore the answer is NO.
Is 8424 divisible by 13? 8424 -> 842+4*4=858 -> 85+4*8=117 -> 11+4*7=39=13*3, therefore the answer is YES.

Maple

A076310:=n->floor(n/10) + 4*(n mod 10); seq(A076310(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2014

Mathematica

Table[Floor[n/10] + 4*Mod[n, 10], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 1}, 80] (* Harvey P. Dale, Sep 30 2015 *)

Prog

(Haskell)
a076310 n =  n' + 4 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013
(PARI) a(n) = n\10 + 4*(n % 10); \\ Michel Marcus, Jan 31 2014
(Magma) [Floor(n/10)+4*(n mod 10): n in [0..75]]; // Vincenzo Librandi, Feb 27 2016

Crossrefs

Cf. A008595, A076309, A076311, A076312.

Sequence in context: A276079 A311124 A191677 * A161352 A337080 A295774

Adjacent sequences: A076307 A076308 A076309 * A076311 A076312 A076313

Keyword

nonn,changed

Author

Reinhard Zumkeller, Oct 06 2002

Status

approved