A193108 The tetrahedral numbers A000292 mod 10.

1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0

Offset: 1

Author

Chris Fry, Jul 16 2011

Keywords

• nonn
• easy

Comments

Periodic with period 20.

The cycle is symmetric about index 9 in that a(8)+a(10), a(7)+a(11), etc are all congruent to 0 mod 10.

If the first diagonal of Pascal's triangle is given index 0 this sequence is the 3rd diagonal of Pascal's triangle modulo 10, or the binomial coefficients C(n+2,3)mod 10. Note that the last three terms in the cycle are 0.

The Pisano period lengths of A000292 (mod m) are 1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40.., for m>=1. This sequence describes the case m=10. - R. J. Mathar, Oct 25 2011

Links

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

Crossrefs

Cf. A000292, A008954.

Programs

Mathematica

Table[Mod[Binomial[n+2,3],10],{n,1,21}]

Formula

a(n) = a(n-20).

G.f. -x*(1+4*x+5*x^4+6*x^5+4*x^6+5*x^8+6*x^10+4*x^11+5*x^12+6*x^15+9*x^16) / ( (x-1)*(1+x^4+x^3+x^2+x)*(1+x)*(1-x+x^2-x^3+x^4)*(1+x^2)*(x^8-x^6+x^4-x^2+1) ). - R. J. Mathar, Oct 25 2011

a(n) = 55 -a(n-1) -a(n-2) … -a(n-18) -a(n-19). - Ant King, Oct 19 2012

Extensions

Edited by N. J. A. Sloane, Jul 16 2011