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

1,2

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

Table of n, a(n) for n=1..100.

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).

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

Mathematica

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

Crossrefs

Cf. A000292, A008954.

Sequence in context: A273515 A308225 A021718 * A212044 A290335 A341795

Adjacent sequences: A193105 A193106 A193107 * A193109 A193110 A193111

Keyword

nonn,easy

Author

Chris Fry, Jul 16 2011

Extensions

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

Status

approved