A074867 a(n) = M(a(n-1)) + M(a(n-2)) where a(1)=a(2)=1 and M(k) is the product of the digits of k in base 10.

1, 1, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 7, 7, 14, 11, 5, 6, 11, 7, 8, 15, 13, 8, 11, 9, 10, 9, 9, 18, 17, 15, 12, 7, 9, 16, 15, 11, 6, 7, 13, 10, 3, 3, 6, 9, 15, 14, 9, 13, 12, 5, 7, 12, 9, 11, 10, 1, 1, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 7, 7, 14, 11, 5, 6, 11, 7, 8, 15, 13

Offset: 1

Author

Felice Russo, Sep 11 2002

Keywords

• base
• easy
• nonn

Comments

Periodic with least period 60. - Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003

From Hieronymus Fischer, Jul 01 2007: (Start)

The digital product analog (in base 10) of the Fibonacci recurrence.

a(n) and Fib(n)=A000045(n) are congruent modulo 10 which implies that (a(n) mod 10) is equal to (Fib(n) mod 10) = A003893(n). Thus (a(n) mod 10) is periodic with the Pisano period A001175(10)=60.

a(n)==A131297(n) modulo 10 (A131297(n)=digital sum analog base 11 of the Fibonacci recurrence).

For general bases p>1, we have the inequality 1<=a(n)<=2p-2 (for n>0). Actually, a(n)<=18.

(End)

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045, A010073, A010074, A010075, A010076, A131294, A131295, A131296, A131297, A131318, A131319, A131320.

Programs

Mathematica

nxt[{a_,b_}]:={b,Times@@IntegerDigits[a]+Times@@IntegerDigits[b]}; Transpose[ NestList[nxt,{1,1},90]][[1]] (* Harvey P. Dale, Feb 01 2015 *)

Formula

From Hieronymus Fischer, Jul 01 2007: (Start)

a(n) = a(n-1)+a(n-2)-10*(floor(a(n-1)/10)+floor(a(n-2)/10)). This is valid, since a(n)<100.

a(n) = ds_10(a(n-1))+ds_10(a(n-2))-(floor(a(n-1)/10)+floor(a(n-2)/10)) where ds_10(x) is the digital sum of x in base 10.

a(n) = (a(n-1)mod 10)+(a(n-2)mod 10) = A010879(a(n-1))+A010879(a(n-2)).

a(n) = A131297(n) if A131297(n)<=10.

a(n) = Fib(n)-10*sum{1A000045(n).

a(n) = A000045(n)-10*sum{1A000045(n-k+1)*A059995(a(k))}. (End)

Extensions

More terms from Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003

Definition adapted to offset by Georg Fischer, Jun 18 2021