A112677 Sum of digits of the sum of the previous 4 terms.

1, 1, 1, 1, 4, 7, 4, 7, 4, 4, 10, 7, 7, 10, 7, 4, 10, 4, 7, 7, 10, 10, 7, 7, 7, 4, 7, 7, 7, 7, 10, 4, 10, 4, 10, 10, 7, 4, 4, 7, 4, 10, 7, 10, 4, 4, 7, 7, 4, 4, 4, 10, 4, 4, 4, 4, 7, 10, 7, 10, 7, 7, 4, 10, 10, 4, 10, 7, 4, 7, 10, 10, 4, 4, 10, 10, 10, 7, 10, 10, 10, 10

Offset

0,5

Comments

This is to the tetranacci sequence as A112661 is to the tribonacci and as A030132 is to Fibonacci. A000288 is the tetranacci sequence (A000078) but starting with values (1,1,1,1). Andrew Carmichael Post (andrewpost(AT)gmail.com) wrote the program that generated this sequence and showed that for any 4 initial integers a(0),a(1),a(2),a(3) the length of the cycle eventually entered is a factor of 312. For instance, starting with (6,6,6,6) continues in a cycle of length 1 since SOD(6+6+6+6) = SOD(24) = 6; and 1 divides 312. For the SOD(tribonacci) which is A112661, the length of any cycle eventually entered is a factor of 78.

All terms for n >= 4 are 4, 7, or 10. The sequence has period 78; the 78 terms after the initial 1,1,1,1 repeat forever. - Nathaniel Johnston, May 04 2011

Links

Table of n, a(n) for n=0..81.

Formula

a(0)=a(1)=a(2)=a(3)=1. a(n) = SumDigits(a(n-1) + a(n-2) + a(n-3) + a(n-4)).

a(n) = SumDigits(A000288(n)).

a(n) = A007953(a(n-1) + a(n-2) + a(n-3) + a(n-4)). - Nathaniel Johnston, May 04 2011

Example

a(0)=a(1)=a(2)=a(3)=1.
a(4) = SOD(1+1+1+1) = SOD(4) = 4.
a(5) = SOD(1+1+1+4) = SOD(7) = 7.
a(10) = SOD(4+7+4+4) = SOD(19) = 10, note that we do not iterate SOD to reduce 10 to 1.

Maple

A112677 := proc(n) option remember: if(n<=3)then return 1:fi: return add(d, d=convert(procname(n-1) + procname(n-2) + procname(n-3) + procname(n-4), base, 10)): end: seq(A112677(n), n=0..100); # Nathaniel Johnston, May 04 2011

Mathematica

nxt[{a_, b_, c_, d_}]:={b, c, d, Total[IntegerDigits[a+b+c+d]]}; Transpose[ NestList[ nxt, {1, 1, 1, 1}, 90]][[1]] (* or *) PadRight[{1, 1, 1, 1}, 120, {10, 10, 10, 10, 4, 7, 4, 7, 4, 4, 10, 7, 7, 10, 7, 4, 10, 4, 7, 7, 10, 10, 7, 7, 7, 4, 7, 7, 7, 7, 10, 4, 10, 4, 10, 10, 7, 4, 4, 7, 4, 10, 7, 10, 4, 4, 7, 7, 4, 4, 4, 10, 4, 4, 4, 4, 7, 10, 7, 10, 7, 7, 4, 10, 10, 4, 10, 7, 4, 7, 10, 10, 4, 4, 10, 10, 10, 7}](* Harvey P. Dale, Mar 05 2016 *)

Crossrefs

Cf. A000078, A000288, A004090, A007953, A010888, A030132, A112661.

Sequence in context: A358189 A195362 A106739 * A010712 A259499 A254398

Adjacent sequences: A112674 A112675 A112676 * A112678 A112679 A112680

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Dec 30 2005

Extensions

Name corrected by Nathaniel Johnston, May 04 2011

Status

approved