A111072 Write the digit string 0123456789, repeated infinitely many times. Then, starting from the first "0" digit at the left end, move to the right by one digit (to the "1"), then two digits (to the "3"), then three digits (to the "6"), four digits ("0"), five digits ("5"), and so on. Partial sums of the digits thus reached are 0, 1, 4, 10, 10, 15, ...

0, 1, 4, 10, 10, 15, 16, 24, 30, 35, 40, 46, 54, 55, 60, 60, 66, 69, 70, 70, 70, 71, 74, 80, 80, 85, 86, 94, 100, 105, 110, 116, 124, 125, 130, 130, 136, 139, 140, 140, 140, 141, 144, 150, 150, 155, 156, 164, 170, 175, 180, 186, 194, 195, 200, 200, 206, 209, 210

Offset

0,3

Comments

The first differences 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, etc. are in A008954.

References

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 62.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

J. Bokowski & N. J. A. Sloane, Emails, June 1994

Formula

a(n+1) = a(n) + (a(n) - a(n-1) + (n+1) mod 10) mod 10, with a(0)=0, a(1)=1.

G.f.: x*(x^12+3*x^11+6*x^10+5*x^8+5*x^6+5*x^4+6*x^2+3*x+1) / (x^16 -x^15 -x^11 +x^10 +x^6 -x^5 -x +1). - Alois P. Heinz, Jan 23 2021

Example

a(9) = 35 because a(8) - a(7) + (9 mod 10) = 30 - 24 + 9 = 15 and a(8) + (15 mod 10) = 30 + 5 = 35.
Jumping we move to the numbers 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, etc. Summing the numbers we obtain 0, 0+1 = 1, 1+3 = 4, 4+6 = 10, 10+0 = 10, 10+5 = 16, etc.

Maple

ANM:=proc(N) global anplus1, anminus1; local an, i, anpolus; anminus1:=0; an:=1; print (anminus1, an); for i from 2 by 1 to N do anplus1:=an+((an-anminus1+ i mod 10) mod 10); print(anplus1); anminus1:=an; an:=anplus1; od; end: ANM(100);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
      [0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0]
      [1+irem(n, 20)])
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 23 2021

Mathematica

Fold[Append[#1, #1[[-1]] + Mod[(#1[[-1]] - #1[[-2]] + Mod[#2, 10]), 10]] &, {0, 1}, Range[2, 58]] (* Michael De Vlieger, Nov 05 2017 *)

Crossrefs

Cf. A008954.

Sequence in context: A081547 A264272 A264257 * A189895 A310333 A180862

Adjacent sequences: A111069 A111070 A111071 * A111073 A111074 A111075

Keyword

nonn,base,easy

Author

Giorgio Balzarotti & Paolo P. Lava, Oct 10 2005

Status

approved