A378316 a(1) = 1, a(2) = 2. For n > 2, a(n) is the sum of the digits of a(n-2) plus the sum of the same digits occurring in all terms a(i); 1 <= i <= n-2.

1, 2, 1, 2, 2, 4, 6, 4, 6, 8, 12, 8, 11, 16, 5, 24, 5, 22, 10, 14, 7, 24, 7, 36, 14, 27, 33, 39, 9, 21, 18, 30, 35, 15, 33, 32, 24, 49, 52, 59, 51, 66, 48, 36, 68, 72, 88, 56, 56, 94, 105, 85, 64, 119, 110, 70, 18, 35, 91, 93, 83, 108, 119, 109, 104, 114, 73, 79

Offset

1,2

Comments

Definition related to that of the paint sprayer sequence A279818 so we may see similar behaviors and plots for this sequence.

From Michael De Vlieger, Dec 08 2024: (Start)

Let c(d) be the cardinality of digits d in a(k), k = 1..n-m. Then a(n) is the sum of d*c(d) across the set of (distinct) digits in a(n-m), m = 2. Then it is easy to see that zeros have no effect on the sums.

Scatterplot shows "rays" associated with distinct digits in a(n-2), therefore there are 2^(b-1) rays. Since b = 10, there are 512 rays in the plot, just as there are in A279818.

A279818 is the case regarding m = 1.

A378360(n) is the sum of (d+[d=0])*c(d) across the set of digits in a(n-m), m = 1, starting instead with a(1) = 0, so as to count zeros, and thus the scatterplot shows 1024 rays. (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Example

From Michael De Vlieger, Dec 08 2024: (Start)
Let c(d) be the number of digits d in a(k), k = 1..n-2.
a(3) = 1 since a(1) = 1.
a(4) = 2 since a(2) = 2.
a(5) = 2 since a(1) = a(3) = 1, i.e., 1*c(1) = 1*2 = 2.
a(6) = 4 since a(2) = a(4) = 2, i.e., 2*c(2) = 2*2 = 4.
a(7) = 6 since a(2) = a(4) = a(5) = 2, i.e., 2*c(2) = 2*3 = 6.
a(13) = 11 since a(11) = 12, and 1*c(1) + 2*c(2) = 1*3 + 2*4 = 3+8 = 11.
a(15) = 5 since a(13) = 11, and 1*c(1) = 1*5 = 5; note, there is 1 distinct digit d = 1, but two 1's in a(13), etc. (End)

Mathematica

nn = 120; a[1] = i = 1; a[2] = j = 2; c[_] := 0;
Do[(k = Total@ Map[#1*(c[#1] += #2) & @@ # &, #]) &@
  Tally@ IntegerDigits[i];
  Set[{a[n], i, j}, {k, j, k}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Dec 08 2024 *)

Crossrefs

Cf. A007953, A279818, A378359, A378360.

Sequence in context: A325246 A193691 A089408 * A350287 A208888 A258783

Adjacent sequences: A378313 A378314 A378315 * A378317 A378318 A378319

Keyword

nonn,base,easy

Author

David James Sycamore, Nov 23 2024

Status

approved