A212306 - OEIS

A212306

Starting with the positive numbers, each element subtracts its value, a(n), from the next a(n) elements.

1, 1, 2, 2, 1, 3, 4, 1, 1, 5, 2, 5, 1, 3, 2, 6, 1, 11, 2, 1, 1, 4, 8, 1, 1, 2, 6, 1, 3, 13, 1, 9, 5, 7, 1, 1, 2, 2, 6, 3, 3, 17, 1, 17, 5, 7, 1, 1, 2, 2, 6, 3, 3, 8, 1, 4, 5, 7, 1, 18, 6, 18, 14, 1, 1, 9, 2, 7, 1, 3, 2, 1, 1, 7, 2, 17, 1, 17, 20, 1, 19, 9, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

When calculating this sequence, each element affects a different number of subsequent terms, so there is no known direct formula for the n-th term.

a(A258353(n)) = 1; a(A258354(n)) = n and a(m) != n for m < A258354(n). - Reinhard Zumkeller, May 27 2015

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..10000

FORMULA

a(n) is n - the sum of the terms such that a(i) + i >= n.

Each term a(n) is 1 plus the sum of the terms such that a(i) + i + 1 = n.

EXAMPLE

Starting with A000027, the first term is 1 so we subtract 1 from the next 1 terms so the second term becomes 1. Now again we subtract 1 from the next 1 terms and the third term becomes 2. Subtract 2 from the next two terms of A000027 (4 and 5) to get 2 and 3. Subtract 2 from the next 2 terms (the 3 from 5 and 6) to get 1 and 4. The next term is 4 - 1 = 3. To carry on subtract 3 from the next 3 terms.

MAPLE

b:= proc(n) option remember; local l, t;

if n=1 then [1$2] else l:= b(n-1); t:= n-l[2];

zip((x, y)->x+y, [t$t+1], subsop(1=NULL, 2=0, l), 0) fi

end:

a:= n-> b(n)[1]:

seq(a(n), n=1..100); # Alois P. Heinz, Nov 12 2013

MATHEMATICA

b[n_] := b[n] = Module[{l, t, x, y, m}, If[n == 1, {1, 1}, l = b[n-1]; t = n-l[[2]]; x = Array[t&, t+1]; y = ReplacePart[l, {1 -> Sequence[], 2 -> 0}]; m = Max[Length[x], Length[y]]; Thread[PadRight[x, m] + PadRight[y, m]]]]; a[n_] := b[n] // First; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

PROG

(PARI) {a(n)=local(A=vector(n+1, j, j)); for(k=1, n+1, A = Vec(Ser(A) - x^k*A[k]*(1-x^A[k])/(1-x) +x*O(x^n))); A[n]} \\ Paul D. Hanna, Nov 12 2013

for(n=1, 100, print1(a(n), ", "))

(PARI) /* Vector returns 1000 terms: */

{A=vector(1000, j, j); for(k=1, #A, A = Vec(Ser(A) - x^k*A[k]*(1-x^A[k])/(1-x) +x*O(x^#A))); A} \\ Paul D. Hanna, Nov 12 2013

(Haskell)

a212306 n = a212306_list !! (n-1)

a212306_list = f [1..] where

f (x:xs) = x : f ((map (subtract x) us) ++ vs)

where (us, vs) = splitAt x xs

-- Reinhard Zumkeller, Dec 16 2013

CROSSREFS