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A212306 Starting with the positive numbers, each element subtracts its value, a(n), from the next a(n) elements. 3
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-Fran├žois 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

Generated from A000027.

Cf. A258353, A258354.

Sequence in context: A181512 A076019 A071453 * A080242 A183927 A035317

Adjacent sequences:  A212303 A212304 A212305 * A212307 A212308 A212309

KEYWORD

nonn,nice

AUTHOR

Gabriel Stauth, Oct 26 2013

STATUS

approved

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Last modified October 21 17:57 EDT 2020. Contains 337919 sequences. (Running on oeis4.)