A095114 a(1)=1. a(n) = a(n-1) + (number of elements of {a(1),...,a(n-1)} that are <= n-1).

1, 2, 4, 6, 9, 12, 16, 20, 24, 29, 34, 39, 45, 51, 57, 63, 70, 77, 84, 91, 99, 107, 115, 123, 132, 141, 150, 159, 168, 178, 188, 198, 208, 218, 229, 240, 251, 262, 273, 285, 297, 309, 321, 333, 345, 358, 371, 384, 397, 410, 423, 437, 451, 465, 479, 493, 507, 522

Offset

1,2

Comments

Every positive integer is either of the form a(n)+n-1 or of the form a(n+1)-a(n)+n, but not both.

The sequence a(n)+n-1 is A109512. - Robert Price, Apr 16 2013

The sequence a(n+1)-a(n)+n is A224731. - Robert Price, Apr 16 2013

Equals A001463 + 1, the partial sums of Golomb's sequence A001462. - Ralf Stephan, May 28 2004

a(n) is the position of the first occurrence of n in A001462, i.e., A001462(a(n)) = n and A001462(m) < n for m < a(n). - Reinhard Zumkeller, Feb 09 2012 [Explanation added and first inequality corrected from A001462(m) < m by Glen Whitney, Oct 06 2015]

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Example

3 elements of {a(1),...,a(4)} are <= 4, so a(5) = a(4) + 3 = 9.

Maple

a[1]:= 1; m:= 0;
for n from 2 to 100 do
  if a[m+1] <= n-1 then m:= m+1 fi;
  a[n]:= a[n-1]+m;
od:
seq(a[i], i=1..100); # Robert Israel, Oct 07 2015

Mathematica

a[1]=1; a[n_]:=a[n]=a[n-1]+Length[Select[a/@Range[n-1], #<n&]]

Prog

(PARI) a(n) = sum(k=1, n-1, t(k)) + 1;
t(n)=local(A, t, i); if(n<3, max(0, n), A=vector(n); t=A[i=2]=2; for(k=3, n, A[k]=A[k-1]+if(t--==0, t=A[i++ ]; 1)); A[n]);
vector(100, n, a(n)) \\ Altug Alkan, Oct 06 2015
(Haskell)
a095114 n = a095114_list !! (n-1)
a095114_list = 1 : f [1] 1 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + length [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012

Crossrefs

Equals A001463(n) + 1.

Cf. A109512, A224731.

Sequence in context: A105434 A145196 A061536 * A062048 A194174 A168048

Adjacent sequences: A095111 A095112 A095113 * A095115 A095116 A095117

Keyword

nonn

Author

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

Status

approved