A109512 -id:A109512 - cp's OEIS frontend

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

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

nonn

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]

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

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

Equals A001463(n) + 1.

Cf. A109512, A224731.

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

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

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

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

A224731 b(n+1) - b(n) + n where b(n) = A095114(n).

Original entry on oeis.org

2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75

Offset: 1

Robert Price, Apr 16 2013

nonn

This sequence, requested in A095114, is a companion to A109512.

Every positive integer is either of the form a(n)+n-1 (A109512) or of the form a(n+1)-a(n)+n (A224731), but not both.

Robert Price, Table of n, a(n) for n = 1..9999

Cf. A095114.

A342711 Partial sums of A000267.

Original entry on oeis.org

1, 3, 6, 9, 13, 17, 22, 27, 32, 38, 44, 50, 57, 64, 71, 78, 86, 94, 102, 110, 119, 128, 137, 146, 155, 165, 175, 185, 195, 205, 216, 227, 238, 249, 260, 271, 283, 295, 307, 319, 331, 343, 356, 369, 382, 395, 408, 421, 434, 448, 462, 476, 490, 504, 518, 532, 547

Offset: 0

Slav E. Angelov, Mar 19 2021

nonn

a(n) = A109512(n+1) for n = 0..23.
This sequence is the partial sums of A000267, which in turn is the partial sums of A240025.
It can be used to obtain a formula for the n-th term of A342712.

Cf. A000267, A240025, A342712.

Accumulate @ Array[Floor @ Sqrt[4*# + 1] &, 100, 0] (* Amiram Eldar, Mar 19 2021 *)

a(n) = sum(i=0, n, sqrtint(4*i+1)); \\ Michel Marcus, Mar 19 2021

cp's OEIS Frontend

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

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A224731 b(n+1) - b(n) + n where b(n) = A095114(n).

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A342711 Partial sums of A000267.

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