A126022 -id:A126022 - cp's OEIS frontend

A129299 a(1)=1, a(n) = a(n-1) + (sum of the earlier terms of the sequence which are <= n).

1, 2, 5, 8, 16, 24, 32, 48, 64, 80, 96, 112, 128, 144, 160, 192, 224, 256, 288, 320, 352, 384, 416, 472, 528, 584, 640, 696, 752, 808, 864, 952, 1040, 1128, 1216, 1304, 1392, 1480, 1568, 1656, 1744, 1832, 1920, 2008, 2096, 2184, 2272, 2408, 2544, 2680, 2816

Offset: 1

Leroy Quet, Apr 08 2007

nonn

			The terms that are <= 9 are a(1) through a(4). So a(9) = a(8) + a(1) + a(2) + a(3) + a(4) = 48 + 1 + 2 + 5 + 8 = 64.

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

Cf. A129300, A126022, A095114.

a129299 n = a129299_list !! (n-1)
a129299_list = 1 : f [1] 2 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + sum [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012

a[1]:=1: for n from 2 to 60 do b:=a[n-1]: for j from 1 to n-1 do if a[j]<=n then b:=b+a[j] else b:=b: fi: od: a[n]:=b: od: seq(a[n],n=1..60); # Emeric Deutsch, Apr 10 2007

s={1};Do[AppendTo[s,s[[-1]]+Total[Select[s,#<=n&]]],{n,2,51}];s (* James C. McMahon, Jan 20 2025 *)

More terms from Emeric Deutsch, Apr 10 2007

A129300 a(0)=1. a(n) = a(n-1) + (sum of the terms of the sequence which are <= n).

Original entry on oeis.org

1, 2, 5, 8, 11, 19, 27, 35, 51, 67, 83, 110, 137, 164, 191, 218, 245, 272, 299, 345, 391, 437, 483, 529, 575, 621, 667, 740, 813, 886, 959, 1032, 1105, 1178, 1251, 1359, 1467, 1575, 1683, 1791, 1899, 2007, 2115, 2223, 2331, 2439, 2547, 2655, 2763, 2871, 2979

Offset: 0

Leroy Quet, Apr 08 2007

nonn

			The terms that are <= 9 are a(0) through a(3). So a(9) = a(8) + a(0) + a(1) + a(2) + a(3) = 51 + 1 + 2 + 5 + 8 = 67.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A129299, A126022, A095114.

a129300 n = a129300_list !! (n-1)
a129300_list = 1 : f [1] 1 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + sum [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012

a[0]:=1: for n from 1 to 60 do b:=0: for j from 0 to n-1 do if a[j]<=n then b:=b+a[j] else fi od: a[n]:=a[n-1]+b: od: seq(a[n],n=0..60); # Emeric Deutsch, Apr 12 2007

Nest[Function[{s,n},Append[s,s[[-1]]+Total[TakeWhile[s,#<=n&]]]]@@{#,Length[#]}&,{1}, 50] (* James C. McMahon, Jan 20 2025 *)

More terms from Emeric Deutsch, Apr 12 2007

A368784 a(0) = 1. For n > 0, a(n) is the smallest integer k > n such that (Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) < 1/n.

Original entry on oeis.org

1, 2, 4, 7, 10, 13, 17, 21, 25, 30, 35, 40, 45, 50, 56, 62, 68, 74, 81, 87, 94, 101, 108, 115, 122, 130, 138, 145, 153, 162, 170, 178, 187, 195, 204, 213, 222, 231, 240, 250, 259, 269, 279, 289, 298, 309, 319, 329, 339, 350, 361, 371, 382, 393, 404, 415, 427, 438

Offset: 0

Felix Huber, Feb 15 2024

(Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) = n*(n + 1)/((k - n)*(n + 1 + k)) < 1/n. It follows that k > -1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2.

			a(3) = 7, because (1 + 2 + 3)/(4 + 5 + 6 + 7) = 3/11 < 1/3 and (1 + 2 + 3)/(4 + 5 + 6) = 2/5 > 1/3.

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A003067, A061465, A126022, A130243.

A368784 := n -> floor(-1/2 + 1/2*sqrt(4*n^3 + 8*n^2 + 4*n + 1)) + 1;
seq(A368784(n), n = 0 .. 57);

a[n_]:= Floor[-1/2 + Sqrt[4*n^3 + 8*n^2 + 4*n + 1]/2] + 1; Array[a,58,0] (* Stefano Spezia, Feb 17 2024 *)

a(n) = floor(-1/2 + sqrt(4*n^3 + 8*n^2 + 4*n + 1)/2) + 1.

a(n) = round(sqrt(n*(n+1)^2 + 1/4)). - Chai Wah Wu, Mar 11 2024

cp's OEIS Frontend

A129299 a(1)=1, a(n) = a(n-1) + (sum of the earlier terms of the sequence which are <= n).

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A129300 a(0)=1. a(n) = a(n-1) + (sum of the terms of the sequence which are <= n).

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Haskell

Maple

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Extensions

A368784 a(0) = 1. For n > 0, a(n) is the smallest integer k > n such that (Sum_{i = 1..n} i)/(Sum_{i = n + 1..k} i) < 1/n.

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