A075123 -id:A075123 - cp's OEIS frontend

A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95, 128, 152, 182, 212, 258, 316, 374, 413, 476, 531, 546, 608, 717, 798, 862, 965, 1060, 1161, 1307, 1386, 1435, 1556, 1722, 1834, 1934, 2058, 2261, 2497, 2699, 2874, 3061, 3197, 3332, 3629, 3712, 3868, 4140, 4447, 4640

Offset: 1

Dan Hoey

nonn

a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j)-a(k) for distinct i and j with 1 <= i,j,k <= n-1. [Comment corrected by Jean-Paul Delahaye, Oct 02 2020.]

Klaus Brockhaus and Dan Hoey, Table of n, a(n) for n = 1..2839
Index entries for B_2 sequences.

Cf. A010672.
Cf. A033627, A003278, A026471, A066720, A075122, A075123, A133097.
Cf. A036241, A349777.

from itertools import islice
def agen(): # generator of terms
    aset, sset, k = set(), set(), 0
    while True:
        k += 1
        while any(k+an in sset for an in aset): k += 1
        yield k; sset.update(k+an for an in aset); aset.add(k)
print(list(islice(agen(), 51))) # Michael S. Branicky, Feb 05 2023

a(n) = A010672(n-1)+1.

A026471 a(n) = least positive integer > a(n-1) and not of the form a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 13, 14, 15, 25, 26, 27, 37, 38, 48, 49, 50, 60, 61, 71, 72, 73, 83, 84, 94, 95, 96, 106, 107, 117, 118, 119, 129, 130, 140, 141, 142, 152, 153, 163, 164, 165, 175, 176, 186, 187, 188, 198, 199, 209, 210, 211, 221, 222, 232, 233, 234, 244, 245, 255

Offset: 1

Clark Kimberling

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A003278, A033627, A075122, A075123.

{1, 5, 13} union {n congruent 2, 3, 4, 14, 15 mod 23}, proved by Matthew Akeran. - Ralf Stephan, Nov 15 2004

G.f.: (9*x^11-7*x^10+9*x^8+7*x^5+x^4+x^3+x^2+x+1)*x/(x^6-x^5-x+1). - Alois P. Heinz, Aug 06 2018

Edited by Floor van Lamoen, Sep 02 2002

A075122 a(n) = least positive integer > a(n-1) and not a(i)+2*a(j) for 1<=i

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 17, 18, 19, 20, 21, 22, 23, 24, 29, 31, 32, 33, 34, 73, 94, 96, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137

Offset: 1

Floor van Lamoen, Sep 02 2002

nonn

Cf. A033627, A003278, A026471, A075123.

A234717 a(n) = floor(n/(exp(1/(2*n))-1)).

Original entry on oeis.org

1, 7, 16, 30, 47, 69, 94, 124, 157, 195, 236, 282, 331, 385, 442, 504, 569, 639, 712, 790, 871, 957, 1046, 1140, 1237, 1339, 1444, 1554, 1667, 1785, 1906, 2032, 2161, 2295, 2432, 2574, 2719, 2869, 3022, 3180, 3341, 3507

Offset: 1

Richard R. Forberg, Dec 29 2013

Equations of this general form: (n/(exp(1/(r*n))-1)) have a fractional portion that converges to one or more rational fractions if r is rational. They have second differences that are nearly constant before the floor function, and repeat in patterns when calculated after the floor function.
The fractional portion of this equation (before the floor function) oscillates between two fractions that converge towards 1/24 and 13/24.
Second differences of a(n) = repeat{3,5}.
First differences of a(n) = A075123(n+3).
Partial sums of a(n) = A033951(n).

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A033951, A075123.

Table[Floor[n/(Exp[1/(2 n)] - 1)], {n, 100}] (* Wesley Ivan Hurt, Apr 01 2022 *)

From Ralf Stephan, Mar 28 2014: (Start)
a(n) = (1/4)*(8n^2 - 2n - 1 + (-1)^n).
G.f.: x*(2*x^2 + 5*x + 1)/((1-x^2)*(1-x)^2). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Wesley Ivan Hurt, Apr 01 2022
E.g.f.: (x*(4*x + 3)*cosh(x) + (4*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Nov 23 2023

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A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

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A026471 a(n) = least positive integer > a(n-1) and not of the form a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

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A075122 a(n) = least positive integer > a(n-1) and not a(i)+2*a(j) for 1<=i

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A234717 a(n) = floor(n/(exp(1/(2*n))-1)).

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