A026471 -id:A026471 - 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.

A003663 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 6.

Original entry on oeis.org

1, 6, 8, 10, 12, 15, 17, 19, 24, 26, 28, 33, 35, 37, 42, 44, 46, 51, 53, 55, 60, 62, 64, 69, 71, 73, 78, 80, 82, 87, 89, 91, 96, 98, 100, 105, 107, 109, 114, 116, 118, 123, 125, 127, 132, 134, 136, 141, 143, 145, 150, 152, 154, 159, 161, 163, 168, 170, 172, 177, 179

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Numbers congruent to {1, 6, 8} mod 9 plus the number 12.

R. K. Guy, "s-Additive sequences", preprint, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
S. R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A199162, A060469, A060470, A060471, A060472.

Cf. A003662, A005408, A026471, A026474, A033627, A051039, A051040, A244151.

I:=[1,6,8,10,12,15,17,19,24]; [n le 9 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Feb 22 2018

f[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {j}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#, 2] &, {1, 6}, 65] (* Robert G. Wilson v, Jul 05 2014 *)
LinearRecurrence[{1,0,1,-1},{1,6,8,10,12,15,17,19,24},70] (* Harvey P. Dale, Jul 25 2018 *)

From Chai Wah Wu, Feb 21 2018: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 9.
G.f.: x*(2*x^8 + x^5 - 3*x^4 + x^3 + 2*x^2 + 5*x + 1)/(x^4 - x^3 - x + 1). (End)

Name clarified by David A. Corneth, Mar 13 2023

A075123 a(n) is the least positive integer > a(n-1) and a(n) is not 2*a(i)+a(j) for 1<=i

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 17, 22, 25, 30, 33, 38, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 89, 94, 97, 102, 105, 110, 113, 118, 121, 126, 129, 134, 137, 142, 145, 150, 153, 158, 161, 166, 169, 174, 177, 182, 185, 190, 193, 198, 201, 206, 209, 214, 217, 222, 225, 230

Offset: 1

Floor van Lamoen, Sep 02 2002

a(n) = A047452(n-2) for n > 3 because of first formula. - Georg Fischer, Oct 19 2018

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

Cf. A003278, A026471, A033627, A047452, A075122.

Join[{1,2,3},Table[4n-10-Mod[n,2],{n,4,60}]] (* or *)
LinearRecurrence[ {1,1,-1},{1,2,3,6,9,14},60] (* Harvey P. Dale, Oct 28 2012 *)

def A075123(n): return (n-2<<2)-2-(n&1) if n>3 else n # Chai Wah Wu, Mar 30 2024

a(n) = 4n - 10 - (n mod 2), for n>3. - Ralf Stephan, Nov 16 2004
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3. - Harvey P. Dale, Oct 28 2012
G.f.: x*(1+x+2*x^3+2*x^4+2*x^5)/((1+x)*(1-x)^2). - Georg Fischer, May 15 2019

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.

A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 2, 0, 1, 8, 3, 4, 5, 0, 1, 16, 3, 8, 5, 3, 0, 1, 32, 3, 16, 5, 6, 7, 0, 1, 64, 3, 32, 5, 12, 7, 4, 0, 1, 128, 3, 64, 5, 24, 7, 8, 9, 0, 1, 256, 3, 128, 5, 48, 7, 16, 9, 5, 0, 1, 512, 3, 256, 5, 96, 7, 32, 9, 10, 11, 0, 1, 1024, 3, 512, 5, 192, 7, 64, 9, 20, 11, 6, 0, 1, 2048, 3, 1024, 5

Offset: 0

Paul Curtz, Nov 18 2009

The array is constructed multiplying the even-indexed A026741(k) by 2^n, and keeping the odd-indexed A026471(k) as they are.

Connections to the hydrogen spectrum: The squares of the second row are T(1,k)^2 = A001477(k)^2 = A000290(k) which are the denominators of the Lyman lines (see A171522). The squares of the row T(2,k) are in A154615, denominators of the Balmer series. Row T(3,k) is related to A106833 and A061038.

			The array starts in row n=0 with columns k>=0 as:
0,1,1,3,2,5,3,7,4, A026741
0,1,2,3,4,5,6,7,8, A001477
0,1,4,3,8,5,12,7,16, A022998
0,1,8,3,16,5,24,7,32, A144433
0,1,16,3,32,5,48,7,64,
0,1,32,3,64,5,96,7,128,

A168068 := proc(n,k) if type(k,'odd') then k; else 2^(n-1)*k ; end if; end proc: # R. J. Mathar, Jan 22 2011

A244750 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 2, 3, 4}.

Original entry on oeis.org

0, 2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144

Offset: 1

N. J. A. Sloane and Robert G. Wilson v, Jul 05 2014

			a(5) cannot be 5=2+3. It cannot be 6=2+4. It cannot be 7=3+4, and becomes a(5)=8.
a(6) cannot be 9=2+3+4. It cannot be 10=2+8. It cannot be 11=3+8. It cannot be 12 = 4+8. It cannot be 13=2+3+8. It cannot be 14=2+4+8. It cannot be 15=3+4+8, and becomes a(6)=16.

R. K. Guy, "s-Additive sequences," preprint, 1994.

Steven R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.

Cf. A003662, A003663, A005408, A026471, A026474, A033627, A051039, A051040, A244151, A244749.

Cf. A060469, A060470, A060471, A060472.

A244750:= proc(n)
    option remember;
    if n <= 4 then
        op(n,[0,2,3,4]);
    else
        prev := {seq(procname(k),k=1..n-1)} ;
        for a from procname(n-1)+1 do
            awrks := true ;
            for asub in combinat[choose](prev) do
                if add(p,p=asub) = a then
                    awrks := false;
                    break;
                end if;
            end do:
            if awrks then
                return a;
            end if;
        end do:
    end if;
end proc:
for n from 1 do
    print(A244750(n)) ;
end do: # R. J. Mathar, Jul 12 2014

 f[s_List] := f[n] = Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#] &, {0, 2, 3, 4}, 16]

Corrected by R. J. Mathar, Jul 12 2014

A026475 a(1)=1, a(2)=3, otherwise a(n) = least positive integer > a(n-1) and not a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 19, 20, 21, 22, 36, 37, 38, 39, 53, 54, 55, 69, 70, 71, 72, 86, 87, 88, 102, 103, 104, 105, 119, 120, 121, 135, 136, 137, 138, 152, 153, 154, 168, 169, 170, 171, 185, 186, 187, 201, 202, 203, 204, 218, 219, 220, 234, 235, 236, 237, 251, 252, 253

Offset: 1

Clark Kimberling

Without specifying a(2)=3, a(2) would be 2 and sequence would be A026471. - Robert Israel, Aug 27 2018

Robert Israel, Table of n, a(n) for n = 1..10000
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 (1,0,0,0,0,0,1,-1).

Cf. A026471.

1, 3, 4, 5, 6, 7, 19, seq(op([20, 21, 22, 36, 37, 38, 39]+k*[33$7]),k=0..10); # Robert Israel, Aug 30 2018

LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,3,4,5,6,7,19,20,21,22,36,37,38,39,53},60] (* Harvey P. Dale, Jan 04 2022 *)

Vec(x*(1 + 2*x + x^2 + x^3 + x^4 + x^5 + 12*x^6 - x^8 + 13*x^10 - 11*x^13 + 13*x^14) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^40)) \\ Colin Barker, Oct 10 2019

a(n+7) = a(n) + 33 for n >= 8. - Robert Israel, Aug 27 2018
From Colin Barker, Oct 10 2019: (Start)
G.f.: x*(1 + 2*x + x^2 + x^3 + x^4 + x^5 + 12*x^6 - x^8 + 13*x^10 - 11*x^13 + 13*x^14) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8) for n>15.
(End)

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.

Original entry on oeis.org

2, 5, 6, 9, 10, 28, 29, 85, 86, 256, 257, 769, 770, 2308, 2309, 6925, 6926, 20776, 20777, 62329, 62330, 186988, 186989, 560965, 560966, 1682896, 1682897, 5048689, 5048690, 15146068, 15146069, 45438205, 45438206, 136314616, 136314617, 408943849, 408943850, 1226831548, 1226831549

Offset: 1

N. J. A. Sloane and Robert G. Wilson v, Jul 05 2014

This sequence differs from A003664.

			The numbers 11-27 are not in the sequence since some combination of the previous terms add to it. example 17=2+5+10.
The number 28 however is a term since no combination of the previous terms cannot be found which sum to 28.

R. K. Guy, "s-Additive sequences," preprint, 1994.

Steven R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.
Index entries for linear recurrences with constant coefficients, signature (-1,3,3).

Cf. A003662, A003663, A005408, A026471, A026474, A033627, A051039, A051040, A244151, A244750.

Cf. A060469 - A060472.

f[s_List] := f[n] = Block[{k = s[[-1]] + 1, ss = Union[ Plus @@@ Subsets[s]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#] &, {2, 5}, 20] (* or *)
b = LinearRecurrence[{4, -3}, {9, 28}, 18]; Join[{2, 5, 6}, Riffle[b, b + 1]]
Join[{2, 5, 6},LinearRecurrence[{-1, 3, 3},{9, 10, 28},36]] (* Ray Chandler, Aug 03 2015 *)

Vec(x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2)/((x+1)*(3*x^2-1)) + O(x^100)) \\ Colin Barker, Jul 11 2014

a(2n) = 4a(2n - 2) - 3a(2n - 4) and a(2n +1) = a(2n) +1, for n>2.
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) for n>6. - Colin Barker, Jul 11 2014
G.f.: x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2) / ((x+1)*(3*x^2-1)). - Colin Barker, Jul 11 2014

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.

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A003663 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 6.

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A075123 a(n) is the least positive integer > a(n-1) and a(n) is not 2*a(i)+a(j) for 1<=i

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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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A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

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A244750 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 2, 3, 4}.

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A026475 a(1)=1, a(2)=3, otherwise a(n) = least positive integer > a(n-1) and not a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

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A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.

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