A051039 -id:A051039 - cp's OEIS frontend

A026474 a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j) or a(i)+a(j)+a(k) for 1<=i

1, 2, 4, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 218, 225, 232, 239, 246, 253, 260, 267, 274, 281, 288, 295, 302, 309, 316, 323, 330, 337, 344, 351, 358, 365, 372

Offset: 1

Clark Kimberling

All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - Henry Bottomley, Feb 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Stoehr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A026472, A026476, A033627, A051039, A051040.

[1,2,4] cat [7*n+1: n in [1..60]]; // Vincenzo Librandi, Oct 18 2013

Join[{1,2,4,8},Range[15,500,7]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
CoefficientList[Series[(1 + x^2 + 2 x^3 + 3 x^4)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{2,-1},{1,2,4,8,15},60] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n>3,7*n-20,2^(n-1)) \\ Charles R Greathouse IV, Sep 17 2015

Starts 1, 2, 4 then the numbers 7*(n-3)+1.

a(n) = 7*n-20 for n>3. a(n) = 2*a(n-1)-a(n-2) for n>5. G.f.: x*(1+x^2+2*x^3+3*x^4)/(1-x)^2. - Colin Barker, Sep 19 2012

More terms from Eric W. Weisstein

A051040 5-Stohr sequence.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 63, 94, 125, 156, 187, 218, 249, 280, 311, 342, 373, 404, 435, 466, 497, 528, 559, 590, 621, 652, 683, 714, 745, 776, 807, 838, 869, 900, 931, 962, 993, 1024, 1055, 1086, 1117, 1148, 1179, 1210, 1241, 1272, 1303, 1334, 1365, 1396, 1427

Offset: 1

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Stoehr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A062001, A033627, A026474, A051039.

Join[{1,2,4,8,16,32}, Range[63, 2000, 31]] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
LinearRecurrence[{2,-1},{1,2,4,8,16,32,63},60] (* Harvey P. Dale, Sep 14 2015 *)

a(n)=if(n>6,31*n-154,2^(n-1)) \\ Charles R Greathouse IV, Jun 20 2024

Terms are 2^(n-1) up to a(6) and then 31n-154.

a(n) = 31*n-154 for n>5. a(n) = 2*a(n-1)-a(n-2) for n>7. G.f.: x*(15*x^6+8*x^5+4*x^4+2*x^3+x^2+1)/(x-1)^2. - Colin Barker, Sep 19 2012

Corrected by Henry Bottomley, May 29 2001

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

A276206 a(0) = a(1) = a(2) = a(3) = 0. For n>3 a(n) is the smallest nonnegative integer such that there is no arithmetic progression i,j,k,m,n (of length 5) such that a(i)+a(j)+a(k)+a(m) = a(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 4, 4, 4, 4, 1, 4, 4, 4, 4, 1, 4, 4, 4, 4, 1, 4, 4, 4, 4, 1, 2

Offset: 0

Michal Urbanski, Aug 24 2016

nonn

The sequence has the same set of values as A051039 (4-Stohr sequence)
Conjecture 1:
One can calculate a(n) in a following, non-recursive way, using the quinary representation of n.
Let n>=0 be an integer. We consider two cases:
1 There is no digit 4 in the quinary representation of n
Then a(n)=0
2 There is a digit 4 in the quinary representation of n
Let i be the number of the position (counting from right) of the rightmost digit 4 in quinary representation of n, then a(n)=A051039(i).
For example let n=22. The quinary representation of 22 is 42. The rightmost digit 4 in the number 42 is on the second position (counting from right), so a(22) = A051039(2) = 2
Conjecture 2:
The sequence can be generated in a following way:
Start with a zero. Take five consecutive copies of all you have, replace all zeros in the fifth copy with the next value of A051039, repeat
Both of these conjectures can be generalized for similarly defined sequences where the length of the arithmetic progression in the definition (in A276206 it is 5) is a prime number, see A276204.
The assumption about primality is essential, for complex lengths of the arithmetic progression the sequence is more irregular, see A276205.

			For n = 23 we have that:
a(23)>0, because a(3)+a(8)+a(13)+a(18)=0 and 3,8,13,18,23 is an arithmetic progression.
a(23)>1, because a(7)+a(11)+a(15)+a(19)=1 and 7,11,15,19,23 is an arithmetic progression.
There is no such arithmetic progression i,j,k,m,23 that a(i)+a(j)+a(k)+a(m)=2, so a(23) = 2.

Michal Urbanski, Table of n, a(n) for n = 0..199999

Cf. A276204 (length 3), A276205 (length 4), A276207 (any length).

Cf. A051039 (4-Stohr sequence).

A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 7, 4, 2, 1, 6, 10, 8, 4, 2, 1, 7, 13, 15, 8, 4, 2, 1, 8, 16, 22, 16, 8, 4, 2, 1, 9, 19, 29, 31, 16, 8, 4, 2, 1, 10, 22, 36, 46, 32, 16, 8, 4, 2, 1, 11, 25, 43, 61, 63, 32, 16, 8, 4, 2, 1, 12, 28, 50, 76, 94, 64, 32, 16, 8, 4, 2, 1, 13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1

Offset: 1

Henry Bottomley, May 29 2001

			Array begins as:
  1, 2, 3, 4,  5,  6,  7,   8,   9, ... A000027;
  1, 2, 4, 7, 10, 13, 16,  19,  22, ... A033627;
  1, 2, 4, 8, 15, 22, 29,  36,  43, ... A026474;
  1, 2, 4, 8, 16, 31, 46,  61,  76, ... A051039;
  1, 2, 4, 8, 16, 32, 63,  94, 125, ... A051040;
  1, 2, 4, 8, 16, 32, 64, 127, 190, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 255, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
  1, 2, 4, 8, 16, 32, 64, 128, 256, ... ;
Antidiagonal triangle begins as:
   1;
   2,  1;
   3,  2,  1;
   4,  4,  2,  1;
   5,  7,  4,  2,   1;
   6, 10,  8,  4,   2,   1;
   7, 13, 15,  8,   4,   2,  1;
   8, 16, 22, 16,   8,   4,  2,  1;
   9, 19, 29, 31,  16,   8,  4,  2,  1;
  10, 22, 36, 46,  32,  16,  8,  4,  2, 1;
  11, 25, 43, 61,  63,  32, 16,  8,  4, 2, 1;
  12, 28, 50, 76,  94,  64, 32, 16,  8, 4, 2, 1;
  13, 31, 57, 91, 125, 127, 64, 32, 16, 8, 4, 2, 1;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Rows include A000027, A033627, A026474, A051039, A051040.
Diagonals include A000079, A000225, A033484, A036563, A048487.
Cf. A048488, A048489, A048490, A048491, A139634, A139635, A139697.
A048483 can be seen as half this table.

T[n_, k_]:= If[kG. C. Greubel, May 03 2022 *)

def A062001(n,k):
    if (kA062001(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, May 03 2022

If k <= n+1 then A(n, k) = 2^(k-1), while if k > n+1, A(n, k) = (2^n - 1)*(k - n) + 1 (array).
T(n, k) = A(k, n-k+1) (antidiagonals).
T(2*n-1, n) = A000079(n-1), n >= 1.
T(2*n, n) = A000079(n), n >= 1.
T(2*n+1, n) = A000225(n+1), n >= 1.
T(2*n+2, n) = A033484(n), n >= 1.
T(2*n+3, n) = A036563(n+3), n >= 1.
T(2*n+4, n) = A048487(n), n >= 1.
From G. C. Greubel, May 03 2022: (Start)
T(n, k) = (2^k - 1)*(n-2*k+1) + 1 for k < n/2, otherwise 2^(n-k).
T(2*n+5, n) = A048488(n), n >= 1.
T(2*n+6, n) = A048489(n), n >= 1.
T(2*n+7, n) = A048490(n), n >= 1.
T(2*n+8, n) = A048491(n), n >= 1.
T(2*n+9, n) = A139634(n), n >= 1.
T(2*n+10, n) = A139635(n), n >= 1.
T(2*n+11, n) = A139697(n), n >= 1. (End)

A193911 Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

Original entry on oeis.org

1, 3, 7, 14, 25, 43, 69, 110, 167, 255, 375, 558, 805, 1179, 1681, 2438, 3451, 4975, 7011, 10070, 14153, 20283, 28461, 40734, 57103, 81663, 114415, 163550, 229069, 327355, 458409, 654998, 917123, 1310319, 1834587, 2620998, 3669553, 5242395, 7339525, 10485230

Offset: 1

Jeffrey R. Goodwin, Aug 08 2011

			Portion of the first three rows:
A033627, 2-Stohr  1  2  4  7
A026474, 3-Stohr  1  2  4  8
A051039, 4-Stohr  1  2  4  8
Thus a(1)=1, a(2)=2+1=3, and a(3)=4+2+1=7.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Stöhr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,1,4,-2).

A193911=t={0,1}; Do[AppendTo[t,t[[-2]]+t[[-1]]]; AppendTo[t,2*t[[-2]]],{n,41}]; Drop[Nest[Accumulate,t,2],1] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
LinearRecurrence[{2,2,-6,1,4,-2},{1,3,7,14,25,43},40] (* Harvey P. Dale, Jun 20 2015 *)

All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - Henry Bottomley, Feb 04 2000
So we get the sums from the piecewise function:
for odd n>=1, a(n)=2^((n+1)/2)-n+((n+1)/2)-2+Sum_{i=0..((n+1)/2)-1}(2*i+1)*(2^(((n+1)/2)-i) -1);
for even n>=2, a(n)=2^((n/2)+2)-n-4+Sum_{i=0..(n/2)-1}(2*i+1)*(2^((n/2)-i) -1). - Jeffrey R. Goodwin, Aug 09 2011
Let odd m>=3, then a(n)=a(m)-A000295(((m+1)/2)+1), where n>=2 is even. - Jeffrey R. Goodwin, Aug 09 2011
Let even m>=2, then a(n)=a(m)-A077802(m/2)=a(m)-A095151(m/2), where n>=1 is odd. - Jeffrey R. Goodwin, Aug 09 2011
From Alexander R. Povolotsky, Aug 09 2011: (Start)
G.f.: x*(1 + x - x^2)/((-1 + x)^3*(-1 - x + 2*x^2 + 2*x^3)).
a(n+4) = -2*a(n)+3*a(n+2)+n+5.
a(n) = 1/8*(2^(n/2+2)*((10-7*sqrt(2))*(-1)^n+10+7*sqrt(2))-(-1)^n-2*n*(n+12)-79). (End)

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

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

A026474 a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j) or a(i)+a(j)+a(k) for 1<=i

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A051040 5-Stohr sequence.

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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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A276206 a(0) = a(1) = a(2) = a(3) = 0. For n>3 a(n) is the smallest nonnegative integer such that there is no arithmetic progression i,j,k,m,n (of length 5) such that a(i)+a(j)+a(k)+a(m) = a(n).

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A062001 Table by antidiagonals of n-Stohr sequences: T(n,k) is least positive integer not the sum of at most n distinct terms in the n-th row from T(n,1) through to T(n,k-1).

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A193911 Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

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Mathematica

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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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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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