A033627 -id:A033627 - cp's OEIS frontend

A060470 Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j

1, 2, 3, 4, 5, 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

Offset: 1

Henry Bottomley, Mar 15 2001

Numbers {1,6,8} mod 9 plus {2,3,4,5,12}.

			12 is in the sequence since it is 4+8 and 2+10 but no other sum of two distinct terms.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A003044, A033627, A060469, A060471, A060472. Virtually identical to A003663.

I:=[1,2,3,4,5,6,8,10,12,15,17,19,24]; [n le 13 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Feb 22 2018

[n le 9 select Floor(n^2/12+n/2+3/4) else 2*n+3*Floor(n/3+2/3)-17: n in [1..65]]; // Bruno Berselli, Feb 22 2018

f[s_List, j_Integer] := Block[{cnt, k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ cnt = Count[ss, k]; cnt == 0 || cnt > 2, k++]; Append[s, k]]; Nest[f[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)
CoefficientList[Series[(2 x^12 + x^9 + x^8 + x^7 + x^6 + x^2 + x + 1) / (x^4 - x^3 - x + 1), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 22 2018 *)

From Chai Wah Wu, Feb 21 2018: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 13.
G.f.: x*(2*x^12 + x^9 + x^8 + x^7 + x^6 + x^2 + x + 1)/(x^4 - x^3 - x + 1). (End)
a(n) = 2*n + 3*floor(n/3 + 2/3) - 17 for n>9. - Bruno Berselli, Feb 22 2018

A060472 Smallest positive a(n) such that the number of solutions to a(n)=a(j)+a(k), j

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 36, 38, 40, 42, 44, 51, 53, 55, 57, 64, 66, 68, 70, 72, 77, 79, 81, 83, 85, 92, 94, 96, 98, 105, 107, 109, 111, 118, 120, 122, 124, 126, 131, 133, 135, 137, 139, 146, 148, 150, 152, 159, 161, 163

Offset: 1

Henry Bottomley, Mar 15 2001

nonn

Numbers {1,3,10,12,14,16,18,23,25,27,29,31,38,40,42,44,51,53} mod 54 plus {2,4,5,6,7,8,9,20,36}.

			12 is in the sequence since it is 5+7, 4+8, 3+9 and 2+10 but no other sum of two distinct terms.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A033627, A060469, A060470, A060471, A060472.

f[s_List, j_Integer] := Block[{cnt, k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ cnt = Count[ss, k]; cnt == 0 || cnt > 4, k++]; Append[s, k]]; Nest[f[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)
CoefficientList[Series[(5 x^39 - 3 x^38 + 3 x^34 - x^33 + 5 x^29 + x^27 + x^26 + 6 x^25 + x^24 + x^23 + x^22 + x^21 + 4 x^20 + x^19 + x^18 + 2 x^17 + 2 x^16 + 3 x^15 + 2 x^14 + 2 x^13 + 2 x^12 + 2 x^11 + 2 x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) / (x^19 - x^18 - x + 1), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 22 2018 *)

From Chai Wah Wu, Feb 21 2018: (Start)
a(n) = a(n-1) + a(n-18) - a(n-19) for n > 40.
G.f.: x*(5*x^39 - 3*x^38 + 3*x^34 - x^33 + 5*x^29 + x^27 + x^26 + 6*x^25 + x^24 + x^23 + x^22 + x^21 + 4*x^20 + x^19 + x^18 + 2*x^17 + 2*x^16 + 3*x^15 + 2*x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 2*x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/(x^19 - x^18 - x + 1). (End)

A276204 a(0) = a(1) = 0. For n>1 a(n) is the smallest nonnegative integer such that there is no arithmetic progression k,m,n (of length 3) such that a(k)+a(m) = a(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 7, 7, 1, 7, 7, 1, 2, 2, 1, 7, 7, 1, 7, 7, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0

Offset: 0

Michal Urbanski, Aug 24 2016

The sequence has the same set of values as A033627.
The sequence has a kind of a "triple rhythm", compare the distribution of zeros to the Cantor set.
Conjecture 1:
One can calculate a(n) in a following, non-recursive way, using the ternary representation of n.
Let n>=0 be an integer. We consider two cases:
1. There is no digit 2 in the ternary representation of n. Then a(n) = 0.
2. There is a digit 2 in the ternary representation of n.
Let i be the number of the position (counting from right) of the rightmost digit 2 in ternary representation of n, then a(n) = A033627(i).
For example: let n = 19. The ternary representation of 19 is 201. The rightmost digit 2 in the number 201 is on the third position (counting from right), so a(19) = A033627(3) = 4.
Conjecture 2:
The sequence can be generated in a following way:
Start with a zero. Take three consecutive copies of all you have, replace all zeros in the third copy with the next value of A033627, repeat.
Both of these conjectures can be generalized for similarly defined sequences where the length of the arithmetic progression in the definition (in A276204 it is 3) is a prime number, see A276206.
The assumption about primality is essential, for complex lengths of the arithmetic progression the sequence is irregular, see A276205.

			For n = 6 we have that:
a(6)>0, because a(0)+a(3)=0 and 0,3,6 is an arithmetic progression.
a(6)>1, because a(4)+a(5)=0 and 4,5,6 is an arithmetic progression.
There is no such arithmetic progression k,m,6 that a(k)+a(m) = 2, so a(6) = 2.

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

Cf. A276205 (length 4), A276206 (length 5), A276207 (any length).

Cf. A033627, A229037.

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

A060471 Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 23, 25, 30, 32, 34, 36, 41, 43, 45, 47, 52, 54, 56, 58, 63, 65, 67, 69, 74, 76, 78, 80, 85, 87, 89, 91, 96, 98, 100, 102, 107, 109, 111, 113, 118, 120, 122, 124, 129, 131, 133, 135, 140, 142, 144, 146, 151, 153, 155, 157

Offset: 1

Henry Bottomley, Mar 15 2001

nonn

Numbers {1,3,8,10} mod 11 plus {2,4,5,6,7,16}. Note that while the cases for "zero", "one", "two" and "four" essentially involve a third of the natural numbers, this case for "three" involves 4/11.

			12 is in the sequence since it is 5+7, 4+8 and 2+10 but no other sum of two distinct terms.

Cf. A033627, A060469, A060470, A060471, A060472.

f[s_List, j_Integer] := Block[{cnt, k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ cnt = Count[ss, k]; cnt == 0 || cnt > 3, k++]; Append[s, k]]; Nest[f[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)

From Chai Wah Wu, Feb 21 2018: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 17.
G.f.: x*(2*x^16 + x^12 + x^11 + x^10 + x^9 + x^8 + x^3 + x^2 + x + 1)/(x^5 - x^4 - x + 1). (End)

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

A257941 Lexicographically earliest sequence of positive integers such that the terms and their absolute first differences are all distinct and no term is the sum of two distinct earlier terms.

Original entry on oeis.org

1, 3, 7, 12, 18, 26, 9, 20, 34, 24, 39, 55, 22, 45, 66, 28, 47, 72, 85, 49, 76, 108, 68, 99, 53, 82, 112, 70, 114, 149, 74, 122, 172, 93, 145, 203, 101, 160, 95, 162, 216, 118, 187, 224, 141, 214, 143, 235, 139, 195, 281, 164, 241, 329, 166, 260, 170, 283, 168

Offset: 1

Eric Angelini and Alois P. Heinz, May 13 2015

The sequence of absolute first differences begins: 2, 4, 5, 6, 8, 17, 11, 14, 10, 15, 16, 33, 23, 21, 38, 19, 25, 13, 36, 27, 32, 40, ... .

The sequence is 0-additive.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
E. Angelini et al., 0-additive and first differences and follow-up messages on the SeqFan list, May 13 2015
Eric Weisstein's World of Mathematics, s-Additive Sequence

Cf. A005228, A030124, A033627, A095115, A140778, A257944.

s:= proc() false end: b:= proc() false end:
a:= proc(n) option remember; local i, k;
      if n=1 then b(1):= true; 1
    else for k while b(k) or s(k) or
         (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
         for i to n-1 do s(a(i)+k):= true od;
         b(k), b(abs(a(n-1)-k)):= true$2; k
      fi
    end:
seq(a(n), n=1..101);

s[] = False; b[] = False;
a[n_] := a[n] = Module[{i, k}, If[n == 1, b[1] = True; 1, For[k = 1, b[k] || s[k] || Function[t, b[t] || t == k][Abs[a[n-1]-k]], k++]; For[i = 1, i <= n-1, i++, s[a[i]+k] = True]; {b[k], b[Abs[a[n-1]-k]]} = {True, True}; k]];
Array[a, 101] (* Jean-François Alcover, Oct 28 2020, after Maple *)

A244151 0-additive sequence: start with a(1) = 2; thereafter, a(n) = smallest number not already in sequence which is not the sum of any previous two terms.

Original entry on oeis.org

2, 3, 4, 8, 9, 14, 15, 20, 21, 26, 27, 32, 33, 38, 39, 44, 45, 50, 51, 56, 57, 62, 63, 68, 69, 74, 75, 80, 81, 86, 87, 92, 93, 98, 99, 104, 105, 110, 111, 116, 117, 122, 123, 128, 129, 134, 135, 140, 141, 146, 147, 152, 153, 158, 159, 164, 165, 170, 171, 176, 177, 182, 183, 188, 189, 194, 195, 200, 201, 206, 207, 212, 213

Offset: 1

Leonardo Sznajder, Jun 21 2014

As A033627, but first term is 2.

4 and the numbers in A047243. - Joerg Arndt, Jun 22 2014

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

Cf. A033627, A047243.

f[s_List] := Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s, {2}]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[f, {2}, 70] (* Robert G. Wilson v, Jun 23 2014 *)

Vec(x*(x^5+3*x^3-x^2+x+2)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Jun 26 2014

a(2n) = 6(n-1)+2 & a(2n+1) = 6(n-1)+3 for n>1. - Robert G. Wilson v, Jun 23 2014
From Colin Barker, Jun 26 2014: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 6.
G.f.: x*(x^5 + 3*x^3 - x^2 + x + 2)/((x - 1)^2*(x + 1)). (End)
E.g.f.: (x^3 + 3*x^2 + 30*x + 24)/6 + (3*x - 4)*cosh(x) + 3*(x - 2)*sinh(x). - Stefano Spezia, Apr 15 2023

Added terms >= 20, Joerg Arndt, Jun 22 2014

A301451 Numbers congruent to {1, 7} mod 9.

Original entry on oeis.org

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268

Offset: 1

Bruno Berselli, Mar 21 2018

First bisection of A056991, second bisection of A242660.

The squares of the terms of A174396 are the squares of this sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A274406: numbers congruent to {0, 8} mod 9.
Cf. A193910: numbers congruent to {2, 6} mod 9.
Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;

Magma
```
&cat [[9*n+1, 9*n+7]: n in [0..40]];
```

Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)

Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018

[2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]

[n for n in (1..300) if n % 9 in (1,7)]

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.
a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019

cp's OEIS Frontend

A060470 Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j

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A060472 Smallest positive a(n) such that the number of solutions to a(n)=a(j)+a(k), j

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A276204 a(0) = a(1) = 0. For n>1 a(n) is the smallest nonnegative integer such that there is no arithmetic progression k,m,n (of length 3) such that a(k)+a(m) = a(n).

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A051040 5-Stohr sequence.

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A060471 Smallest positive a(n) such that number of solutions to a(n)=a(j)+a(k) j

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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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A257941 Lexicographically earliest sequence of positive integers such that the terms and their absolute first differences are all distinct and no term is the sum of two distinct earlier terms.

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