A060469 -id:A060469 - 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)

A003662 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 8, 11, 13, 16, 18, 23, 25, 28, 30, 35, 37, 40, 42, 47, 49, 52, 54, 59, 61, 64, 66, 71, 73, 76, 78, 83, 85, 88, 90, 95, 97, 100, 102, 107, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 143, 145, 148, 150, 155, 157, 160, 162, 167, 169, 172, 174, 179, 181, 184

Offset: 1

N. J. A. Sloane, Mira Bernstein

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).

Colin Barker, Table of n, a(n) for n = 1..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A060469.

Cf. A003666.

Sort[Join[{8},Select[Range[200],MemberQ[{1,4,6,11},Mod[#,12]]&]]] (* Harvey P. Dale, Apr 26 2011 *)

Vec(x*(2*x^8+x^6-x^5+2*x^4+2*x^3+2*x^2+3*x+1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 27 2015

Numbers congruent to {1, 4, 6, 11} mod 12 plus the number 8.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7. - Colin Barker, Feb 27 2015
G.f.: x*(2*x^8 + x^6 - x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 3*x + 1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 27 2015

Name clarified by David A. Corneth, Mar 13 2023

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

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

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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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A003662 a(n) is smallest number != a(j) + a(k), j < k and a(1) = 1, a(2) = 4.

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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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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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