A037257 -id:A037257 - cp's OEIS frontend

A225385 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives P.

1, 3, 9, 20, 38, 64, 100, 148, 209, 284, 374, 480, 603, 745, 908, 1093, 1301, 1533, 1790, 2074, 2386, 2727, 3098, 3500, 3934, 4401, 4902, 5438, 6011, 6623, 7275, 7968, 8703, 9481, 10303, 11170, 12083, 13043, 14052, 15111, 16221, 17383, 18598, 19867, 21191, 22571, 24008, 25503, 27057, 28671, 30347, 32086, 33890, 35760, 37697, 39702, 41776, 43920

Offset: 1

N. J. A. Sloane, May 15 2013

nonn

In contrast to A225376-A225378, here it is not required (and not true) that each number should appear just once in P union Q union R. On the other hand, again in contrast to A225376-A225378, here it is obvious that P, Q, R are infinite.

The first three numbers that are repeated are 284, 2074, 3500, which appear in both P and Q. There may be no others. Of course R is disjoint from P and Q, by definition.

Cf. A225386, A225387, A005228, A030124, A037257, A225376, A225377, A225378.

# Based on Christopher Carl Heckman's program for A225376.
f:=proc(N) local h,dh,ddh,S,mex,i;
h:=1,3,9; dh:=2,6; ddh:=4; mex:=5; S:={h,dh,ddh};
for i from 4 to N do
while mex in S do S:=S minus {mex}; mex:=mex+1; od;
ddh:=ddh,mex; dh:=dh,dh[-1]+mex; h:=h,h[-1]+dh[-1];
S:=S union {h[-1], dh[-1], ddh[-1]};
mex:=mex+1;
od;
RETURN([[h],[dh],[ddh]]);
end;
f(100);

f[N_] := Module[{P = {1, 3, 9}, Q = {2, 6}, R = {4}, S, mex = 5, i},
  S = Join[P, Q, R];
  For[i = 4, i <= N, i++,
   While[MemberQ[S, mex], S = S~Complement~{mex}; mex++];
   AppendTo[R, mex];
   AppendTo[Q, Q[[-1]] + mex];
   AppendTo[P, P[[-1]] + Q[[-1]]];
   S = S~Union~{P[[-1]], Q[[-1]], R[[-1]]}; mex++];
P];
f[100] (* Jean-François Alcover, Mar 06 2023, after Maple code *)

A225386 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives Q.

Original entry on oeis.org

2, 6, 11, 18, 26, 36, 48, 61, 75, 90, 106, 123, 142, 163, 185, 208, 232, 257, 284, 312, 341, 371, 402, 434, 467, 501, 536, 573, 612, 652, 693, 735, 778, 822, 867, 913, 960, 1009, 1059, 1110, 1162, 1215, 1269, 1324, 1380, 1437, 1495, 1554, 1614, 1676, 1739, 1804, 1870, 1937, 2005, 2074, 2144

Offset: 1

N. J. A. Sloane, May 15 2013

nonn

In contrast to A225376-A225378, here it is not required (and not true) that each number should appear just once in P union Q union R. On the other hand, again in contrast to A225376-A225378, here it is obvious that P, Q, R are infinite.

The first three numbers that are repeated are 284, 2074, 3500, which appear in both P and Q. There may be no others. Of course R is disjoint from P and Q, by definition.

Cf. A225385, A225387, A005228, A030124, A037257, A225376, A225377, A225378.

Maple
```
See A225385.
```

A225387 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives R.

Original entry on oeis.org

4, 5, 7, 8, 10, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

N. J. A. Sloane, May 15 2013

nonn

In contrast to A225376-A225378, here it is not required (and not true) that each number should appear just once in P union Q union R. On the other hand, again in contrast to A225376-A225378, here it is obvious that P, Q, R are infinite.

The first three numbers that are repeated are 284, 2074, 3500, which appear in both P and Q. There may be no others. Of course R is disjoint from P and Q, by definition.

Cf. A225385, A225386, A005228, A030124, A037257, A225376, A225377, A225378.

Maple
```
See A225385.
```

A073158 a() = 1,2,4,7,... [ A073158 ], adjacent sums = 3,6,11... [ A073159 ] and 2nd adjacent sums = 9,17,... [ A073160 ] are disjoint but not monotonic; adjoin next free number to A073158 to construct three sequences that together include all positive integers.

Original entry on oeis.org

1, 2, 4, 7, 5, 8, 10, 14, 15, 19, 16, 20, 21, 22, 26, 28, 27, 30, 32, 33, 37, 38, 40, 39, 44, 45, 47, 46, 49, 50, 51, 52, 56, 58, 59, 61, 60, 64, 66, 67, 72, 68, 73, 74, 76, 80, 81, 82, 85, 86, 87, 88, 90, 94, 96, 97, 98, 100, 105, 104, 106, 107, 110, 111, 113

Offset: 1

Paul D. Hanna, Jul 29 2002

a(n)/n is asymptotic to 7n/4.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A037257, A035311, A073159, A073160.

A073158:_1, 2, 4, 7, 5, 8, 10, 14, 15, 19, 16, 20, 21, 22, 26, 28, 27, ... A073159:__3, 6, 11, 12, 13, 18, 24, 29, 34, 35, 36, 41, 43, 48, 54, 55, ... A073160:___9, 17, 23, 25, 31, 42, 53, 63, 69, 71, 77, 84, 91, 102, 109, ...

a(35) onward corrected by Sean A. Irvine, Nov 19 2024

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A225385 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives P.

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A225386 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives Q.

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Maple

A225387 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives R.

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Maple

A073158 a() = 1,2,4,7,... [ A073158 ], adjacent sums = 3,6,11... [ A073159 ] and 2nd adjacent sums = 9,17,... [ A073160 ] are disjoint but not monotonic; adjoin next free number to A073158 to construct three sequences that together include all positive integers.

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