A056233 -id:A056233 - cp's OEIS frontend

A057153 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives first row of array.

1, 3, 9, 26, 73, 194, 485, 1150, 2617, 5779, 12497, 26653, 56355, 118545, 248605, 520365, 1087853, 2272770, 4748677, 9930475, 20800122, 43658310, 91836751, 193549010, 408449129, 862455193, 1820771343, 3840569196, 8089327515

Offset: 1

Jonas Wallgren, Jul 30 2000

This is the limit of the family of sequences starting with A005228, A037257, A037260. Adjoin next free number as the first element in a new level of differences unless it would produce a duplicate in which case ignore.

			Array begins
1 3 9 26 73 194 ...
.2 6 17 47 121 ...
. 4 11 30 74 ...
.. 7 19 44 ...
... 12 25 ...
.... 13 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..150

Cf. A057154 (numbers not used), A052474 (main diagonal), A056230 (array). See also A005228, A037257, A037260, A057267, A056231, A056232, A056233, A056234.
Cf. A133544.
Cf. A200379 (first differences).

a057153 n = a056230_tabl !! (n-1) !! (n-1)
-- Reinhard Zumkeller, Nov 19 2011

Row sums of triangle A133544. - Gary W. Adamson, Sep 14 2007

More terms from Rob Speer (rob(AT)twcny.rr.com) and Loren Merritt, Aug 14 2000

A052474 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives main diagonal.

Original entry on oeis.org

1, 2, 4, 7, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 32, 35, 36, 39, 42, 48, 49, 50, 51, 53, 54, 57, 58, 59, 61, 62, 64, 66, 68, 76, 77, 78, 80, 82, 86, 87, 89, 91, 92, 93, 98, 103, 109, 110, 118, 124, 127, 128, 129, 132, 133, 135, 136, 139, 140, 141, 142, 143, 149

Offset: 1

Jonas Wallgren, Jul 30 2000

			Array begins
1 3 9 26 73 194 ...
.2 6 17 47 121 ...
. 4 11 30 74 ...
.. 7 19 44 ...
... 12 25 ...
.... 13 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..150

Cf. A057153 (first row), A057154 (numbers not used), A056230 (array), A056231, A056232, A056233, A056234.

a052474 n = a056230_tabl !! (n-1) !! 0
-- Reinhard Zumkeller, Nov 19 2011

A056230 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives array read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 7, 11, 17, 26, 12, 19, 30, 47, 73, 13, 25, 44, 74, 121, 194, 14, 27, 52, 96, 170, 291, 485, 15, 29, 56, 108, 204, 374, 665, 1150, 16, 31, 60, 116, 224, 428, 802, 1467, 2617, 18, 34, 65, 125, 241, 465, 893, 1695, 3162, 5779, 20, 38, 72, 137, 262

Offset: 1

Jonas Wallgren, Jul 30 2000

			Array begins
1 3 9 26 73 194 ...
.2 6 17 47 121 ...
. 4 11 30 74 ...
.. 7 19 44 ...
... 12 25 ...
.... 13 ...

Reinhard Zumkeller, Antidiagonals n=1..150 of array, flattened

Cf. A057153 (first row), A200379 (second row), A052474 (main diagonal), A057154 (numbers not used), A056231, A056232, A056233, A056234.

import Data.List (intersect, union, (\\))
a056230 n k = a056230_tabl !! (n-1) !! (k-1)
a056230_tabl = [1] : f [1] [2..] [1] where
   f adiag (a:as) us | null (adiag' `intersect` us) =
                       adiag' : f adiag' (as \\ adiag') (us `union` adiag')
                     | otherwise = f adiag as us
                     where adiag' = scanl (+) a adiag
-- Reinhard Zumkeller, Nov 19 2011

A056231 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives row 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 10, 12, 13, 14, 16, 19, 20, 21, 23, 28, 31, 32, 34, 36, 37, 38, 43, 45, 46, 48, 49, 50, 53, 54, 55, 56, 58, 60, 61, 62, 64, 67, 68, 69, 71, 76, 77, 78, 79, 82, 83, 84, 86, 87, 89, 92, 96, 98, 100, 101, 102, 104, 105, 106, 108, 113, 115

Offset: 1

N. J. A. Sloane, E. M. Rains, Aug 22 2000

			Array begins
1 2 4 7 8 10 12 ...
.3 6 11 15 18 ...
. 9 17 26 33 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A056232, A056233, A056234. See also A057153, A052474, A057154, A056230.

import Data.List (transpose)
a056231 n = ([1,2] ++ threeRows !! 0) !! (n-1)
a056232 n = ([3] ++ threeRows !! 1) !! (n-1)
a056233 n = threeRows !! 2 !! (n-1)
threeRows = transpose $ f [4..] [1,2,3] [2,1] [3] [] where
   f (u:free) used us vs ws
       | u `notElem` used &&
         v `notElem` used &&
         w `notElem` used = [u, v, w] :
                            f free (w:v:u:used) (u:us) (v:vs) (w:ws)
       | otherwise        = f free used us vs ws
       where v = u + head us; w = v + head vs
-- Reinhard Zumkeller, Oct 18 2011

a := [1,2,4]; an := 3; b := [3,6]; bn := 2; c := [9]; cn := 1; h := array(1..10000); h[1] := 1; h[2] := 1; h[3] := 1; h[4] := 1; h[6] := 1; h[9] := 1; m := []; k := 5;
for i from 1 to 200 do for n from k to k+100 do n1 := a[an]+n; n2 := b[bn]+n1; if h[n]<>1 and h[n1]<>1 and h[n2]<>1 then h[n] := 1; h[n1] := 1; h[n2] := 1; an := an+1; bn := bn+1; cn := cn+1; a := [op(a), n]; b := [op(b), n1]; c := [op(c), n2]; k := n+1; break;
else if h[n]<>1 then m := [op(m), n]; fi; fi; od; od; a; b; c; m;

a = {1, 2, 4}; an = 3; b = {3, 6}; bn = 2; c = {9}; cn = 1; Clear[h]; h[_] = 0; h[1] = h[2] = h[3] = h[4] = h[6] = h[9] = 1; m = {}; k = 5;
For[i = 1, i <= 200, i++, For[n = k, n <= k + 100, n++, n1 = a[[an]] + n; n2 = b[[bn]] + n1; If[h[n] != 1 && h[n1] != 1 && h[n2] != 1, h[n] = 1; h[n1] = 1; h[n2] = 1; an++; bn++; cn++; AppendTo[a, n]; AppendTo[b, n1]; AppendTo[c, n2]; k = n+1; Break[], If[h[n] != 1, AppendTo[m, n]]]]];
{a, b, c, m} (* Jean-François Alcover, Dec 17 2019, translated from Maple *)

A056232 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives row 2.

Original entry on oeis.org

3, 6, 11, 15, 18, 22, 25, 27, 30, 35, 39, 41, 44, 51, 59, 63, 66, 70, 73, 75, 81, 88, 91, 94, 97, 99, 103, 107, 109, 111, 114, 118, 121, 123, 126, 131, 135, 137, 140, 147, 153, 155, 157, 161, 165, 167, 170, 173, 176, 181, 188, 194, 198, 201, 203

Offset: 1

N. J. A. Sloane, E. M. Rains, Aug 22 2000

			Array begins
1 2 4 7 8 10 12 ...
.3 6 11 15 18 ...
. 9 17 26 33 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A056231, A056233, A056234. See also A057153, A052474, A057154, A056230.

Haskell
```
cf. A056231.
```

A056234 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives numbers not used.

Original entry on oeis.org

5, 24, 29, 42, 72, 90, 93, 112, 120, 125, 138, 158, 172, 175, 192, 197, 200, 205, 208, 213, 218, 230, 235, 264, 282, 285, 302, 305, 310, 321, 324, 329, 333, 364, 372, 375, 378, 386, 416, 430, 439, 452, 455, 477, 496, 504, 509, 522, 542, 556

Offset: 1

N. J. A. Sloane, E. M. Rains, Aug 22 2000

			Array begins
1 2 4 7 8 10 12 ...
.3 6 11 15 18 ...
. 9 17 26 33 ...

Zak Seidov and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A056231, A056232, A056233. See also A057153, A052474, A057154, A056230.

a056234 n = a056234_list !! (n-1)
a056234_list = notUsed 1 a056231_list a056232_list a056233_list where
   notUsed x us'@(u:us) vs'@(v:vs) ws'@(w:ws)
    | x == u = notUsed (x + 1) us vs' ws'
    | x == v = notUsed (x + 1) us' vs ws'
    | x == w = notUsed (x + 1) us' vs' ws
    | otherwise = x : notUsed (x + 1) us' vs' ws'
-- Reinhard Zumkeller, Nov 07 2011

A057154 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives numbers not used.

Original entry on oeis.org

5, 8, 10, 24, 28, 33, 37, 40, 46, 63, 69, 70, 83, 85, 94, 95, 102, 105, 106, 112, 113, 114, 119, 131, 145, 147, 148, 152, 161, 165, 166, 175, 181, 197, 203, 207, 215, 221, 235, 236, 239, 247, 253, 254, 267, 270, 276, 286, 289, 290, 296, 307, 309, 315, 317, 322

Offset: 1

Jonas Wallgren, Jul 30 2000

			Array begins
1 3 9 26 73 194 ...
.2 6 17 47 121 ...
. 4 11 30 74 ...
.. 7 19 44 ...
... 12 25 ...
.... 13 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..514

Cf. A057153 (first row), A052474 (main diagonal), A056230 (array), A056231, A056232, A056233, A056234.

import Data.List (intersect, union, (\\))
a057154 n = a057154_list !! (n-1)
a057154_list = g [1] [2..] [1] where
   g ds (a:as) us
     | null (ds' `intersect` us) = g ds' (as \\ ds') (us `union` ds')
     | otherwise = a : g ds as us
     where ds' = scanl (+) a ds
-- Reinhard Zumkeller, Nov 19 2011

More terms from Rob Speer (rob(AT)twcny.rr.com) and Loren Merritt, Aug 14 2000

Further terms from Larry Reeves (larryr(AT)acm.org), May 09 2001

cp's OEIS Frontend

A057153 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives first row of array.

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A052474 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives main diagonal.

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A056230 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives array read by antidiagonals.

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A056231 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives row 1.

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A056232 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives row 2.

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A056234 Form an array with 3 rows: row 1 begins with 1; all rows are increasing; each entry is sum of 2 entries above it; each number appears at most once; smallest unused number is appended to first row if possible. Sequence gives numbers not used.

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A057154 Construct difference array so that (1) first row begins with 1, (2) every row is monotonic increasing, (3) no number appears more than once, (4) smallest number not yet used begins a new row. Sequence gives numbers not used.

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