A033291 -id:A033291 - cp's OEIS frontend

A192735 Left edge of the triangle in A033291.

1, 2, 6, 16, 30, 54, 91, 136, 198, 280, 374, 492, 637, 798, 990, 1216, 1462, 1746, 2071, 2420, 2814, 3256, 3726, 4248, 4825, 5434, 6102, 6832, 7598, 8430, 9331, 10272, 11286, 12376, 13510, 14724, 16021, 17366, 18798, 20320, 21894, 23562, 25327, 27148, 29070

Offset: 1

Reinhard Zumkeller, Jul 08 2011

nonn

a(n) = A087483(n-2) * n.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A192736.

Haskell
```
See A033291.
```

A192735 := proc(n)
    A087483(n-2)*n ;
end proc:
seq(A192735(n),n=1..10) ; # R. J. Mathar, Aug 10 2017

G.f.: x*(x^2+1)*(2*x^4+4*x^3+2*x^2+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 31 2013

A192736 Right edge of the triangle in A033291.

Original entry on oeis.org

1, 4, 12, 28, 50, 84, 133, 192, 270, 370, 484, 624, 793, 980, 1200, 1456, 1734, 2052, 2413, 2800, 3234, 3718, 4232, 4800, 5425, 6084, 6804, 7588, 8410, 9300, 10261, 11264, 12342, 13498, 14700, 15984, 17353, 18772, 20280, 21880, 23534, 25284, 27133, 29040

Offset: 1

Reinhard Zumkeller, Jul 08 2011

nonn

a(n) = A007980(n-1) * n.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A192735.

Haskell
```
See A033291.
```

G.f.: x*(2*x^5+2*x^4+6*x^3+5*x^2+2*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 31 2013

A045975 Take the first odd integer and multiple of 1, the next 2 even integers and multiples of 2, the next 3 odd integers and multiples of 3, the next 4 even integers and multiples of 4, ...

Original entry on oeis.org

1, 2, 4, 9, 15, 21, 24, 28, 32, 36, 45, 55, 65, 75, 85, 90, 96, 102, 108, 114, 120, 133, 147, 161, 175, 189, 203, 217, 224, 232, 240, 248, 256, 264, 272, 280, 297, 315, 333, 351, 369, 387, 405, 423, 441, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 561, 583, 605, 627, 649, 671, 693

Offset: 1

Fang-kuo Huang (gsyps(AT)ms17.hinet.net)

A generalized Connell sequence.

			Triangle begins:
    1;
    2,   4;
    9,  15,  21;
   24,  28,  32,  36;
   45,  55,  65,  75,  85;
   90,  96, 102, 108, 114, 120;
  133, 147, 161, 175, 189, 203, 217;
  ...

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A001614, A033291.

Seen as a triangle read by rows: cf. A204558 (row sums), A005917 (central terms), A204556 (left edge), A204557 (right edge).

a045975 n k = a045975_tabl !! (n-1) !! (k-1)
a045975_row n = a045975_tabl !! (n-1)
a045975_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys | even k    = take k ms
        | otherwise = take k $ filter odd ms
     ms = filter ((== 0) . (`mod` k)) xs
-- Reinhard Zumkeller, Jan 18 2012

first[n_?EvenQ] := (n - 1)*n^2/2; first[n_?OddQ] := n*(n^2 - 2n + 3)/2; row[n_] := (ro = {first[n]}; next = first[n] + n; While[ Length[ro] < n, If[Mod[next , 2] == Mod[n, 2], AppendTo[ro, next]]; next = next + n]; ro); Flatten[ Table[row[n], {n, 1, 11}]](* Jean-François Alcover, Jun 08 2012 *)

More terms from James Sellers

Keyword tabl added by Reinhard Zumkeller, Jan 18 2012

A071977 Triangle in which first row is {1}; to get n-th row take first n numbers greater than last number in previous row which are relatively prime to n.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 29, 31, 35, 37, 41, 43, 44, 45, 46, 47, 48, 50, 51, 53, 55, 57, 59, 61, 63, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115

Offset: 1

Amarnath Murthy, Jun 18 2002

			Triangle begins 1; 3 5; 7 8 10; 11 13 15 17; 18 19 21 22 23; 25 29 31 35 37 41; ....

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened

Diagonals give A071978, A071979.

A033291.

a071977 n k = a071977_tabl !! (n-1) !! (k-1)
a071977_row n = a071977_tabl !! (n-1)
a071977_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 1) . (gcd k)) xs
-- Reinhard Zumkeller, Jan 18 2012

a = {1}; k = 2; Do[i = 1; While[i < n + 1, If[ GCD[k, n] == 1, a = Append[a, k]; i++ ]; k++ ], {n, 2, 12}]; a

Edited by Robert G. Wilson v, Jun 28 2002

A128716 Triangle where the n-th row, of n terms in order, contains consecutive multiples of n. The smallest term of row n is the smallest integer greater than or equal to the largest term of row (n-1), for n >= 2.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 12, 16, 20, 24, 25, 30, 35, 40, 45, 48, 54, 60, 66, 72, 78, 84, 91, 98, 105, 112, 119, 126, 128, 136, 144, 152, 160, 168, 176, 184, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 363, 374, 385, 396

Offset: 1

Leroy Quet, Jun 12 2007

If we instead had the triangle where the smallest term of row n is the smallest integer strictly greater than the largest term of row (n-1), for n >= 2, then we would have sequence A033291.

			Triangle starts
    1;
    2,   4;
    6,   9,  12;
   12,  16,  20,  24;
   25,  30,  35,  40,  45;
   48,  54,  60,  66,  72,  78;
   84,  91,  98, 105, 112, 119, 126;
  128, 136, 144, 152, 160, 168, 176, 184;
  189, 198, 207, 216, 225, 234, 243, 252, 261;

Cf. A033291.

A128716 := proc(n,k) option remember ; if n = 1 then 1 ; elif k = 1 then n*ceil(A128716(n-1,n-1)/n) ; else A128716(n,k-1)+n ; fi ; end: for n from 1 to 11 do for k from 1 to n do printf("%d,",A128716(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007

T(n,k+1) = T(n,k) + n for 1 <= k < n. T(n,1) = n*ceiling(T(n-1,n-1)/n) for n >= 2. - R. J. Mathar, Nov 01 2007

More terms from R. J. Mathar, Nov 01 2007

cp's OEIS Frontend

A192735 Left edge of the triangle in A033291.

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A192736 Right edge of the triangle in A033291.

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Haskell

Formula

A045975 Take the first odd integer and multiple of 1, the next 2 even integers and multiples of 2, the next 3 odd integers and multiples of 3, the next 4 even integers and multiples of 4, ...

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A071977 Triangle in which first row is {1}; to get n-th row take first n numbers greater than last number in previous row which are relatively prime to n.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A128716 Triangle where the n-th row, of n terms in order, contains consecutive multiples of n. The smallest term of row n is the smallest integer greater than or equal to the largest term of row (n-1), for n >= 2.

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Author

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Comments

Examples

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Programs

Maple

Formula

Extensions