A319556 -id:A319556

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A094727

Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

+10
22

1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

All numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

LINKS

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

Bruno Berselli, Illustration of the initial terms.

László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

FORMULA

T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.

T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.

T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.

T(A005408(n), n) = A016777(n), n >= 0.

Sum_{k=1..n} T(n, k) = A000326(n) (row sums).

T(n, k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006

As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009

T(n, k) = n+k-1, n >= k >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]

T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012

From Boris Putievskiy, Jan 16 2013: (Start)

a(n) = A002260(n) + A003056(n).

a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

From G. C. Greubel, Mar 10 2024: (Start)

T(3*n-3, n) = A016813(n-1).

T(4*n-4, n) = A016861(n-1).

Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).

Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).

Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)

EXAMPLE

Triangle begins:

2, 3;

3, 4, 5;

4, 5, 6, 7;

5, 6, 7, 8, 9;

6, 7, 8, 9, 10, 11;

7, 8, 9, 10, 11, 12, 13;

8, 9, 10, 11, 12, 13, 14, 15;

9, 10, 11, 12, 13, 14, 15, 16, 17;

... - Philippe Deléham, Mar 30 2013

MATHEMATICA

Table[n + Range[0, n-1], {n, 12}]//Flatten (* Michael De Vlieger, Dec 16 2016 *)

PROG

(Magma) z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

(Haskell)

a094727 n k = n + k

a094727_row n = a094727_tabl !! (n-1)

a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]

-- Reinhard Zumkeller, Jul 22 2012

(SageMath) flatten([[n+k for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, Mar 10 2024

CROSSREFS

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.

Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.

Cf. A094728, A214604, A214661, A319556.

Cf. A128076 (rows reversed).

KEYWORD

nonn,tabl,easy

AUTHOR

Reinhard Zumkeller, May 24 2004

STATUS

approved

A128622

Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

+10
2

1, 1, 2, 3, 2, 3, 3, 4, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

FORMULA

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.

Sum_{k=1..n} T(n, k) = A014848(n) (row sums).

From G. C. Greubel, Mar 14 2024: (Start)

T(n, k) = n - (1 - (-1)^(n+k))/2 = n - (n+k mod 2).

T(n, 1) = A109613(n+1).

T(n, n) = A000027(n).

T(2*n-1, n) = A042963(n).

T(3*n-1, n) = A016777(n+1).

T(4*n-3, n) = A047461(n).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A000326(floor((n+1)/2)).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A123684(floor((n+1)/2)). (End)

EXAMPLE

First few rows of the triangle are:

1, 2;

3, 2, 3;

3, 4, 3, 4;

5, 4, 5, 4, 5;

5, 6, 5, 6, 5, 6;

7, 6, 7, 6, 7, 6, 7;

...

MATHEMATICA

Table[n - Mod[n+k, 2], {n, 16}, {k, n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

PROG

(Magma) [n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024

(SageMath) flatten([[n - ((n+k)%2) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 14 2024

CROSSREFS

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.

Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Mar 14 2007

EXTENSIONS

More terms added by G. C. Greubel, Mar 14 2024

STATUS

approved

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