A152717 -id:A152717 - cp's OEIS frontend

A004197 Triangle read by rows. T(n, k) = n - k if n - k < k, otherwise k.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2

Offset: 0

David W. Wilson

Table of min(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
Highest power of 6 that divides A036561. - Fred Daniel Kline, May 29 2012
Triangle T(n,k) read by rows: T(n,k) = min(k,n-k). - Philippe Deléham, Feb 25 2014

			From _Philippe Deléham_, Feb 25 2014: (Start)
Top left corner of table:
  0 0 0 0
  0 1 1 1
  0 1 2 2
  0 1 2 3
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 1, 1, 0;
  0, 1, 2, 1, 0;
  0, 1, 2, 2, 1, 0;
  0, 1, 2, 3, 2, 1, 0;
  0, 1, 2, 3, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0;
  ... (End)

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Similar to but strictly different from A087062 and A261684.
Row sums give A002620. - Reinhard Zumkeller, Jul 27 2005
Positions of zero are given in A117142. - Ridouane Oudra, Apr 30 2019
Cf. A144464, A152714, A152716, A152717.

a004197 n k = a004197_tabl !! n !! k
a004197_tabl = map a004197_row [0..]
a004197_row n = hs ++ drop (1 - n `mod` 2) (reverse hs)
   where hs = [0..n `div` 2]
-- Reinhard Zumkeller, Aug 14 2011

T := (n, k) -> if n - k < k then n - k else k fi:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 07 2023

Flatten[Table[IntegerExponent[2^(n - k) 3^k, 6], {n, 0, 20}, {k, 0, n}]] (* Fred Daniel Kline, May 29 2012 *)

T(x,y)=min(x,y) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A003983(n) - 1.
G.f.: x*y/((1-x)*(1-y)*(1-x*y)). - Franklin T. Adams-Watters, Feb 06 2006
2^T(n,k) = A144464(n,k), 3^T(n,k) = A152714(n,k), 4^T(n,k) = A152716(n,k), 5^T(n,k) = A152717(n,k). - Philippe Deléham, Feb 25 2014
a(n) = (1/2)*(t - 1 - abs(t^2 - 2*n - 1)), where t = floor(sqrt(2*n+1)+1/2). - Ridouane Oudra, May 03 2019

Mathematica program fixed by Harvey P. Dale, Nov 26 2020

Name edited by Peter Luschny, May 07 2023

A144464 Triangle T(n,m) read by rows: T(n,m) = 2^min(m,n-m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 8, 4, 2, 1, 1, 2, 4, 8, 8, 4, 2, 1, 1, 2, 4, 8, 16, 8, 4, 2, 1, 1, 2, 4, 8, 16, 16, 8, 4, 2, 1, 1, 2, 4, 8, 16, 32, 16, 8, 4, 2, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 09 2008

			The triangle starts in row n=0 as:
{1},
{1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 4, 2, 1},
{1, 2, 4, 4, 2, 1},
{1, 2, 4, 8, 4, 2, 1},
{1, 2, 4, 8, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 16, 8, 4, 2, 1},
{1, 2, 4, 8, 16, 32, 16, 8, 4, 2, 1}

Cf. A004197, A152714, A152716, A152717.

Clear[f, t]; f[n_, m_] = If[m <= Floor[n/2], m, n - m]; Table[Table[f[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

T(n,m)=1<Charles R Greathouse IV, Jan 15 2012

Row sums: sum_{m=0..n} T(n,m) = A027383(n).

T(n,k) = 2^A004197(n,k). - Philippe Deléham, Feb 25 2014

Offset corrected by the Associate Editors of the OEIS, Sep 11 2009

Better name by Philippe Deléham, Feb 25 2014

A152716 Triangle T(n,k) read by rows: T(n,k) = 4^min(k, n-k) = 4^A004197(n,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 4, 16, 4, 1, 1, 4, 16, 16, 4, 1, 1, 4, 16, 64, 16, 4, 1, 1, 4, 16, 64, 64, 16, 4, 1, 1, 4, 16, 64, 256, 64, 16, 4, 1, 1, 4, 16, 64, 256, 256, 64, 16, 4, 1, 1, 4, 16, 64, 256, 1024, 256, 64, 16, 4, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 11 2008

Row sums are: {1, 2, 6, 10, 26, 42, 106, 170, 426, 682, 1706,...} = A061547(n+2).

			{1},
{1, 1},
{1, 4, 1},
{1, 4, 4, 1},
{1, 4, 16, 4, 1},
{1, 4, 16, 16, 4, 1},
{1, 4, 16, 64, 16, 4, 1},
{1, 4, 16, 64, 64, 16, 4, 1},
{1, 4, 16, 64, 256, 64, 16, 4, 1},
{1, 4, 16, 64, 256, 256, 64, 16, 4, 1},
{1, 4, 16, 64, 256, 1024, 256, 64, 16, 4, 1}

Cf. A004197, A144464, A152714, A152717.

Clear[a, k, m]; k = 4; a[0] = {1}; a[1] = {1, 1};
a[n_] := a[n] = Join[{1}, k*a[n - 2], {1}];
Table[a[n], {n, 0, 10}];
Flatten[%]

T(n,k) = 4^min(k, n-k). - Philippe Deléham, Feb 25 2014

T(n,k) = A144464(n,k)^2. - Philippe Deléham, Feb 26 2014

Better name by Philippe Deléham, Feb 25 2014

A152714 Triangle read by rows: T(n,k) = 3^min(k, n-k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 9, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 27, 9, 3, 1, 1, 3, 9, 27, 27, 9, 3, 1, 1, 3, 9, 27, 81, 27, 9, 3, 1, 1, 3, 9, 27, 81, 81, 27, 9, 3, 1, 1, 3, 9, 27, 81, 243, 81, 27, 9, 3, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 11 2008

			Triangle begins
  {1},
  {1, 1},
  {1, 3, 1},
  {1, 3, 3,  1},
  {1, 3, 9,  3,  1},
  {1, 3, 9,  9,  3,   1},
  {1, 3, 9, 27,  9,   3,  1},
  {1, 3, 9, 27, 27,   9,  3,  1},
  {1, 3, 9, 27, 81,  27,  9,  3, 1},
  {1, 3, 9, 27, 81,  81, 27,  9, 3, 1},
  {1, 3, 9, 27, 81, 243, 81, 27, 9, 3, 1}

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A004197, A144464, A152716, A152717, A062318 (row sums).

[[3^(Min(k,n-k)): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Sep 01 2018

Clear[a, k, m]; k = 3; a[0] = {1}; a[1] = {1, 1};
a[n_] := a[n] = Join[{1}, k*a[n - 2], {1}];
Table[a[n], {n, 0, 10}];
Flatten[%]
Table[3^(Min[k, n - k]), {n, 0, 100}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2018 *)

for(n=0,15, for(k=0,n, print1(3^(min(k,n-k)), ", "))) \\ G. C. Greubel, Sep 01 2018

T(n,k) = 3^min(k, n-k) = 3^A004197(n,k). - Philippe Deléham, Feb 25 2014

Better name by Philippe Deléham, Feb 25 2014

A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062, 45776367187, 76293945312

Offset: 0

Philippe Deléham, Feb 25 2014

Row sums of triangle in A152717.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Cf. A057651, A125831, A152717, A198306.

LinearRecurrence[{1,5,-5},{1,2,7},40] (* Harvey P. Dale, Jul 18 2024 *)

G.f.: (1+x)/((1-x)*(1-5*x^2)).
a(n) = Sum_{k=0..n} A152717(n,k).
a(2*n) = A057651(n).
a(2*n+1) = A125831(n+1) = 2*A003463(n+1).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3), a(0) = 1, a(1) = 2, a(2) = 7.
a(n) = A198306(n+1) for n > 1. - Georg Fischer, Oct 23 2018

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A004197 Triangle read by rows. T(n, k) = n - k if n - k < k, otherwise k.

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A144464 Triangle T(n,m) read by rows: T(n,m) = 2^min(m,n-m).

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A152716 Triangle T(n,k) read by rows: T(n,k) = 4^min(k, n-k) = 4^A004197(n,k).

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A152714 Triangle read by rows: T(n,k) = 3^min(k, n-k).

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A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

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