A133819 -id:A133819 - cp's OEIS frontend

A141719 Triangle online: 4, A140978(n+1) - A133819 .

4, 8, 8, 12, 15, 12, 16, 24, 21, 16, 20, 35, 32, 27, 20, 24, 48, 45, 40, 33, 24, 28, 63, 60, 55, 48, 39, 28, 32, 80, 77, 72, 65, 56, 45, 32, 36, 99, 96, 91, 84, 75, 64, 51, 36, 40, 120, 117, 112, 105, 96, 85, 72, 57, 40, 44

Offset: 1

Paul Curtz, Sep 11 2008

4 is A140978(0).

Triangle: 4; 8, 8; 12, 15, 12; 16, 24, 21, 16; 20, 35, 32, 27, 20; .

A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.

Original entry on oeis.org

3, 8, 5, 15, 12, 7, 24, 21, 16, 9, 35, 32, 27, 20, 11, 48, 45, 40, 33, 24, 13, 63, 60, 55, 48, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 2

Wolfdieter Lang, Jul 20 2006

The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1 >= n, else 0, are used to compute the frequencies of the spectral lines of the H-atom according to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) an approximation for the Rydberg frequency. R' indicates, that the correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses for the electron and proton, has been used for the Rydberg constant R_infinity. c:=299792458 m/s is, per definition, the velocity of light in vacuo (see A003678).
In order to compute the wave length of the spectral lines approximately one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies one uses approximately E(m,n) = r(m,n)*13.599 eV (electron Volts).
The author was inspired by Dewdney's book to compile this table and related ones.
For the approximate frequencies, energies and wavelengths of the first members of the Lyman (n=1, m>=2), Balmer (n=2, m>=3), Paschen (n=3, m>=4), Brackett (n=4, m>=5) and Pfund (n=5, m>=6) series see the W. Lang link under A120072.
Frenicle wrote this as a(n+1) = A140978(n) - A133819(n-1). - Paul Curtz, Aug 19 2008
This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - Raphie Frank, Dec 30 2012

			Triangle begins
  [ 3];
  [ 8, 5];
  [15, 12,  7];
  [24, 21, 16,  9];
  [35, 32, 27, 20, 11];
  ...

A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.

Stanislav Sykora, Table of n, a(n) for n = 2..79801
W. Lang: First ten rows and more.
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11.
Wikipedia, Spin (physics)
Wikipedia, Hydrogen spectral series

Row sums give A016061(n-1), n>=2.

Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).

ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)

nmax=400;a=vector(1+nmax*(nmax-1)\2);idx=1;for(n=2,nmax,for(k=1,n-1,a[idx]=n*n-k*k;idx++)) \\ Stanislav Sykora, Feb 17 2014

T(n,k)=n^2-k^2;
for (n=1,10,for(k=1,n-1, print1(T(n,k),", ")));
\\ Joerg Arndt, Feb 23 2014

a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012

A055096 Triangle read by rows, sums of 2 distinct nonzero squares: T(n,k) = k^2+n^2, (n>=2, 1 <= k <= n-1).

Original entry on oeis.org

5, 10, 13, 17, 20, 25, 26, 29, 34, 41, 37, 40, 45, 52, 61, 50, 53, 58, 65, 74, 85, 65, 68, 73, 80, 89, 100, 113, 82, 85, 90, 97, 106, 117, 130, 145, 101, 104, 109, 116, 125, 136, 149, 164, 181, 122, 125, 130, 137, 146, 157, 170, 185, 202, 221, 145, 148, 153, 160

Offset: 2

Antti Karttunen, Apr 04 2000

Discovered by Bernard Frénicle de Bessy (1605?-1675). - Paul Curtz, Aug 18 2008
Terms that are not hypotenuses in primitive Pythagorean triangles, are replaced by 0 in A222946. - Reinhard Zumkeller, Mar 23 2013
This triangle T(n,k) gives the circumdiameters for the Pythagorean triangles with a = (n+1)^2 - k^2, b = 2*(n+1)*k and c = (n+1)^2 + k^2 (see the Floor van Lamoen entries or comments A063929, A063930, A002283, A003991). See also the formula section. Note that not all Pythagorean triangles are covered, e.g., (9,12,15) does not appear. - Wolfdieter Lang, Dec 03 2014

			The triangle T(n, k) begins:
n\k   1   2   3   4   5   6   7   8   9  10  11 ...
2:    5
3:   10  13
4:   17  20  25
5:   26  29  34  41
6:   37  40  45  52  61
7:   50  53  58  65  74  85
8:   65  68  73  80  89 100 113
9:   82  85  90  97 106 117 130 145
10: 101 104 109 116 125 136 149 164 181
11: 122 125 130 137 146 157 170 185 202 221
12: 145 148 153 160 169 180 193 208 225 244 265
...
13: 170 173 178 185 194 205 218 233 250 269 290 313,
14: 197 200 205 212 221 232 245 260 277 296 317 340 365,
15: 226 229 234 241 250 261 274 289 306 325 346 369 394 421,
16: 257 260 265 272 281 292 305 320 337 356 377 400 425 452 481,
...
Formatted and extended by _Wolfdieter Lang_, Dec 02 2014 (reformatted Jun 11 2015)
The successive terms are (1^2+2^2), (1^2+3^2), (2^2+3^2), (1^2+4^2), (2^2+4^2), (3^2+4^2), ...

Reinhard Zumkeller, Rows n = 2..121 of triangle, flattened
M. de Frénicle, Méthode pour trouver la solutions des problèmes par les exclusions, in: "Divers ouvrages de mathématiques et de physique, par Messieurs de l'Académie royale des sciences", Paris, 1693, pp 1-44.
Antti Karttunen, Larger table, showing also locations of 4k+1 primes and squares
Eric Weisstein's World of Mathematics, Congruum Problem.
Index entries for sequences related to sums of squares

Sorting gives A024507. Count of divisors: A055097, Möbius: A055132. For trinv, follow A055088.
Cf. A001844 (right edge), A002522 (left edge), A033429 (central column).
Cf. A000290, A033583, A079273, A190816, A226141, A331987.

a055096 n k = a055096_tabl !! (n-1) !! (k-1)
a055096_row n = a055096_tabl !! (n-1)
a055096_tabl = zipWith (zipWith (+)) a133819_tabl a140978_tabl
-- Reinhard Zumkeller, Mar 23 2013

[n^2+k^2: k in [1..n-1], n in [2..15]]; // G. C. Greubel, Apr 19 2023

sum2distinct_squares_array := (n) -> (((n-((trinv(n-1)*(trinv(n-1)-1))/2))^2)+((trinv(n-1)+1)^2));

T[n_, k_]:= (n+1)^2 + k^2; Table[T[n, k], {n,15}, {k,n}]//Flatten (* Jean-François Alcover, Mar 16 2015, after Reinhard Zumkeller *)

def A055096(n,k): return n^2 + k^2
flatten([[A055096(n,k) for k in range(1,n)] for n in range(2,16)]) # G. C. Greubel, Apr 19 2023

a(n) = sum2distinct_squares_array(n).
T(n, 1) = A002522(n).
T(n, n-1) = A001844(n-1).
T(2*n-2, n-1) = A033429(n-1).
T(n,k) = A133819(n,k) + A140978(n,k) = (n+1)^2 + k^2, 1 <= k <= n. - Reinhard Zumkeller, Mar 23 2013
T(n, k) = a*b*c/(2*sqrt(s*(s-1)*(s-b)*(s-c))) with s =(a + b + c)/2 and the substitution a = (n+1)^2 - k^2, b = 2*(n+1)*k and c = (n+1)^2 + k^2 (the circumdiameter for the considered Pythagorean triangles). - Wolfdieter Lang, Dec 03 2014
From Bob Selcoe, Mar 21 2015:  (Start)
T(n,k) = 1 + (n-k+1)^2 + Sum_{j=0..k-2} (4*j + 2*(n-k+3)).
T(n,k) = 1 + (n+k-1)^2 - Sum_{j=0..k-2} (2*(n+k-3) - 4*j).
Therefore: 4*(n-k+1) + Sum_{j=0..k-2} (2*(n-k+3) + 4*j) = 4*n(k-1) - Sum_{j=0..k-2} (2*(n+k-3) - 4*j). (End)
From G. C. Greubel, Apr 19 2023: (Start)
T(2*n-3, n-1) = A033429(n-1).
T(2*n-4, n-2) = A079273(n-1).
T(2*n-2, n) = A190816(n).
T(3*n-4, n-1) = 10*A000290(n-1) = A033583(n-1).
Sum_{k=1..n-1} T(n, k) = A331987(n-1).
Sum_{k=1..floor(n/2)} T(n-k, k) = A226141(n-1). (End)

Edited: in T(n, k) formula by Reinhard Zumkeller k < n replaced by k <= n. - Wolfdieter Lang, Dec 02 2014

Made definition more precise, changed offset to 2. - N. J. A. Sloane, Mar 30 2015

A048152 Triangular array T read by rows: T(n,k) = k^2 mod n, for 1 <= k <= n, n >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 0

Offset: 1

Clark Kimberling

			Rows:
  0;
  1, 0;
  1, 1, 0;
  1, 0, 1, 0;
  1, 4, 4, 1, 0;
  1, 4, 3, 4, 1, 0;

T. D. Noe, Rows n = 1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Quadratic Residue

Cf. A060036.
Cf. A225126 (central terms).
Cf. A070430 (row 5), A070431 (row 6), A053879 (row 7), A070432 (row 8), A008959 (row 10), A070435 (row 12), A070438 (row 15), A070422 (row 20).
Cf. A046071 (in ascending order, without zeros and duplicates).
Cf. A063987 (for primes, in ascending order, without zeros and duplicates).

a048152 n k = a048152_tabl !! (n-1) !! (k-1)
a048152_row n = a048152_tabl !! (n-1)
a048152_tabl = zipWith (map . flip mod) [1..] a133819_tabl
-- Reinhard Zumkeller, Apr 29 2013

Flatten[Table[PowerMod[k,2,n],{n,15},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

T(n,k) = A133819(n,k) mod n, k = 1..n. - Reinhard Zumkeller, Apr 29 2013

T(n,k) = (T(n,k-1) + (2k+1)) mod n. - Andrés Ventas, Apr 06 2021

A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

3, 7, 12, 13, 19, 27, 21, 28, 37, 48, 31, 39, 49, 61, 75, 43, 52, 63, 76, 91, 108, 57, 67, 79, 93, 109, 127, 147, 73, 84, 97, 112, 129, 148, 169, 192, 91, 103, 117, 133, 151, 171, 193, 217, 243, 111, 124, 139, 156, 175, 196, 219, 244, 271, 300, 133, 147, 163

Offset: 1

Reinhard Zumkeller, Nov 11 2012

			The triangle begins:
row n   T(n,k), 1 <= k <= n
   1:     3
   2:     7   12
   3:    13   19   27
   4:    21   28   37   48
   5:    31   39   49   61   75
   6:    43   52   63   76   91  108
   7:    57   67   79   93  109  127  147
   8:    73   84   97  112  129  148  169  192
   9:    91  103  117  133  151  171  193  217  243
  10:   111  124  139  156  175  196  219  244  271  300
  11:   133  147  163  181  201  223  247  273  301  331  363
  12:   157  172  189  208  229  252  277  304  333  364  397  432

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

Cf. A215646 (row sums), A002061 (left edge, shifted), A033428 (right edge), A003215.

a215631 n k = a215631_tabl !! (n-1) !! (k-1)
a215631_row n = a215631_tabl !! (n-1)
a215631_tabl = zipWith3 (zipWith3 (\u v w -> u + v + w))
                        a093995_tabl a075362_tabl a133819_tabl
-- Reinhard Zumkeller, Nov 11 2012

[[i^2+i*j+j^2: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Jun 07 2015

seq(seq(i^2+i*j+j^2, j=1..i),i=1..10); # Robert Israel, May 10 2015

Table[n^2 + n*k + k^2, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, May 12 2015 *)

for(n=1,15,for(k=1,n,print1(n^2+n*k+k^2,", "))) \\ Derek Orr, May 13 2015

T(n,k) = 2*A070216(n,k) - A215630(n,k).
G.f. for triangle: (3-2*x+3*x*y+x^2-11*x^2*y+4*x^3*y+x^3*y^2+x^4*y^2)*x*y/((1-x)^3*(1-x*y)^3). - Robert Israel, May 10 2015
From Avi Friedlich, May 26 2015: (Start)
T(n,k) = A093995(n,k) + A075362(n,k) + A133819(n,k).
T(k+1,k) = A003215(k).
T(k+2,k) = A003215(k)/2 + A003215(k+1)/2.
T(k+3,k) = A003215(k)/3 + A003215(k+1)/3 + A003215(k+2)/3 and so on. (End)

A215630 Triangle read by rows: T(n,k) = n^2 - n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

1, 3, 4, 7, 7, 9, 13, 12, 13, 16, 21, 19, 19, 21, 25, 31, 28, 27, 28, 31, 36, 43, 39, 37, 37, 39, 43, 49, 57, 52, 49, 48, 49, 52, 57, 64, 73, 67, 63, 61, 61, 63, 67, 73, 81, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 111, 103, 97, 93, 91, 91, 93, 97, 103, 111

Offset: 1

Reinhard Zumkeller, Nov 11 2012

T(n,k) = A093995(n,k) - A075362(n,k) + A133819(n,k) = 2*A070216(n,k) - A215631(n,k), 1 <= k <= n.

			The triangle begins:
.  1:     1
.  2:     3    4
.  3:     7    7    9
.  4:    13   12   13   16
.  5:    21   19   19   21   25
.  6:    31   28   27   28   31   36
.  7:    43   39   37   37   39   43   49
.  8:    57   52   49   48   49   52   57   64
.  9:    73   67   63   61   61   63   67   73   81
. 10:    91   84   79   76   75   76   79   84   91  100
. 11:   111  103   97   93   91   91   93   97  103  111  121
. 12:   133  124  117  112  109  108  109  112  117  124  133  144 .

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

Cf. A004068 (row sums), A002061 (left edge), A000290 (right edge).

Cf. A003215 (central terms).

a215630 n k = a215630_tabl !! (n-1) !! (k-1)
a215630_row n = a215630_tabl !! (n-1)
a215630_tabl = zipWith3 (zipWith3 (\u v w -> u - v + w))
                        a093995_tabl a075362_tabl a133819_tabl

A124258 Triangle whose rows are sequences of increasing and decreasing squares: 1; 1,4,1; 1,4,9,4,1; ...

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16

Offset: 1

Jonathan Vos Post, Dec 16 2006

The triangle A003983 with individual entries squared and each 2nd row skipped.
Analogous to A004737. - Peter Bala, Sep 25 2007
T(n,k) =  min(n,k)^2. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

			Triangle starts
  1;
  1, 4, 1;
  1, 4, 9, 4, 1:
  1, 4, 9, 16, 9, 4, 1:
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...4...4...4...4...4...
  1...4...9...9...9...9...
  1...4...9..16..16..16...
  1...4...9..16..25..25...
  1...4...9..16..25..36...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 4, 1;
  1, 4, 9,  4,  1;
  1, 4, 9, 16,  9,  4,  1;
  1, 4, 9, 16, 25, 16,  9,  4, 1;
  1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1;
  ...
Row number k contains 2*k-1 numbers 1,4,...,(k-1)^2,k^2,(k-1)^2,...,4,1. (End)

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

Cf. A005900 (row sums), A004737, A133819, A133823, A133824, A133825, A003983.

A003983 := proc(n,k) min(n,k) ; end: A124258 := proc(n,k) A003983(n,k)^2 ; end: for d from 1 to 20 by 2 do for c from 1 to d do printf("%d, ",A124258(d+1-c,c)) ; od: od: # R. J. Mathar, Sep 21 2007
# second Maple program:
T:= n-> i^2$i=1..n, (n-i)^2$i=1..n-1:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 15 2022

Flatten[Table[Join[Range[n]^2,Range[n-1,1,-1]^2],{n,10}]] (* Harvey P. Dale, Jun 14 2015 *)

O.g.f.: (1+qx)^2/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 9q^2 + 4q^3 + q^4) + ... . - Peter Bala, Sep 25 2007
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^2. (End)

More terms from R. J. Mathar, Sep 21 2007

Edited by N. J. A. Sloane, Jun 30 at the suggestion of R. J. Mathar

A140978 Repeat (n+1)^2 n times.

Original entry on oeis.org

4, 9, 9, 16, 16, 16, 25, 25, 25, 25, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Paul Curtz, Aug 17 2008

See A093995.
Frenicle writes the entries in the form a(n) = A055096(n)-A133819(n), with the flattened index view of A133819: 4=5-1, 9=10-1, 9=13-4, 16=17-1, 16=20-4, 16=25-9 etc.
Also triangle T(n, k) = (n+1)^2, 1<=k<=n. - Michel Marcus, Feb 03 2013

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, table page 11.

Cf. A000290.

a140978 n k = a140978_tabl !! (n-1) !! (k-1)
a140978_row n = a140978_tabl !! (n-1)
a140978_tabl = map snd $ iterate
               (\(i, xs@(x:_)) -> (i + 2, map (+ i) (x:xs))) (5, [4])
-- Reinhard Zumkeller, Mar 23 2013

Table[PadRight[{},n,(n+1)^2],{n,10}]//Flatten (* Harvey P. Dale, Oct 10 2019 *)

from math import isqrt
def A140978(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1))+1)**2 # Chai Wah Wu, Nov 07 2024

a(n)=(A003057(n+1))^2. - R. J. Mathar, Aug 25 2008

A245717 Triangle read by rows: T(n,k) = gcd(n,k^2), 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 4, 1, 8, 1, 4, 1, 8, 1, 1, 9, 1, 1, 9, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Reinhard Zumkeller, Jul 30 2014

			First rows and their sums (A078430):
.    1:   1                                           1
.    2:   1, 2                                        3
.    3:   1, 1, 3                                     5
.    4:   1, 4, 1, 4                                 10
.    5:   1, 1, 1, 1, 5                               9
.    6:   1, 2, 3, 2, 1,  6                          15
.    7:   1, 1, 1, 1, 1,  1, 7                       13
.    8:   1, 4, 1, 8, 1,  4, 1, 8                    28
.    9:   1, 1, 9, 1, 1,  9, 1, 1, 9                 33
.   10:   1, 2, 1, 2, 5,  2, 1, 2, 1, 10             27
.   11:   1, 1, 1, 1, 1,  1, 1, 1, 1,  1, 11         21
.   12:   1, 4, 3, 4, 1, 12, 1, 4, 3,  4,  1, 12     50

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A050873, A002024, A133819, A078430 (row sums).

a245717 n k = a245717_tabl !! (n-1) !! (k-1)
a245717_row n = a245717_tabl !! (n-1)
a245717_tabl = zipWith (zipWith gcd) a002024_tabl a133819_tabl

Table[GCD[n,k^2],{n,15},{k,n}]//Flatten (* Harvey P. Dale, Nov 05 2022 *)

row(n) = vector(n, k, gcd(n, k^2)); \\ Michel Marcus, Jan 24 2022

A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

Original entry on oeis.org

1, 1, 8, 1, 8, 27, 1, 8, 27, 64, 1, 8, 27, 64, 125, 1, 8, 27, 64, 125, 216, 1, 8, 27, 64, 125, 216, 343, 1, 8, 27, 64, 125, 216, 343, 512, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,8,27,1,8,27,64,..., analogous to A002260.

			Triangle starts
1;
1, 8;
1, 8, 27;
1, 8, 27, 64;
1, 8, 27, 64, 125;

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

Cf. A000537 (row sums), A002260, A019522, A133819, A133821, A133823.

a133820 n k = a133820_tabl !! (n-1) !! (k-1)
a133820_row n = a133820_tabl !! (n-1)
a133820_tabl = map (`take` (tail a000578_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,c},c=Range[nn]^3;Flatten[Table[Take[c,n],{n,10}]]] (* Harvey P. Dale, Mar 05 2014 *)

O.g.f.: (1+4qx+q^2x^2)/((1-x)(1-qx)^4) = 1 + x(1 + 8q) + x^2(1 + 8q + 27q^2) + ... .

Offset changed by Reinhard Zumkeller, Nov 11 2012

cp's OEIS Frontend

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