A028560 -id:A028560 - cp's OEIS frontend

A028562 Palindromes of form k*(k+6).

0, 7, 55, 616, 19591, 76167, 515515, 749947, 7474747, 72999927, 236799997632, 769437734967, 1900289820091, 2998954598992, 5170703070715, 5934592954395, 29175111157192, 51175166157115, 57154688645175, 211106050601112, 570608929806075, 574823545328475

Offset: 1

Patrick De Geest

Michael S. Branicky, Table of n, a(n) for n = 1..35
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Cf. A028560, A028561.

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    for k in count(0):
        if ispal(k*(k+6)):
            yield k*(k+6)
print(list(islice(agen(), 19))) # Michael S. Branicky, Jan 25 2022

a(n) = A028561(n) * (A028561(n) + 6). - Michael S. Branicky, Jan 25 2022

a(20) and beyond from Michael S. Branicky, Jan 25 2022

A132414 Integers n such that n^3 - (n + 2)^2 + n + 4 is a square.

Original entry on oeis.org

-1, 0, 3, 4, 75

Offset: 1

Mohamed Bouhamida, Nov 12 2007

n^3 - (n + 2)^2 + n + 4 = n^3 - n^2 - 3*n. The set of x values of integral solutions to the elliptic curve y^2 = n^3 - n^2 - 3*n (see Magma program) is {-1, 0, 3, 4, 75}. - Klaus Brockhaus, Nov 13 2007

			0^3 - 2^2 + 4 = 0^2, 3^3 - 5^2 + 7 = 3^2, 4^3 - 6^2 + 8 = 6^2 and 75^3 - 77^2 + 79 = 645^2.

Cf. A005563, A028560.

P := PolynomialRing(Integers()); {x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 - n^2 - 3*n)) ])}; /* Klaus Brockhaus, Nov 13 2007 */

[i[0] for i in EllipticCurve([0, -1, 0, -3, 0]).integral_points()] # Seiichi Manyama, Aug 26 2019

A132772 a(n) = n*(n + 30).

Original entry on oeis.org

0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 400, 451, 504, 559, 616, 675, 736, 799, 864, 931, 1000, 1071, 1144, 1219, 1296, 1375, 1456, 1539, 1624, 1711, 1800, 1891, 1984, 2079, 2176, 2275, 2376, 2479, 2584, 2691, 2800, 2911, 3024, 3139, 3256, 3375, 3496, 3619

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A067079, A098849, A098850, A098603, A098847, A098848, A102928, A120071.
Cf. A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767.
Cf. A132768, A132769, A132770, A132771.

Table[n(n+30),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,31,64},50] (* Harvey P. Dale, Mar 06 2015 *)

a(n)=n*(n+30) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+30) for n in (0..50)] # G. C. Greubel, Mar 13 2022

G.f.: x*(31-29*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*n + a(n-1) + 29 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=31, a(2)=64, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 06 2015
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(30)/30 = A001008(30)/A102928(30) = 9304682830147/69872686884000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 225175759291/9981812412000. (End)
E.g.f.: x*(31 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A169603 Triangle T(n,k) = k(4n+k+2), read by rows.

Original entry on oeis.org

0, 0, 7, 0, 11, 24, 0, 15, 32, 51, 0, 19, 40, 63, 88, 0, 23, 48, 75, 104, 135, 0, 27, 56, 87, 120, 155, 192, 0, 31, 64, 99, 136, 175, 216, 259, 0, 35, 72, 111, 152, 195, 240, 287, 336, 0, 39, 80, 123, 168, 215, 264, 315, 368, 423, 0, 43, 88, 135, 184, 235, 288, 343, 400, 459, 520

Offset: 0

Paul Curtz, Dec 03 2009

These are the numerators of 1/(2*n+1)^2 - 1/(2*n+k+1)^2 as they appear in the energies of the hydrogen spectrum, not reduced by common factors with the denominators.

			The array begins as:
  0,  3,  8,  15,  24,  35,  48,  63,  80 ... A005563;
  0,  7, 16,  27,  40,  55,  72,  91, 112 ... A028560;
  0, 11, 24,  39,  56,  75,  96, 119, 144 ... A098603;
  0, 15, 32,  51,  72,  95, 120, 147, 176 ... A098848;
  0, 19, 40,  63,  88, 115, 144, 175, 208 ... A098850;
  0, 23, 48,  75, 104, 135, 168, 203, 240 ... A132764;
  0, 27, 56,  87, 120, 155, 192, 231, 272 ... A132768;
  0, 31, 64,  99, 136, 175, 216, 259, 304 ... A132772;
  0, 35, 72, 111, 152, 195, 240, 287, 336 ...;
The triangle starts as:
  0;
  0,  7;
  0, 11, 24;
  0, 15, 32,  51;
  0, 19, 40,  63,  88;
  0, 23, 48,  75, 104, 135;
  0, 27, 56,  87, 120, 155, 192;
  0, 31, 64,  99, 136, 175, 216, 259;
  0, 35, 72, 111, 152, 195, 240, 287, 336;
  0, 39, 80, 123, 168, 215, 264, 315, 368, 423;
  0, 43, 88, 135, 184, 235, 288, 343, 400, 459, 520;

Charles Janet, Considérations sur la structure du noyau de l'atome, Décembre 1929, N 5, Beauvais, page 39.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A169607 (row sums).

Cf. A005563, A028560, A098603, A098848, A098850, A132764, A132768, A132772.

[k*(4*n+k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 13 2022

Table[k(4n+2+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 08 2021 *)

flatten([[k*(4*n+k+2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 13 2022

T(n,k) = k*(4*n+k+2).

Sum_{k=0..n} T(n,k) = A169607(n) = 7*A000330(n), 7 times the sum of squares.

A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 2, 9, 15, 6, 7, 16, 24, 13, 5, 14, 25, 35, 22, 11, 12, 23, 36, 48, 33, 20, 10, 21, 34, 49, 63, 46, 31, 18, 19, 32, 47, 64, 80, 61, 44, 29, 17, 30, 45, 62, 81, 99, 78, 59, 42, 27, 28, 43, 60, 79, 100, 120, 97, 76, 57, 40, 26, 41, 58, 77, 98, 121

Offset: 1

Boris Putievskiy, Mar 05 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
...
T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
...

			The start of the sequence as a table:
   1,  3,  8, 15, 24, 35, ...
   4,  2,  6, 13, 22, 33, ...
   9,  7,  5, 11, 20, 31, ...
  16, 14, 12, 10, 18, 29, ...
  25, 23, 21, 19, 17, 27, ...
  36, 34, 32, 30, 28, 26, ...
...
The start of the sequence as triangular array read by rows:
   1;
   3,  4;
   8,  2,  9;
  15,  6,  7, 16;
  24, 13,  5, 14, 25;
  35, 22, 11, 12, 23, 36;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.

f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Aug 19 2017 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i >= j:
   result=i*i-2*j+2
else:
   result=j*j-2*i+1

As a table,
  T(n,k) = n*n - 2*k + 2, if n >= k;
  T(n,k) = k*k - 2*n + 1, if n < k.
As a linear sequence,
  a(n) = i*i - 2*j + 2, if i >= j;
  a(n) = j*j - 2*i + 1, if i < j
  where
    i = n - t*(t+1)/2,
    j = (t*t + 3*t + 4)/2 - n,
    t = floor((-1 + sqrt(8*n-7))/2).

A028561 Numbers k such that k*(k+6) is a palindrome.

Original entry on oeis.org

0, 1, 5, 22, 137, 273, 715, 863, 2731, 8541, 486618, 877173, 1378507, 1731746, 2273915, 2436099, 5401396, 7153679, 7560069, 14529486, 23887419, 23975475, 73114035, 84890503, 88837611, 235680755, 235769755, 272515513, 440021417, 782357262, 1414071397, 2352019439

Offset: 1

Patrick De Geest

Michael S. Branicky, Table of n, a(n) for n = 1..35
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)
Erich Friedman, What's Special About This Number? (See entries 863, 8541.)

Cf. A002113, A028560, A028562.

Select[Range[0,24*10^6],PalindromeQ[#(#+6)]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2017 *)

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    for k in count(0):
        if ispal(k*(k+6)):
            yield k
print(list(islice(agen(), 22))) # Michael S. Branicky, Jan 25 2022

a(23) and beyond from Michael S. Branicky, Jan 25 2022

A265017 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with smallest part k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 5, 0, 0, 4, 0, 7, 8, 0, 0, 5, 0, 25, 12, 0, 0, 0, 6, 0, 36, 16, 15, 0, 0, 0, 7, 0, 81, 20, 21, 0, 0, 0, 0, 8, 0, 107, 74, 27, 24, 0, 0, 0, 0, 9, 0, 316, 102, 33, 32, 0, 0, 0, 0, 0, 10, 0, 427, 222, 39, 40, 35, 0, 0, 0, 0, 0, 11

Offset: 0

Alois P. Heinz, Nov 30 2015

			Triangle T(n,k) begins:
00 :  1;
01 :  0,   1;
02 :  0,   0,   2;
03 :  0,   3,   0,   3;
04 :  0,   5,   0,   0,  4;
05 :  0,   7,   8,   0,  0,  5;
06 :  0,  25,  12,   0,  0,  0, 6;
07 :  0,  36,  16,  15,  0,  0, 0, 7;
08 :  0,  81,  20,  21,  0,  0, 0, 0, 8;
09 :  0, 107,  74,  27, 24,  0, 0, 0, 0, 9;
10 :  0, 316, 102,  33, 32,  0, 0, 0, 0, 0, 10;
11 :  0, 427, 222,  39, 40, 35, 0, 0, 0, 0,  0, 11;
12 :  0, 869, 286, 153, 48, 45, 0, 0, 0, 0,  0,  0, 12;

Alois P. Heinz, Rows n = 0..100, flattened
Richard P. Stanley, Parking Functions, 2011.

Row sums give A265016.
Column k=0 gives A000007.
Main diagonal gives A028310, first lower diagonal is A000004.
T(2n+1,n) gives A005563.
T(2n+2,n) gives A028347(n+2).
T(2n+3,n) gives A028560.

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)-> `if`(i*(i+1)/2n, 0, g(n-i, i-1, [i, l[]])))):
T:= n-> (f-> seq(coeff(f, x, i), i=0..n))(g(n$2, [])):
seq(T(n), n=0..16);

p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,
     If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];
g[n_, i_, l_] := If[i(i+1)/2 < n, 0, If[n == 0, p[l]*x^
     If[l == {}, 0, l[[1]]], g[n, i - 1, l] +
     If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666

Offset: 0

Omar E. Pol, Aug 28 2007

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Table[n*(n + 31), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

a(n)=n*(n+31) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

cp's OEIS Frontend

A028562 Palindromes of form k*(k+6).

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A132414 Integers n such that n^3 - (n + 2)^2 + n + 4 is a square.

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A132772 a(n) = n*(n + 30).

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A169603 Triangle T(n,k) = k*(4*n+k+2), read by rows.

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Magma

Mathematica

Sage

Formula

A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

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Mathematica

Python

Formula

A028561 Numbers k such that k*(k+6) is a palindrome.

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Python

Extensions

A265017 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into distinct parts with smallest part k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

Mathematica

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

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A169603 Triangle T(n,k) = k(4n+k+2), read by rows.