A001105 -id:A001105 - cp's OEIS frontend

A070216 Triangle T(n, k) = n^2 + k^2, 1 <= k <= n, read by rows.

2, 5, 8, 10, 13, 18, 17, 20, 25, 32, 26, 29, 34, 41, 50, 37, 40, 45, 52, 61, 72, 50, 53, 58, 65, 74, 85, 98, 65, 68, 73, 80, 89, 100, 113, 128, 82, 85, 90, 97, 106, 117, 130, 145, 162, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200, 122, 125, 130, 137, 146, 157

Offset: 1

Charles Northup (cnorthup(AT)esc6.net), May 07 2002

The formula yields squares of hypotenuses of right triangles having integer side lengths (A000404), but with duplicates (cf. A024508) and not in increasing order. - M. F. Hasler, Apr 05 2016

			a(3,2)=13 because 3^2+2^2=13.
Triangle begins:
2;
5, 8;
10, 13, 18;
17, 20, 25, 32;
26, 29, 34, 41, 50;
37, 40, 45, 52, 61, 72;
50, 53, 58, 65, 74, 85, 98;
65, 68, 73, 80, 89, 100, 113, 128;
82, 85, 90, 97, 106, 117, 130, 145, 162;
101, 104, 109, 116, 125, 136, 149, 164, 181, 200; ...
- _Vincenzo Librandi_, Apr 30 2014

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

Not a permutation of sequence A000404 (which has no duplicates).

Cf. A002522 (left edge), A001105 (right edge), A219054 (row sums).

a070216 n k = a070216_tabl !! (n-1) !! (k-1)
a070216_row n = a070216_tabl !! (n-1)
a070216_tabl = zipWith (zipWith (\u v -> (u + v) `div` 2))
                       a215630_tabl a215631_tabl
-- Reinhard Zumkeller, Nov 11 2012

[n^2+k^2: k in [1..n], n in [1..15]]; // Vincenzo Librandi, Apr 30 2014

t[n_,k_]:=n^2 + k^2; Table[t[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Apr 30 2014 *)

T(n, k) = n^2+k^2;
for (n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ Altug Alkan, Mar 24 2016

from math import isqrt
def A070216(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return (a*(a*(a*(a-2)-(m:=n<<2)+5)+m)>>2)+n**2 # Chai Wah Wu, Jun 20 2025

a(n, k) = n^2 + k^2, 1 <= k <= n.
T(n,k) = (A215630(n,k) + A215631(n,k)) / 2, 1 <= k <=n. - Reinhard Zumkeller, Nov 11 2012
T(n,k) = A002024(n,k)^2 + A002260(n,k)^2. - David Rabahy, Mar 24 2016

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2002

Edited and corrected by M. F. Hasler, Apr 05 2016

A224665 T(n,k)=Number of n X n 0..k matrices with each 2X2 subblock idempotent.

Original entry on oeis.org

2, 3, 8, 4, 12, 32, 5, 16, 50, 78, 6, 20, 72, 108, 196, 7, 24, 98, 142, 260, 428, 8, 28, 128, 180, 332, 542, 916, 9, 32, 162, 222, 412, 668, 1126, 1858, 10, 36, 200, 268, 500, 806, 1356, 2230, 3678, 11, 40, 242, 318, 596, 956, 1606, 2634, 4336, 7096, 12, 44, 288, 372

Offset: 1

R. H. Hardin Apr 14 2013

Table starts
....2....3....4.....5.....6.....7.....8....9...10...11...12...13..14..15.16.17
....8...12...16....20....24....28....32...36...40...44...48...52..56..60.64
...32...50...72....98...128...162...200..242..288..338..392..450.512.578
...78..108..142...180...222...268...318..372..430..492..558..628.702
..196..260..332...412...500...596...700..812..932.1060.1196.1340
..428..542..668...806...956..1118..1292.1478.1676.1886.2108
..916.1126.1356..1606..1876..2166..2476.2806.3156.3526
.1858.2230.2634..3070..3538..4038..4570.5134.5730
.3678.4336.5046..5808..6622..7488..8406.9376
.7096.8246.9480.10798.12200.13686.15256

			Some solutions for n=3 k=4
..1..1..4....1..0..0....1..1..3....1..0..0....1..1..1....1..1..3....1..1..2
..0..0..0....1..0..0....0..0..0....1..0..0....0..0..0....0..0..0....0..0..0
..3..1..1....1..0..0....0..0..0....0..0..1....1..1..1....4..1..1....2..1..1

R. H. Hardin, Table of n, a(n) for n = 1..132

Column 1 is A224543(n-1)
Row 1 is A000027(n+1)
Row 2 is A008574(n+1)
Row 3 is A001105(n+3)

Empirical for columns k=1..7:
k=1..7: a(n) = 6*a(n-1) -12*a(n-2) +5*a(n-3) +12*a(n-4) -12*a(n-5) -3*a(n-6) +6*a(n-7) -a(n-9) for n>10
Empirical for row n:
n=1: a(n) = 0*n^2 + 1*n + 1
n=2: a(n) = 0*n^2 + 4*n + 4
n=3: a(n) = 2*n^2 + 12*n + 18
n=4: a(n) = 2*n^2 + 24*n + 52
n=5: a(n) = 4*n^2 + 52*n + 140
n=6: a(n) = 6*n^2 + 96*n + 326
n=7: a(n) = 10*n^2 + 180*n + 726
n=8: a(n) = 16*n^2 + 324*n + 1518
n=9: a(n) = 26*n^2 + 580*n + 3072
n=10: a(n) = 42*n^2 + 1024*n + 6030
n=11: a(n) = 68*n^2 + 1796*n + 11594
n=12: a(n) = 110*n^2 + 3128*n + 21912

A345959 Numbers whose prime indices have alternating sum -1.

Original entry on oeis.org

6, 15, 24, 35, 54, 60, 77, 96, 135, 140, 143, 150, 216, 221, 240, 294, 308, 315, 323, 375, 384, 437, 486, 540, 560, 572, 600, 667, 693, 726, 735, 864, 875, 884, 899, 960, 1014, 1147, 1176, 1215, 1232, 1260, 1287, 1292, 1350, 1500, 1517, 1536, 1715, 1734, 1748

Offset: 1

Gus Wiseman, Jul 12 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with even Omega (A001222) and exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

			The initial terms and their prime indices:
    6: {1,2}
   15: {2,3}
   24: {1,1,1,2}
   35: {3,4}
   54: {1,2,2,2}
   60: {1,1,2,3}
   77: {4,5}
   96: {1,1,1,1,1,2}
  135: {2,2,2,3}
  140: {1,1,3,4}
  143: {5,6}
  150: {1,2,3,3}
  216: {1,1,1,2,2,2}
  221: {6,7}
  240: {1,1,1,1,2,3}

These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The k = 1 version is A001105.
The k > 0 version is A026424.
These are the positions of -1's in A316524.
The k = 2 version is A345960.
The k = -2 version is A345962.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[primeMS[#]]==-1&]

A069011 Triangle with T(n,k) = n^2 + k^2.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200

Offset: 0

Henry Bottomley, Apr 02 2002

For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
A227481(n) = number of squares in row n. - Reinhard Zumkeller, Oct 11 2013
Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - Stefano Spezia, Aug 07 2025

			Triangle T(n,k) begins:
    0;
    1,   2;
    4,   5,   8;
    9,  10,  13,  18;
   16,  17,  20,  25,  32;
   25,  26,  29,  34,  41,  50;
   36,  37,  40,  45,  52,  61,  72;
   49,  50,  53,  58,  65,  74,  85,  98;
   64,  65,  68,  73,  80,  89, 100, 113, 128;
   81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
  100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
Column k=0 gives A000290.
Main diagonal gives A001105.
Row sums give A132124.
T(2n,n) gives A033429.

a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
   f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
-- Reinhard Zumkeller, Oct 11 2013

Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Aug 07 2025 *)

T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013

G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Aug 04 2025

A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190

Offset: 0

Reinhard Zumkeller, Nov 18 2004

Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - Franklin T. Adams-Watters, Apr 04 2010

			Triangle begins:
   0
   1   2
   4   6   8
   9  12  15  18
  16  20  24  28  32
  25  30  35  40  45  50
  36  42  48  54  60  66  72
  49  56  63  70  77  84  91  98
  64  72  80  88  96 104 112 120 128

Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.

Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)

row(n) = vector(n+1, k, n*(n+k-1)); \\ Amiram Eldar, May 09 2025

T(n,0) = A000290(n).
T(n,1) = A002378(n) for n > 0.
T(n,2) = A005563(n) for n > 1.
T(n,3) = A028552(n) for n > 2.
T(n,4) = A028347(n+2) for n > 3.
T(n,5) = A028557(n) for n > 4.
T(n,6) = A028560(n) for n > 5.
T(n,7) = A028563(n) for n > 6.
T(n,8) = A028566(n) for n > 7.
T(n,9) = A028569(n) for n > 8.
T(n,10) = A098603(n) for n > 9.
T(n,n-5) = A071355(n-4) for n > 4.
T(n,n-4) = A054000(n-1) for n > 3.
T(n,n-3) = A014107(n) for n > 2.
T(n,n-2) = A046092(n-1) for n > 1.
T(n,n-1) = A000384(n) for n > 0.
T(n,n) = A001105(n).
Row sums give A085789 for n > 0.
G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 03 2025

A195323 a(n) = 22*n^2.

Original entry on oeis.org

0, 22, 88, 198, 352, 550, 792, 1078, 1408, 1782, 2200, 2662, 3168, 3718, 4312, 4950, 5632, 6358, 7128, 7942, 8800, 9702, 10648, 11638, 12672, 13750, 14872, 16038, 17248, 18502, 19800, 21142, 22528, 23958, 25432, 26950, 28512, 30118, 31768, 33462, 35200, 36982, 38808

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 22, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195318 in the same spiral.

Surface area of a rectangular prism with dimensions n, 2n and 3n. - Wesley Ivan Hurt, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195149.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195322.
Cf. A000290, A001105, A008604, A033584, A195313, A195318.

[22*n^2 : n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195323:=n->22*n^2: seq(A195323(n), n=0..80); # Wesley Ivan Hurt, Apr 10 2015

22Range[0,50]^2 (* or *) LinearRecurrence[{3,-3,1},{0,22,88},50] (* Harvey P. Dale, Sep 19 2011 *)

vector(50,n,22*(n-1)^2) \\ Derek Orr, Apr 10 2015

a(n) = 22*A000290(n) = 11*A001105(n) = 2*A033584(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 19 2011
G.f.: 22*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Apr 10 2015
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 22*x*(1 + x)*exp(x).
a(n) = n*A008604(n) = A195149(2*n). (End)

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0

Offset: 0

Ilya Gutkovskiy, Aug 07 2017

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give:  A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019

Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

A305258 List of y-coordinates of a point moving in a smooth counterclockwise spiral rotated by Pi/4.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, May 29 2018

sign

			Sequence gives y-coordinate of the n-th point of the following spiral:
   d:
   4 |                  32  49
     |                 /   \   \
   3 |              33  18  31  48
     |             /   /   \   \   \
   2 |          34  19   8  17  30  47
     |         /   /   /   \   \   \   \
   1 |      35  20   9   2   7  16  29  46
     |     /   /   /   /   \   \   \   \   \
   0 |  36  21  10   3   0---1   6  15  28  45
     |     \   \   \   \       /   /   /   /
  -1 |      37  22  11   4---5  14  27  44
     |         \   \   \      /    /   /
  -2 |          38  23  12--13  26  43
     |             \   \       /   /
  -3 |              39  24--25  42
     |                 \       /
  -4 |                  40--41
       _______________________________________
  x:    -4  -3  -2  -1   0   1   2   3   4   5

Hugo Pfoertner, Illustration of spiral.
Index entries for sequences related to coordinates of 2D curves

A010751 gives sequence of x-coordinates.
Cf. A053616.
Cf. A000384, A001105, A002024, A014105, A046092.

up=-1;print1(x=0,", ");for(stride=1,12,up=-up;x+=stride;y=x+stride+1;for(k=x,y-1,print1(up*min(k-x,y-k), ", "))) \\ Hugo Pfoertner, Jun 02 2018

a(n) = A053616(n)*sign(sin(Pi*(1+sqrt(1+8*n))/2)), so that abs(a(n)) = A053616(n).

a(n) = A010751(n-floor((1/2)*(sqrt(2n-1)+1))). - William McCarty, Jul 29 2021

A096033 Difference between leg and hypotenuse in primitive Pythagorean triangles.

Original entry on oeis.org

1, 2, 8, 9, 18, 25, 32, 49, 50, 72, 81, 98, 121, 128, 162, 169, 200, 225, 242, 288, 289, 338, 361, 392, 441, 450, 512, 529, 578, 625, 648, 722, 729, 800, 841, 882, 961, 968, 1058, 1089, 1152, 1225, 1250, 1352, 1369, 1458, 1521, 1568, 1681, 1682, 1800, 1849

Offset: 1

Lekraj Beedassy, Jun 16 2004

Consists of the odd squares and the halves of the even squares. - Andrew Weimholt, Sep 07 2010

Question: Do we have a(n) mod 2 = A004641(n)? - David A. Corneth, Jan 02 2019

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 170.

David A. Corneth, Table of n, a(n) for n = 1..10099
James M. Parks, Computing Pythagorean Triples, arXiv:2107.06891 [math.GM], 2021.
James M. Parks, On the Curved Patterns Seen in the Graph of PPTs, arXiv:2104.09449 [math.CO], 2021.

Cf. A001105, A004641, A016754, A094904, A245058.

nmax = 100;
Union[2 Range[nmax]^2, (2 Range[0, Ceiling[nmax/Sqrt[2]]] + 1)^2] (* Jean-François Alcover, Jan 01 2019 *)

upto(n) = vecsort(concat(vector((sqrtint(n)+1)\2, i, (2*i-1)^2), vector(sqrtint(n\2), i, 2*i^2))) \\ David A. Corneth, Jan 02 2019

Union of A001105 (integers of form 2*n^2) and A016754 (the odd squares).

Sum_{n>=1} 1/a(n) = 5*Pi^2/24 = 10 * A245058. - Amiram Eldar, Feb 14 2021

Corrected and extended by Matthew Vandermast and Ray Chandler, Jun 17 2004

Erroneous comment deleted by Andrew Weimholt, Sep 07 2010

A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 66, 24, 1, 50, 192, 146, 32, 1, 72, 450, 608, 258, 40, 1, 98, 912, 1970, 1408, 402, 48, 1, 128, 1666, 5336, 5890, 2720, 578, 56, 1, 162, 2816, 12642, 20256, 14002, 4672, 786, 64, 1, 200, 4482, 27008, 59906, 58728, 28610, 7392, 1026, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    66,    146,     258,      402,       578, ... A010006;
  1,  32,   192,    608,    1408,     2720,      4672, ... A019560;
  1,  50,   450,   1970,    5890,    14002,     28610, ... A019561;
  1,  72,   912,   5336,   20256,    58728,    142000, ... A019562;
  1,  98,  1666,  12642,   59906,   209762,    596610, ... A019563;
  1, 128,  2816,  27008,  157184,   658048,   2187520, ... A019564;
  1, 162,  4482,  53154,  374274,  1854882,   7159170, ... A035746;
  1, 200,  6800,  97880,  822560,  4780008,  21278640, ... A035747;
  1, 242,  9922, 170610, 1690370, 11414898,  58227906, ... A035748;
  1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749;
  ...
Antidiagonals, T(n, k), begins as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   66,   24;
  1,  50,  192,  146,   32;
  1,  72,  450,  608,  258,   40;
  1,  98,  912, 1970, 1408,  402,  48;
  1, 128, 1666, 5336, 5890, 2720, 578, 56;

G. C. Greubel, Antidiagonals n = 2..50, flattened
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A022144, A010006, A019560, A019561, A019562, A019563, A019564, A035746, A035747, A035748, A035749, A035750 - A035787.
Columns include A001105, A035598, A035600, A035602, A035604, A035606.
Main diagonal is in A103885.
Cf. A103881, A103993, A108998.

A103884:= func< n,k | k eq 0 select 1 else 2*(&+[2^j*Binomial(n-k,j+1)*Binomial(2*k-1,j) : j in [0..2*k-1]]) >;
[A103884(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, May 23 2023

nmin = 2; nmax = 11; t[n_, 0]= 1; t[n_, k_]:= 2n*Hypergeometric2F1[1-2k, 1-n, 2, 2]; tnk= Table[ t[n, k], {n, nmin, nmax}, {k, 0, nmax-nmin}]; Flatten[ Table[ tnk[[ n-k+1, k ]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after formula *)

def A103884(n,k): return 1 if k==0 else 2*sum(2^j*binomial(n-k,j+1)*binomial(2*k-1,j) for j in range(2*k))
flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, May 23 2023

A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).
G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.
T(n, k) = A(n-k, k) (antidiagonals).
T(n, n-2) = A022144(n-2).
T(n, k) = 2*(n-k)*Hypergeometric2F1([1+k-n, 1-2*k], [2], 2), T(n, 0) = 1, for n >= 2, 0 <= k <= n-2. - G. C. Greubel, May 23 2023
From  Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Chebyshev_T(n, (1 + x)/(1 - x)), where Chebyshev_T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
T(n+1,k) = T(n+1,k-1)  + 2*T(n,k) + 2*T(n,k-1) + T(n-1,k) - T(n-1,k-1). (End)

Definition clarified by N. J. A. Sloane, May 25 2023

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