A167667 -id:A167667 - cp's OEIS frontend

A268904 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.

0, 3, 0, 12, 36, 0, 36, 168, 240, 0, 96, 696, 1584, 1344, 0, 240, 2664, 9720, 12960, 6912, 0, 576, 9720, 54936, 118584, 98496, 33792, 0, 1344, 34344, 299088, 1004184, 1347192, 715392, 159744, 0, 3072, 118584, 1585800, 8250912, 17194680, 14644152, 5038848

Offset: 1

R. H. Hardin, Feb 15 2016

Table starts
.0.......3........12..........36............96............240..............576
.0......36.......168.........696..........2664...........9720............34344
.0.....240......1584........9720.........54936.........299088..........1585800
.0....1344.....12960......118584.......1004184........8250912.........66210264
.0....6912.....98496.....1347192......17194680......214142760.......2611960344
.0...33792....715392....14644152.....282550680.....5344944120......99308573208
.0..159744...5038848...154472184....4513169016...129834259704....3679171151832
.0..737280..34712064..1594323000...70609114584..3091414865040..133712637011640
.0.3342336.235146240.16185567096.1087342615224.72488795124312.4788143315276472

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

R. H. Hardin, Table of n, a(n) for n = 1..287

Row 1 is A167667(n-1).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 8*a(n-1) -16*a(n-2)
k=3: a(n) = 12*a(n-1) -36*a(n-2)
k=4: a(n) = 18*a(n-1) -81*a(n-2) for n>3
k=5: a(n) = 30*a(n-1) -261*a(n-2) +540*a(n-3) -324*a(n-4)
k=6: a(n) = 50*a(n-1) -805*a(n-2) +4662*a(n-3) -12150*a(n-4) +14580*a(n-5) -6561*a(n-6)
k=7: [order 8]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -4*a(n-2)
n=2: a(n) = 6*a(n-1) -9*a(n-2) for n>4
n=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4) for n>6
n=4: [order 6] for n>12
n=5: [order 14] for n>18
n=6: [order 18] for n>26
n=7: [order 54] for n>60

A269035 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 3, 0, 12, 24, 0, 36, 48, 120, 0, 96, 216, 348, 504, 0, 240, 672, 2166, 2136, 1944, 0, 576, 2208, 9528, 18384, 12228, 7128, 0, 1344, 6912, 44760, 115656, 146064, 67104, 25272, 0, 3072, 21408, 198816, 785124, 1326576, 1114848, 357756, 87480, 0, 6912, 65280

Offset: 1

R. H. Hardin, Feb 18 2016

Table starts
.0......3.......12.........36...........96............240.............576
.0.....24.......48........216..........672...........2208............6912
.0....120......348.......2166.........9528..........44760..........198816
.0....504.....2136......18384.......115656.........785124.........4998648
.0...1944....12228.....146064......1326576.......13031664.......119790816
.0...7128....67104....1114848.....14710368......208867428......2783857776
.0..25272...357756....8277072....159397596.....3266423688.....63310818360
.0..87480..1867560...60218112...1698064656....50155587360...1416701634552
.0.297432..9593844..431354928..17853542544...759280601376..31304407671636
.0.997272.48665904.3052215072.185754411168.11364951702132.684763778434512

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

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 2 is A268633.

Row 1 is A167667(n-1).

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 6*a(n-1) -9*a(n-2) for n>3
k=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4) for n>5
k=4: a(n) = 14*a(n-1) -57*a(n-2) +56*a(n-3) -16*a(n-4) for n>5
k=5: [order 12] for n>13
k=6: [order 18] for n>19
k=7: [order 38] for n>39
Empirical for row n:
n=1: a(n) = 4*a(n-1) -4*a(n-2)
n=2: a(n) = 4*a(n-1) -8*a(n-3) -4*a(n-4)
n=3: a(n) = 6*a(n-1) -a(n-2) -28*a(n-3) -4*a(n-4) +16*a(n-5) -4*a(n-6) for n>8
n=4: [order 12] for n>14
n=5: [order 20] for n>22
n=6: [order 46] for n>48
n=7: [order 92] for n>94

A268639 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or vertically adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 3, 3, 12, 24, 12, 36, 120, 120, 36, 96, 504, 840, 504, 96, 240, 1944, 5178, 5178, 1944, 240, 576, 7128, 29772, 47640, 29772, 7128, 576, 1344, 25272, 163878, 412740, 412740, 163878, 25272, 1344, 3072, 87480, 875592, 3440052, 5419992, 3440052, 875592

Offset: 1

R. H. Hardin, Feb 09 2016

Table starts
....0......3........12..........36............96.............240
....3.....24.......120.........504..........1944............7128
...12....120.......840........5178.........29772..........163878
...36....504......5178.......47640........412740.........3440052
...96...1944.....29772......412740.......5419992........68710116
..240...7128....163878.....3440052......68710116......1328460312
..576..25272....875592....27906474.....849572724.....25093766490
.1344..87480...4578186...221913216...10310685036....465757993812
.3072.297432..23548164..1737860310..123340687488...8527096170390
.6912.997272.119570574.13445785116.1458578214948.154406753980596

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

R. H. Hardin, Table of n, a(n) for n = 1..420

Column 1 is A167667(n-1).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -4*a(n-2)
k=2: a(n) = 6*a(n-1) -9*a(n-2) for n>3
k=3: a(n) = 10*a(n-1) -29*a(n-2) +20*a(n-3) -4*a(n-4)
k=4: [order 6] for n>7
k=5: [order 10]
k=6: [order 14] for n>15
k=7: [order 26]

A268774 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 3, 3, 12, 12, 12, 36, 32, 32, 36, 96, 100, 112, 100, 96, 240, 248, 446, 446, 248, 240, 576, 620, 1524, 2296, 1524, 620, 576, 1344, 1456, 5214, 10340, 10340, 5214, 1456, 1344, 3072, 3380, 17000, 46312, 64112, 46312, 17000, 3380, 3072, 6912, 7656, 54822

Offset: 1

R. H. Hardin, Feb 13 2016

Table starts
....0.....3.....12.......36........96.........240..........576..........1344
....3....12.....32......100.......248.........620.........1456..........3380
...12....32....112......446......1524........5214........17000.........54822
...36...100....446.....2296.....10340.......46312.......198114........837848
...96...248...1524....10340.....64112......387146......2258084......12951796
..240...620...5214....46312....387146.....3104544.....24222418.....185142872
..576..1456..17000...198114...2258084....24222418....255353744....2624246370
.1344..3380..54822...837848..12951796...185142872...2624246370...36091542548
.3072..7656.173244..3472210..73011192..1393319226..26623649020..491176316484
.6912.17148.541910.14245712.406925194.10357051740.266457432340.6585970939900

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

R. H. Hardin, Table of n, a(n) for n = 1..612

Column 1 is A167667(n-1).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -4*a(n-2)
k=2: a(n) = 2*a(n-1) +3*a(n-2) -4*a(n-3) -4*a(n-4) for n>5
k=3: a(n) = 4*a(n-1) +2*a(n-2) -16*a(n-3) -a(n-4) +12*a(n-5) -4*a(n-6) for n>8
k=4: [order 8] for n>10
k=5: [order 12] for n>14
k=6: [order 16] for n>18
k=7: [order 28] for n>30

A268798 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 3, 3, 12, 22, 12, 36, 78, 78, 36, 96, 234, 248, 234, 96, 240, 652, 950, 950, 652, 240, 576, 1714, 3384, 4800, 3384, 1714, 576, 1344, 4360, 11948, 23994, 23994, 11948, 4360, 1344, 3072, 10820, 41248, 117062, 168740, 117062, 41248, 10820, 3072, 6912, 26366

Offset: 1

R. H. Hardin, Feb 13 2016

Table starts
....0.....3......12.......36.........96.........240...........576
....3....22......78......234........652........1714..........4360
...12....78.....248......950.......3384.......11948.........41248
...36...234.....950.....4800......23994......117062........561116
...96...652....3384....23994.....168740.....1158904.......7801688
..240..1714...11948...117062....1158904....11138352.....104971262
..576..4360...41248...561116....7801688...104971262....1384570516
.1344.10820..140698..2652936...51781418...974000420...17967375416
.3072.26366..474472.12405748..339641264..8927994302..230262982692
.6912.63346.1586038.57490444.2206871084.81031120788.2921020155826

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

R. H. Hardin, Table of n, a(n) for n = 1..637

Column 1 is A167667(n-1).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -4*a(n-2)
k=2: a(n) = 2*a(n-1) +3*a(n-2) -2*a(n-3) -6*a(n-4) -4*a(n-5) -a(n-6) for n>7
k=3: [order 10] for n>12
k=4: [order 16] for n>19
k=5: [order 26] for n>29
k=6: [order 42] for n>45
k=7: [order 68] for n>71

A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 0, 4, 5, 1, 0, 0, 6, 7, 1, 0, 0, 0, 8, 9, 1, 0, 0, 0, 0, 10, 11, 1, 0, 0, 0, 0, 0, 12, 13, 1, 0, 0, 0, 0, 0, 0, 14, 15, 1, 0, 0, 0, 0, 0, 0, 0, 16, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 23, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Nov 08 2009

Row sums = A111284(n+1), Diagonal sums = A109613(n).

			Triangle begins :
1 ;
1, 1 ;
2, 3, 1 ;
0, 4, 5, 1 ;
0, 0, 6, 7, 1 ;
0, 0, 0, 8, 9, 1 ;
0, 0, 0, 0, 10, 11, 1 ; ...

Cf. A000012, A005408, A005843

T(n,k) = 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(3,0) = 0, T(3,1) = 4. - Philippe Deléham, Feb 18 2012
G.f.: (1+(1-y)*x+(2+y)*x^2)/(1-y*x)^2. - Philippe Deléham, Feb 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A130779(n), A111284(n+1), A167667(n), A167682(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 18 2012

A204206 Triangle based on (1,3/2,2) averaging array.

Original entry on oeis.org

3, 5, 7, 9, 12, 15, 17, 21, 27, 31, 33, 38, 48, 58, 63, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 511, 513, 522, 558, 642, 768, 894, 978, 1014, 1023, 1025, 1035, 1080, 1200, 1410, 1662, 1872

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
3
5...7
9...12...15
17..21...27...31
33..38...48...58...63
65..71...86...106..121..127

Cf. A204201.

a = 1; r = 3/2; b = 2;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[3 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]   (* A204206 triangle *)
Flatten[u]     (* A204206 sequence *)

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A000225(n+1).
Sum_{k=1..n} T(n,k) = A167667(n).
T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=3, T(2,1)=5, T(2,2)=7, T(n,k)=0 if k<1 or if k>n. (End)

A092393 Triangle read by rows: T(n,k) = (n+k)*binomial(n,k) (for k=0..n-1).

Original entry on oeis.org

1, 2, 6, 3, 12, 15, 4, 20, 36, 28, 5, 30, 70, 80, 45, 6, 42, 120, 180, 150, 66, 7, 56, 189, 350, 385, 252, 91, 8, 72, 280, 616, 840, 728, 392, 120, 9, 90, 396, 1008, 1638, 1764, 1260, 576, 153, 10, 110, 540, 1560, 2940, 3780, 3360, 2040, 810, 190, 11, 132, 715

Offset: 1

Benoit Cloitre, Mar 21 2004

			Triangle starts:
1;
2, 6;
3, 12, 15;
4, 20, 36,  28;
5, 30, 70,  80,  45;
6, 42, 120, 180, 150, 66;
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Cf. A029635.

A092393 := proc(n,k)
    (n+k)*binomial(n,k) ;
end proc:
seq(seq( A092393(n,k),k=0..n-1),n=1..12) ; # R. J. Mathar, Nov 02 2023

A092393row[n_]:=Table[(n+k)Binomial[n,k],{k,0,n-1}];Array[A092393row,10]  (* Paolo Xausa, Nov 02 2023 *)

PARI
```
T(n,k)=binomial(n,k)*(n+k)
```

First column = positive integers;
second column = A002378;
third column = A077414;
main diagonal (i.e., T(n,n) = (n+n)*binomial(n,n) = 2n, which is not included in this sequence) = even integers;
second diagonal = A000384.
Row sums = 1, 8, 30, 88, 230,... = A167667(n)-2n. - R. J. Mathar, Nov 02 2023

A203150 (n-1)-st elementary symmetric function of the first n terms of (1,2,1,2,1,2,1,2,1,2,...)=A000034.

Original entry on oeis.org

1, 3, 5, 12, 16, 36, 44, 96, 112, 240, 272, 576, 640, 1344, 1472, 3072, 3328, 6912, 7424, 15360, 16384, 33792, 35840, 73728, 77824, 159744, 167936, 344064, 360448, 737280, 770048, 1572864, 1638400, 3342336, 3473408, 7077888, 7340032

Offset: 1

Clark Kimberling, Dec 29 2011

			Let esf abbreviate "elementary symmetric function".  Then
0th esf of {1}:  1
1st esf of {1,2}:  1+2=3
2nd esf of {1,2,1} is 1*2+1*1+2*1=5

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000034, A167667 (bisection?), A053220 (bisection?)

f[k_] := 1 + Mod[k + 1, 2];
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}]  (* A203150 *)

Empirical G.f.: x*(1+3*x+x^2)/(1-4*x^2+4*x^4). - Colin Barker, Jan 03 2012

Conjecture: a(n) = (6*r*n+(1+3*(1-r)*n)*(1-(-1)^n))*r^(n-1)/8, where r=sqrt(2). - Bruno Berselli, Jan 03 2011

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376

Offset: 1

Fedor Igumnov, Jan 28 2014

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Fedor Igumnov, T(n,k) for n = 1..26

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

/* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014

t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

T(n,k) = T(n-1,k) + T(n-1,k+1).

Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

More terms from Bruno Berselli, Jan 28 2014

cp's OEIS Frontend

A268904 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.

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A269035 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two exactly once.

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Formula

A268639 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or vertically adjacent neighbor totalling two exactly once.

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Author

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A268774 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two exactly once.

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A268798 T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two exactly once.

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A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A204206 Triangle based on (1,3/2,2) averaging array.

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A092393 Triangle read by rows: T(n,k) = (n+k)*binomial(n,k) (for k=0..n-1).

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A203150 (n-1)-st elementary symmetric function of the first n terms of (1,2,1,2,1,2,1,2,1,2,...)=A000034.

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A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.