A208354 -id:A208354 - cp's OEIS frontend

A268995 T(n,k)=Number of nXk binary arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two not more than once.

2, 4, 4, 7, 13, 8, 13, 35, 41, 16, 23, 103, 174, 126, 32, 41, 278, 805, 849, 379, 64, 72, 763, 3331, 6009, 4083, 1121, 128, 126, 2037, 14080, 37987, 43512, 19416, 3272, 256, 219, 5421, 57287, 244397, 421450, 308112, 91491, 9449, 512, 379, 14264, 232449, 1506570

Offset: 1

R. H. Hardin, Feb 17 2016

Table starts
....2.....4.......7........13..........23............41..............72
....4....13......35.......103.........278...........763............2037
....8....41.....174.......805........3331.........14080...........57287
...16...126.....849......6009.......37987........244397.........1506570
...32...379....4083.....43512......421450.......4097199........38241770
...64..1121...19416....308112.....4583103......66954420.......946498448
..128..3272...91491...2144780....49084071....1073436321.....22995344760
..256..9449..427863..14730784...519385102...16957258387....550731432312
..512.27049.1988142.100087792..5442503771..264744926212..13040291111728
.1024.76866.9187653.674045392.56571775611.4093941136805.305911647779632

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

R. H. Hardin, Table of n, a(n) for n = 1..799

Column 1 is A000079.

Row 1 is A208354(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3) -a(n-4)
k=3: a(n) = 10*a(n-1) -31*a(n-2) +30*a(n-3) -9*a(n-4)
k=4: a(n) = 16*a(n-1) -88*a(n-2) +200*a(n-3) -208*a(n-4) +96*a(n-5) -16*a(n-6) for n>7
k=5: [order 8] for n>9
k=6: [order 10] for n>12
k=7: [order 14] for n>16
Empirical for row n:
n=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)
n=2: a(n) = 2*a(n-1) +5*a(n-2) -4*a(n-3) -11*a(n-4) -6*a(n-5) -a(n-6)
n=3: [order 9]
n=4: [order 16]
n=5: [order 26]
n=6: [order 42]
n=7: [order 68]

A269075 T(n,k)=Number of nXk binary arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two not more than once.

Original entry on oeis.org

2, 4, 4, 7, 11, 8, 13, 27, 32, 16, 23, 76, 123, 89, 32, 41, 185, 521, 537, 244, 64, 72, 489, 1887, 3288, 2343, 659, 128, 126, 1204, 7477, 17713, 20400, 10167, 1760, 256, 219, 3059, 27042, 102545, 165607, 123976, 43959, 4657, 512, 379, 7539, 102070, 542112

Offset: 1

R. H. Hardin, Feb 19 2016

Table starts
....2.....4.......7........13..........23...........41.............72
....4....11......27........76.........185..........489...........1204
....8....32.....123.......521........1887.........7477..........27042
...16....89.....537......3288.......17713.......102545.........542112
...32...244....2343.....20400......165607......1383105.......10778640
...64...659...10167....123976.....1529241.....18220241......210476400
..128..1760...43959....742688....14011359....236272677.....4064720816
..256..4657..189465...4397376...127528641...3024972401....77785162880
..512.12228..814359..25791040..1154377943..38333973609..1477636398784
.1024.31899.3491691.150081504.10400164377.481701017577.27897108860960

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

R. H. Hardin, Table of n, a(n) for n = 1..721

Column 1 is A000079.
Column 2 is A268744.
Row 1 is A208354(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) -a(n-4)
k=3: a(n) = 10*a(n-1) -31*a(n-2) +24*a(n-3) +21*a(n-4) -18*a(n-5) -9*a(n-6)
k=4: a(n) = 12*a(n-1) -40*a(n-2) +8*a(n-3) +92*a(n-4) -32*a(n-5) -64*a(n-6) for n>7
k=5: [order 12]
k=6: [order 14]
k=7: [order 24] for n>25
Empirical for row n:
n=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)
n=2: a(n) = 2*a(n-1) +5*a(n-2) -6*a(n-3) -9*a(n-4)
n=3: a(n) = 4*a(n-1) +8*a(n-2) -34*a(n-3) -16*a(n-4) +60*a(n-5) -25*a(n-6)
n=4: [order 8]
n=5: [order 14]

A103450 A figurate number triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 22, 22, 9, 1, 1, 11, 35, 50, 35, 11, 1, 1, 13, 51, 95, 95, 51, 13, 1, 1, 15, 70, 161, 210, 161, 70, 15, 1, 1, 17, 92, 252, 406, 406, 252, 92, 17, 1, 1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1, 1, 21, 145, 525, 1170, 1722, 1722, 1170, 525, 145, 21, 1

Offset: 0

Paul Barry, Feb 06 2005

Row coefficients are the absolute values of the coefficients of the characteristic polynomials of the n X n matrices A(n) with A(n){i,i} = 2, i>0, A(n){i,j} = 1, otherwise (starts with (0,0) position).

The triangle can be generated by the matrix multiplication A007318 * A114219s, where A114219s = 0; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,3,1; 0,-1,2,-3,4,1; ... = A097807 * A128229 is a signed variant of A114219. - Gary W. Adamson, Feb 20 2007

			From _Roger L. Bagula_, Oct 21 2008: (Start)
The triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  7,  12,   7,   1;
  1,  9,  22,  22,   9,   1;
  1, 11,  35,  50,  35,  11,   1;
  1, 13,  51,  95,  95,  51,  13,   1;
  1, 15,  70, 161, 210, 161,  70,  15,   1;
  1, 17,  92, 252, 406, 406, 252,  92,  17,  1;
  1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1; ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 9.
G. Chiaselotti, W. Keith, and P. A. Oliverio, Two Self-Dual Lattices of Signed Integer Partitions, Appl. Math. Inf. Sci. 8, No. 6, 3191-3199 (2014), via ResearchGate.

Row sums are A045623.
Columns include: A000326, A002412, A002418, A005408.
Cf. A000045, A208354.

A103450:= func< n,k | k eq 0 select 1 else Binomial(n, k)*(k*(n-k) + n)/n >;
[A103450(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 17 2021

(* First program *)
p[x_, n_]:= p[x, n]= If[n==0, 1, (-1+x)^(n-2)*(1 -(n+1)*x +x^2)];
T[n_, k_]:= T[n,k]= (-1)^(n+k)*SeriesCoefficient[p[x, n], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 21 2008 *)(* corrected by G. C. Greubel, Jun 17 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Binomial[n, k]*(n*(k+1) -k^2)/n];
Table[T[n, k], {n,0,16}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 17 2021 *)

def A103450(n, k): return 1 if (k==0) else binomial(n, k)*(k*(n-k) + n)/n
flatten([[A103450(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 17 2021

T(n, k) = binomial(n-1, k-1)*(k*(n-k) + n)/k with T(n, 0) = 1.
T(n, k) = T(n-1, k-1) + T(n-1, k) + binomial(n-2, k-1) with T(n, 0) = 1.
Column k is generated by (1+k*x)*x^k/(1-x)^(k+1).
Rows are coefficients of the polynomials P(0, x) = 1, P(n, x) = (1+x)^(n-2)*(1 +(n+1)*x + x^2) for n>0.
T(n,k) = Sum_{j=0..n} binomial(k, k-j)*binomial(n-k, j)*(j+1). - Paul Barry, Oct 28 2006
A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1, p(x, 1) = (-1 +x), p(x, 2) = (1 -3*x +x^2), p(x,n) = (-1 +x)^(n-2)*(1 - (n + 1)*x + x^2); T(n, k) = (-1)^(n+k)*coefficient of x^k of ( p(x,n) ). - Roger L. Bagula and Gary W. Adamson, Oct 21 2008
T(2*n+1, n) = A141222(n). - Emanuele Munarini, Jun 01 2012 [corrected by Werner Schulte, Nov 27 2021]
G.f.: is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). - Joerg Arndt, Aug 27 2013
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/5)*((-n+5)*Fibonacci(n+1) + (3*n- 2)*Fibonacci(n)) = A208354(n). - G. C. Greubel, Jun 17 2021
T(2*n, n) = A000984(n) * (n + 2) / 2 for n >= 0. - Werner Schulte, Nov 27 2021

A268750 T(n,k)=Number of nXk binary arrays with some element plus some horizontally or vertically adjacent neighbor totalling two no more than once.

Original entry on oeis.org

2, 4, 4, 7, 11, 7, 13, 32, 32, 13, 23, 89, 143, 89, 23, 41, 244, 623, 623, 244, 41, 72, 659, 2615, 4110, 2615, 659, 72, 126, 1760, 10830, 26334, 26334, 10830, 1760, 126, 219, 4657, 44067, 165019, 255651, 165019, 44067, 4657, 219, 379, 12228, 177429, 1016807

Offset: 1

R. H. Hardin, Feb 12 2016

Table starts
...2.....4.......7........13..........23............41.............72
...4....11......32........89.........244...........659...........1760
...7....32.....143.......623........2615.........10830..........44067
..13....89.....623......4110.......26334........165019........1016807
..23...244....2615.....26334......255651.......2425799.......22577073
..41...659...10830....165019.....2425799......34732937......487682438
..72..1760...44067...1016807....22577073.....487682438....10319681062
.126..4657..177429...6183665...207252725....6746117783...215027310572
.219.12228..707163..37209717..1880654551...92215499119..4425392044505
.379.31899.2796840.221970102.16909709308.1248437108837.90177748184504

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

R. H. Hardin, Table of n, a(n) for n = 1..1404

Column 1 is A208354(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)
k=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) -a(n-4)
k=3: a(n) = 4*a(n-1) +8*a(n-2) -24*a(n-3) -38*a(n-4) +4*a(n-5) +12*a(n-6) -a(n-8)
k=4: [order 10]
k=5: [order 18]
k=6: [order 22]
k=7: [order 42]

A268781 T(n,k) = Number of n X k binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

2, 4, 4, 7, 11, 7, 13, 26, 26, 13, 23, 65, 91, 65, 23, 41, 148, 316, 316, 148, 41, 72, 343, 1031, 1462, 1031, 343, 72, 126, 766, 3354, 6383, 6383, 3354, 766, 126, 219, 1709, 10615, 27531, 38483, 27531, 10615, 1709, 219, 379, 3752, 33344, 115391, 224960

Offset: 1

R. H. Hardin, Feb 13 2016

Table starts
...2....4......7......13........23.........41...........72...........126
...4...11.....26......65.......148........343..........766..........1709
...7...26.....91.....316......1031.......3354........10615.........33344
..13...65....316....1462......6383......27531.......115391........478849
..23..148...1031....6383.....38483.....224960......1288693.......7271509
..41..343...3354...27531....224960....1755113.....13493468.....101738555
..72..766..10615..115391...1288693...13493468....140404442....1425678976
.126.1709..33344..478849...7271509..101738555...1425678976...19400886875
.219.3752.103339.1957904..40511381..758303322..14341399141..262072220011
.379.8195.317958.7940136.223527424.5590121407.142487073304.3491534799847

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

R. H. Hardin, Table of n, a(n) for n = 1..1404

Column 1 is A208354(n+1).

Diagonal is A143870.

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

A269089 T(n,k)=Number of nXk binary arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two not more than once.

Original entry on oeis.org

2, 4, 4, 7, 11, 7, 13, 30, 30, 13, 23, 76, 114, 76, 23, 41, 191, 428, 428, 191, 41, 72, 467, 1531, 2238, 1531, 467, 72, 126, 1127, 5387, 11314, 11314, 5387, 1127, 126, 219, 2686, 18590, 55620, 80422, 55620, 18590, 2686, 219, 379, 6339, 63347, 268289, 555789

Offset: 1

R. H. Hardin, Feb 19 2016

Table starts
...2.....4......7.......13.........23..........41............72............126
...4....11.....30.......76........191.........467..........1127...........2686
...7....30....114......428.......1531........5387.........18590..........63347
..13....76....428.....2238......11314.......55620........268289........1274435
..23...191...1531....11314......80422......555789.......3761534.......25063389
..41...467...5387....55620.....555789.....5372270......50865307......473602013
..72..1127..18590...268289....3761534....50865307.....673690710.....8768989835
.126..2686..63347..1274435...25063389...473602013....8768989835...159449028034
.219..6339.213490..5982734..164926651..4353444165..112658396453..2861259712706
.379.14840.713237.27813229.1074440360.39602482120.1431998499913.50787612264272

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

R. H. Hardin, Table of n, a(n) for n = 1..1404

Column 1 is A208354(n+1).

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

A208514 Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 6, 7, 5, 1, 5, 8, 12, 13, 8, 1, 6, 10, 18, 24, 23, 13, 1, 7, 12, 25, 38, 46, 41, 21, 1, 8, 14, 33, 55, 78, 88, 72, 34, 1, 9, 16, 42, 75, 120, 158, 165, 126, 55, 1, 10, 18, 52, 98, 173, 255, 313, 307, 219, 89, 1, 11, 20, 63, 124, 238

Offset: 1

Clark Kimberling, Feb 28 2012

u(n,n) = Fibonacci(n), A000045
u(n+1,n) = A208354(n)
col 1:  A000012
col 2:  A000027
col 3:  A005843
col 4:  A055998
col 5:  A140090

			First five rows:
1
1...1
1...2...2
1...3...4...3
1...4...6...7...5
First five polynomials u(n,x):
1
1 + x
1 + 2x + 2x^2
1 + 3x + 4x^2 + 3x^3
1 + 4x + 6x^2 + 7x^3 + 5x^4

Cf. A208515.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208514 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208515 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A211164 Number of compositions of n with at most one odd part.

Original entry on oeis.org

1, 1, 1, 3, 2, 8, 4, 20, 8, 48, 16, 112, 32, 256, 64, 576, 128, 1280, 256, 2816, 512, 6144, 1024, 13312, 2048, 28672, 4096, 61440, 8192, 131072, 16384, 278528, 32768, 589824, 65536, 1245184, 131072, 2621440, 262144, 5505024, 524288, 11534336, 1048576, 24117248

Offset: 0

Alois P. Heinz, Jan 30 2013

			a(3) = 3: [3], [1,2], [2,1].
a(4) = 2: [4], [2,2].
a(5) = 8: [5], [3,2], [2,3], [1,4], [4,1], [1,2,2], [2,1,2], [2,2,1].
a(6) = 4: [6], [4,2], [2,4], [2,2,2].
a(8) = 8: [8], [4,4], [2,6], [6,2], [2,2,4], [4,2,2], [2,4,2], [2,2,2,2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4)

Bisection gives: A011782 (even part), A001792 (odd part).

Cf. A208354.

a:= n-> `if`(n<2, 1, 2^iquo(n-2, 2) *
        `if`(irem(n, 2)=0, 1, iquo(n+3, 2))):
seq(a(n), n=0..60);

Vec((1-x)^2*(1+x)*(1+2*x)/(1-2*x^2)^2 + O(x^50)) \\ Colin Barker, May 07 2016

G.f.: -(2*x^4-x^3-3*x^2+x+1)/(-4*x^4+4*x^2-1).
From Colin Barker, May 07 2016: (Start)
a(n) = 2^((n-7)/2+5/2) for n>0 and even.
a(n) = 2^((n-7)/2)*(2*n+6) for n>0 and odd.
a(n) = 4*a(n-2)-4*a(n-4) for n>4.
(End)

A240847 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) - a(n-4) for n>3, a(0)=a(1)=a(3)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, -2, -5, -12, -25, -50, -96, -180, -331, -600, -1075, -1908, -3360, -5878, -10225, -17700, -30509, -52390, -89664, -153000, -260375, -442032, -748775, -1265832, -2136000, -3598250, -6052061, -10164540

Offset: 0

Paul Curtz, Apr 13 2014

F1(m, n) is the difference table of a(n):
    0,   0,   1,   0,   1,   0,   0,  -2, ...
    0,   1,  -1,   1,  -1,   0,  -2,  -3, ...
    1,  -2,   2,  -2,   1,  -2,  -1,  -4, ...
   -3,   4,  -4,   3,  -3,   1,  -3,  -2, ...
    7,  -8,   7,  -6,   4,  -4,   1,  -4, ...
  -15,  15, -13,  10,  -8,   5,  -5,   1, ...
   30, -28,  23, -18,  13, -10,   6,  -6, ...
The recurrence holds for every row and every signed column.
Main diagonal: F1(n, n) = A001477(n).
First upper diagonal: F1(n, n+1) = -A001477(n).
F1(m, n) = F1(m, n-1) + F1(m+1, n-1).
Inverse binomial transform: 0, 0, 1, -3, 7, -15, 30, ... = 0, 0, followed by (-1)^n*A023610(n). Without signs: F2(0, n) = 0, 0, 1, 3, 7, 15, 30, ... = b(n) has the same recurrence.
F1(0, n) + F2(0, n) = 0, followed by A099920(n).
a(n) and b(n) are reciprocal by their inverse binomial transform.
0, followed by A001629(n) is an autosequence.
F1(m, 1) = (-1)^n*A029907(n).
F1(1, n) = 0, 1, -1, 1, -1, followed by -A226432(n+3).
F1(m, 2) = (-1)^n*A208354(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A001629 (main sequence for the recurrence), A067331.

List([0..40], n-> (6*Fibonacci(n-3) - (n-3)*Lucas(1,-1,n-3)[2])/5 ); # G. C. Greubel, Feb 06 2020

[(6*Fibonacci(n-3) - (n-3)*Lucas(n-3))/5: n in [0..40]]; // G. C. Greubel, Feb 06 2020

with(combinat): seq( ((n+3)*fibonacci(n-3) - 2*(n-3)*fibonacci(n-2))/5, n=0..40); # G. C. Greubel, Feb 06 2020

a[n_]:= a[n]= 2*a[n-1] +a[n-2] -2*a[n-3] -a[n-4]; a[0]= a[1]= a[3]= 0; a[2]= 1; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 17 2014 *)
CoefficientList[Series[x^2*(1-2*x)/(1-x-x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,2d+c-2b-a}; NestList[nxt,{0,0,1,0},40][[All,1]] (* Harvey P. Dale, Sep 17 2022 *)

Vec(x^2*(1-2*x)/(1-x-x^2)^2 + O(x^100)) \\ Colin Barker, Apr 13 2014

vector(41, n, my(m=n-1); ((m+3)*fibonacci(m-3) - 2*(m-3)*fibonacci(m-2) )/5 ) \\ G. C. Greubel, Feb 06 2020

[((n+3)*fibonacci(n-3) - 2*(n-3)*fibonacci(n-2))/5 for n in (0..40)] # G. C. Greubel, Feb 06 2020

a(n) = 0, 0, 1, 0, 1, 0, 0, followed by -A067331.
G.f.: x^2*(1-2*x)/(1-x-x^2)^2. - Colin Barker, Apr 13 2014
a(n) = ( (10*n + (3-5*n)*t)*(1+t)^n + (10*n-(3-5*n)*t)*(1-t)^n )/(25*2^n), where t=sqrt(5). - Bruno Berselli, Apr 17 2014
a(n) = (6*Fibonacci(n-3) - (n-3)*Lucas(n-3))/5 = ((n+3)*Fibonacci(n-3) - 2*(n-3)*Fibonacci(n-2))/5. - G. C. Greubel, Feb 06 2020

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 6, 7, 8, 8, 8, 7, 6, 4, 4, 5, 5, 8, 10, 13, 15, 16, 16, 15, 13, 10, 8, 5, 5, 6, 6, 10, 13, 18, 23, 28, 31, 32, 31, 28, 23, 18, 13, 10, 6, 6, 7, 7, 12, 16, 23, 31, 41, 51, 59, 63, 63, 59, 51, 41, 31, 23, 16, 12, 7, 7

Offset: 1

Lechoslaw Ratajczak, Jul 04 2020

The number of terms in row n is 3*n-1 = A016789(n-1).
The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).
Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.

			Triangle begins:
n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20...
1   1  1
2   2  2  2  2  2
3   3  3  4  4  4  4  3  3
4   4  4  6  7  8  8  8  7  6  4  4
5   5  5  8 10 13 15 16 16 15 13 10  8  5  5
6   6  6 10 13 18 23 28 31 32 31 28 23 18 13 10  6  6
7   7  7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12  7  7
...

Superdiagonal 1 is A029907 for n >= 1.
The main diagonal is A208354 for n >= 1.
Subdiagonal 1 is A102702(n-1) for n >= 1.
Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).
Subdiagonal 3 is A191830(n+3) for n >= 1.
Cf. A007318, A051597, A228196.

T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.

cp's OEIS Frontend

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A269089 T(n,k)=Number of nXk binary arrays with some element plus some horizontally, vertically or antidiagonally adjacent neighbor totalling two not more than once.

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A240847 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) - a(n-4) for n>3, a(0)=a(1)=a(3)=0, a(2)=1.

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).