A001629 -id:A001629 - cp's OEIS frontend

A004798 Convolution of Fibonacci numbers 1,2,3,5,... with themselves.

1, 4, 10, 22, 45, 88, 167, 310, 566, 1020, 1819, 3216, 5645, 9848, 17090, 29522, 50793, 87080, 148819, 253610, 431086, 731064, 1237175, 2089632, 3523225, 5930668, 9968122, 16730830, 28045221, 46954360, 78524159, 131181406, 218933030, 365044788, 608135635, 1012268592

Offset: 1

Clark Kimberling

From Emeric Deutsch, Feb 15 2010: (Start)
a(n) is the number of subwords of the form 0000 in all binary words of length n+3 that have no pair of adjacent 1's. Example: a(2)=4 because in the 13 (=A000045(7)) binary words of length 5 that have no pair of adjacent 1's, namely 00000, 00001, 00010, 00100, 00101, 01000, 01001, 01010, 10000, 10001, 10010, 10100, 10101, we have 2 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 + 0 + 0 + 0 = 4 subwords of the form 0000.
a(n) = Sum_{k>=0} k*A171855(n + 3,k). (End)
a(n) is the total number of 0's in all binary words of length n that have no pair of adjacent 1's. Example: a(5) = 45 because in the binary words listed in the above example there are respectively 5 + 4 + 4 + 4 + 3 + 4 + 3 + 3 + 4 + 3 + 3 + 3 + 2 = 45. - Geoffrey Critzer, Jul 22 2013

			a(6) = 45 + 22 + A000045(6+2) = 45 + 22 + 21 = 88. - _Philippe Deléham_, Jan 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A171855, A000045.

Cf. A000032, A001629, A030528, A045925.

List([1..40], n-> (n*Lucas(1,-1,n+3)[2] - 2*Fibonacci(n))/5); # G. C. Greubel, Jul 07 2019

I:=[1,4,10,22]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Apr 08 2018

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-2|1|2>>^n. <<0, 1, 4, 10>>)[1, 1]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 04 2013
# Alternative:
a := n -> n*(hypergeom([-(n+1)/2,-n/2],[-n-1],-4) - hypergeom([(1-n)/2,1-n/2],[-n], -4)): seq(simplify(a(n)), n=1..40); # Peter Luschny, Apr 10 2018

nn=40; Drop[CoefficientList[Series[D[(1+x)/(1-y x -y x^2),y]/.y->1,{x,0,nn}],x],1] (* Geoffrey Critzer, Jul 22 2013 *)
Table[n Fibonacci[n] + 2/5 (n LucasL[n] - Fibonacci[n]), {n, 40}] (* Vladimir Reshetnikov, Sep 27 2016 *)
a[n_] := ListConvolve[f = Fibonacci[Range[2, n+1]], f][[1]]; Array[a, 40] (* Jean-François Alcover, Feb 15 2018 *)
LinearRecurrence[{2, 1, -2, -1}, {1, 4, 10, 22}, 40] (* Vincenzo Librandi, Apr 08 2014 *)

Vec(((1+x)/(1-x-x^2))^2+O(x^66)) \\ Joerg Arndt, Jul 04 2013

[(n*lucas_number2(n+3,1,-1) - 2*fibonacci(n))/5 for n in (1..40)] # G. C. Greubel, Jul 07 2019

O.g.f.: (x+1)^2*x/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = a(n-1) + a(n-2) + Fibonacci(n+2). - Philippe Deléham, Jan 22 2012
O.g.f. is the derivative of A(x,y) with respect to y and then evaluated at y = 1, where A(x,y) is the o.g.f. for A030528. - Geoffrey Critzer, Jul 22 2013
a(n) = A001629(n+1) + A001629(n-1) + 2*A001629(n). - R. J. Mathar, Oct 30 2015
a(n) = n*Fibonacci(n) + (2/5)*(n*Lucas(n) - Fibonacci(n)) = A045925(n) + 2*A001629(n), where Lucas = A000032, Fibonacci = A000045. - Vladimir Reshetnikov, Sep 27 2016
a(n) = Sum_{i=0..floor((n+1)/2)} binomial(n+1-i,i)*(n-i). - John M. Campbell, Apr 07 2018
From Peter Luschny, Apr 10 2018: (Start)
a(n) = n*(hypergeom([-(n+1)/2, -n/2], [-n - 1], -4) - hypergeom([(1-n)/2, 1 - n/2], [-n], -4)).
a(n) = n*A000045(n+2) - A001629(n+1). (End)
E.g.f.: exp(x/2)*(35*x*cosh(sqrt(5)*x/2) + sqrt(5)*(15*x - 4)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A111006 Another version of Fibonacci-Pascal triangle A037027.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144

Offset: 0

Philippe Deléham, Oct 02 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
Triangle read by rows: T(n,n-k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)). - Philippe Deléham, Feb 17 2014
Diagonal sums are A013979(n). - Philippe Deléham, Feb 17 2014
T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and 1 X 2 tiles. - Emeric Deutsch, Aug 14 2014

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 5,  5;
  0, 0, 0, 3, 10,  8;
  0, 0, 0, 1,  9, 20, 13;
  0, 0, 0, 0,  4, 22, 38,  21;
  0, 0, 0, 0,  1, 14, 51,  71,  34;
  0, 0, 0, 0,  0,  5, 40, 111, 130,  55;
  0, 0, 0, 0,  0,  1, 20, 105, 233, 235,  89;
  0, 0, 0, 0,  0,  0,  6,  65, 256, 474, 420, 144;

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938, A128100 (reversed version).

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A114197, A162741, A228074.

a111006 n k = a111006_tabl !! n !! k
a111006_row n = a111006_tabl !! n
a111006_tabl =  map fst $ iterate (\(us, vs) ->
   (vs, zipWith (+) (zipWith (+) ([0] ++ us ++ [0]) ([0,0] ++ us))
                    ([0] ++ vs))) ([1], [0,1])
-- Reinhard Zumkeller, Aug 15 2013

T(0, 0) = 1, T(n, k) = 0 for k < 0 or for n < k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k). T(n, n) = A000045(n+1). T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2). - Paul Barry, Oct 04 2005
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively. - Philippe Deléham, Dec 02 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 respectively. - Philippe Deléham, Feb 17 2014

A281205 T(n,k)=Number of nXk 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 2, 14, 10, 0, 5, 28, 56, 38, 0, 10, 52, 98, 168, 130, 0, 20, 94, 176, 270, 448, 420, 0, 38, 166, 310, 470, 676, 1120, 1308, 0, 71, 290, 537, 804, 1141, 1588, 2688, 3970, 0, 130, 502, 922, 1358, 1906, 2602, 3604, 6272, 11822, 0, 235, 864, 1573, 2284, 3137

Offset: 1

R. H. Hardin, Jan 17 2017

Table starts
.0.....0.....1.....2.....5....10.....20.....38.....71....130....235.....420
.0.....2....14....28....52....94....166....290....502....864...1480....2526
.0....10....56....98...176...310....537....922...1573...2672...4524....7640
.0....38...168...270...470...804...1358...2284...3834...6432..10786...18080
.0...130...448...676..1141..1906...3137...5160...8510..14084..23379...38894
.0...420..1120..1588..2602..4248...6838..11010..17840..29120..47838...78978
.0..1308..2688..3604..5712..9118..14375..22700..36144..58168..94524..154800
.0..3970..6272..7960.12208.19026..29416..45614..71452.113388.182228..295950
.0.11822.14336.17254.25577.38916..58984..89916.138676.217124.345089..555674
.0.34690.32256.36848.52784.78356.116466.174558.265278.409976.644568.1028978

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

R. H. Hardin, Table of n, a(n) for n = 1..421

Row 1 is A001629(n-1).

Empirical for column k:
k=1: a(n) = 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) = 4*a(n-1) -4*a(n-2) for n>3
k=4: a(n) = 2*a(n-1) +3*a(n-2) -4*a(n-3) -6*a(n-4) +2*a(n-5) +4*a(n-6) -a(n-8)
k=5: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-4) +4*a(n-5) -a(n-8)
k=6: [order 12]
k=7: [order 12]
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) = 3*a(n-1) -a(n-2) -3*a(n-3) +a(n-4) +a(n-5) for n>7
n=3: a(n) = 3*a(n-1) -a(n-2) -3*a(n-3) +a(n-4) +a(n-5) for n>8
n=4: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6) for n>10
n=5: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6) for n>11
n=6: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6) for n>12
n=7: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6) for n>13

A006478 a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 18, 38, 76, 147, 277, 512, 932, 1676, 2984, 5269, 9239, 16104, 27926, 48210, 82900, 142055, 242665, 413376, 702408, 1190808, 2014608, 3401833, 5734251, 9650312, 16216602, 27213182, 45608092, 76345851, 127656829, 213230144, 355817324, 593205284

Offset: 0

N. J. A. Sloane

Partial sums of A001629.
Number of edges in the Fibonacci hypercube FQ(n-2) (defined in the Rispoli and Cosares reference). - Emeric Deutsch, Oct 06 2014
Circuit rank (cyclomatic number) of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 05 2017

			G.f. = x^3 + 3*x^4 + 8*x^5 + 18*x^6 + 38*x^7 + 76*x^8 + 147*x^9 + 277*x^10 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Gray codes for Fibonacci q-decreasing words, arXiv:2010.09505 [cs.DM], 2020.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
K. J. Overholt, Efficiency of the Fibonacci search method, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 92-96.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
F. J. Rispoli and S. Cosares, The Fibonacci hypercube, Australasian J. Combinatorics, 40, 2008, 187-196.
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A000045, A000051, A006479, A175722.

Cf. A001629.

a006478 n = a006478_list !! (n-3)
a006478_list = scanl1 (+) $ drop 2 a001629_list
-- Reinhard Zumkeller, Sep 12 2015

A006478 := proc(n)
  1 + ((n-5)*combinat[fibonacci](n-1)+(3*n-8)*combinat[fibonacci](n)) / 5;
end proc:
seq(A006478(n),n=0..20) ; # R. J. Mathar, Jun 12 2018

CoefficientList[Series[x^3/((1 - x) (1 - x - x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 1, 3, 8}, 20] (* Eric W. Weisstein, Sep 05 2017 *)
Table[1 + (2 (n + 1) Fibonacci[n] + n Fibonacci[n + 1])/5 - Fibonacci[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Sep 05 2017 *)

{a(n) = if( n<0, polcoeff( x^2 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^-n), -n), polcoeff( x^3 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */

a(n) - a(n-1) = A001629(n-1).
a(n) = 1 + ((n-5)*F(n-1) + (3*n-8)*F(n))/5.
G.f.: x^3/((1-x)*(1-x-x^2)^2). - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} F(i)*F(k-i). - Benoit Cloitre, Jan 26 2003
a(n) = A175722(-2-n). - Michael Somos, Mar 11 2014
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5). - Eric W. Weisstein, Sep 05 2017
E.g.f.: exp(x) + exp(x/2)*(5*(3*x - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x - 11)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Jul 24 2022

a(0)-a(2) added and offset changed - N. J. A. Sloane, Jun 19 2021

Programs and b-file adapted by Georg Fischer, Jun 21 2021

A067330 Triangle read by rows of incomplete convolutions of Fibonacci numbers F(n+1) = A000045(n+1), n>=0.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 3, 5, 7, 10, 5, 8, 12, 15, 20, 8, 13, 19, 25, 30, 38, 13, 21, 31, 40, 50, 58, 71, 21, 34, 50, 65, 80, 96, 109, 130, 34, 55, 81, 105, 130, 154, 180, 201, 235, 55, 89, 131, 170, 210, 250, 289, 331, 365, 420, 89, 144, 212, 275, 340, 404, 469, 532, 600, 655, 744, 144, 233, 343, 445

Offset: 0

Wolfdieter Lang, Feb 15 2002

The diagonals d>=0 (d=0: main diagonal) give convolutions of Fibonacci numbers F(n+1), n>=0, with those with d-shifted index: a(d+n,d)=sum(F(k+1)*F(d+n+1-k),k=0..n), n>=0.
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(x*z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. Fibonacci F(n+1), n>=0).
The diagonals give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for d= n-m= 0..9, respectively.
A row with n terms = the dot product of vectors with n terms: (1,1,2,3,...)dot(...3,2,1,1) with carryovers; such that (3, 5, 7, 10) = (1*3=3), (1*2+3=5), (2*1+5=7), (3*1+7=10).

			{1}; {1,2}; {2,3,5}; {3,5,7,10}; ...; p(2,n)= 2+3*x+5*x^2.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A067418 (triangle with rows read backwards).

Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

a(n, m)= sum(F(k+1)*F(n-k+1), k=0..m), n>=m>=0, else 0.
a(n, m)= (((3*m+5)*F(m+1)+(m+1)*F(m))*F(n-m+1)+(m*F(m+1)+2*(m+1)*F(m))*F(n-m))/5.
G.f. for diagonals d=n-m>=0: (x^d)*(F(d+1)+F(d)*x)/(1-x-x^2)^2, with F(n) := A000045(n) (Fibonacci).
a(n, m) = ((-1)^m*F(n-2*m-1)+m*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(m+1)*L(n+2)-A067979(n, m))/5, n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067418 Triangle A067330 with rows read backwards.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 10, 7, 5, 3, 20, 15, 12, 8, 5, 38, 30, 25, 19, 13, 8, 71, 58, 50, 40, 31, 21, 13, 130, 109, 96, 80, 65, 50, 34, 21, 235, 201, 180, 154, 130, 105, 81, 55, 34, 420, 365, 331, 289, 250, 210, 170, 131, 89, 55, 744, 655, 600, 532, 469, 404, 340, 275, 212, 144, 89, 1308, 1164, 1075, 965

Offset: 0

Wolfdieter Lang, Feb 15 2002

The column m (without leading 0's) gives the convolution of Fibonacci numbers F(n+1) := A000045(n+1), n>=0, with those with m-shifted index: a(n+m,m)=sum(F(k+1)*F(m+n+1-k),k=0..n), n>=0, m=0,1,...
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. for Fibonacci F(n+1), n>=0).
The columns give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for m= 0..9, respectively. Row sums give A067988.

			{1}; {2,1}; {5,3,2}; {10,7,5,3}; ...; p(2,n)=5+3*x+2*x^2.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Reverse /@ Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

a(n, m)=A067330(n, n-m), n>=m>=0, else 0.
a(n, m)= (((3*(n-m)+5)*F(n-m+1)+(n-m+1)*F(n-m))*F(m+1)+((n-m)*F(n-m+1)+2*(n-m+1)*F(n-m))*F(m))/5.
G.f. for column m=0, 1, ...: (x^m)*(F(m+1)+F(m)*x)/(1-x-x^2)^2, with F(m) := A000045(m) (Fibonacci).
a(n, m) = ((-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(n-m+1)*L(n+2)-A067990(n, m))/5, n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A268886 T(n,k)=Number of nXk binary arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 1, 0, 2, 5, 0, 5, 14, 20, 0, 10, 54, 84, 71, 0, 20, 158, 501, 462, 235, 0, 38, 475, 2190, 4133, 2418, 744, 0, 71, 1340, 9996, 27130, 31956, 12252, 2285, 0, 130, 3740, 42362, 186732, 317966, 236960, 60666, 6865, 0, 235, 10204, 178400, 1187838, 3283890

Offset: 1

R. H. Hardin, Feb 15 2016

Table starts
.0.....1.......2.........5..........10............20..............38
.0.....5......14........54.........158...........475............1340
.0....20......84.......501........2190..........9996...........42362
.0....71.....462......4133.......27130........186732.........1187838
.0...235....2418.....31956......317966.......3283890........31427480
.0...744...12252....236960.....3596174......55491832.......800733668
.0..2285...60666...1706732....39670270.....911930096.....19876401224
.0..6865..295230..12034000...429588382...14681855846....483987898760
.0.20284.1417452..83485488..4585939726..232688402028..11611969197776
.0.59155.6732102.571836176.48401059362.3642322709900.275345016177616

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

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

Column 2 is A054444(n-1).

Row 1 is A001629.

Empirical for column k:
k=1: a(n) = 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]

A269011 T(n,k)=Number of nXk binary arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two exactly once.

Original entry on oeis.org

0, 1, 0, 2, 4, 0, 5, 8, 15, 0, 10, 36, 46, 48, 0, 20, 88, 305, 224, 145, 0, 38, 272, 1078, 2136, 1066, 420, 0, 71, 696, 4948, 10976, 14240, 4952, 1183, 0, 130, 1900, 18210, 73568, 109058, 91048, 22654, 3264, 0, 235, 4856, 73277, 390064, 1049588, 1053432, 566656

Offset: 1

R. H. Hardin, Feb 17 2016

Table starts
.0.....1.......2.........5.........10...........20.............38
.0.....4.......8........36.........88..........272............696
.0....15......46.......305.......1078.........4948..........18210
.0....48.....224......2136......10976........73568.........390064
.0...145....1066.....14240.....109058......1049588........8134304
.0...420....4952.....91048....1053432.....14382480......164351184
.0..1183...22654....566656...10002542....192100836.....3258530608
.0..3264..102416...3456320...93733440...2516546784....63679868768
.0..8865..458674..20760192..869397882..32481770852..1230707111424
.0.23780.2038328.123186784.7996744280.414339126768.23573013881888

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

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

Column 2 is A093967.

Row 1 is A001629.

Empirical for column k:
k=1: a(n) = 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]
n=6: [order 20]
n=7: [order 32]

A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 5, 2, 1, 12, 10, 6, 2, 1, 20, 20, 13, 7, 2, 1, 33, 38, 29, 16, 8, 2, 1, 54, 71, 60, 39, 19, 9, 2, 1, 88, 130, 122, 86, 50, 22, 10, 2, 1, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1, 376, 744, 894, 806

Offset: 1

Clark Kimberling, Mar 16 2012

For a discussion and guide to related arrays, see A208510.
From Gus Wiseman, Jun 29 2025: (Start)
This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:
  {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
  {2}      {2,3}    {2,3,4}    {2,3,4,5}
  {3}      {3,4}    {3,4,5}
  {4}      {4,5}    {1,2,3,5}
  {5}      {1,2,4}  {1,2,4,5}
  {1,3}    {1,2,5}  {1,3,4,5}
  {1,4}    {1,3,4}
  {1,5}    {1,4,5}
  {2,4}    {2,3,5}
  {2,5}    {2,4,5}
  {3,5}
  {1,3,5}
For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)

			First five rows:
  1
  2    1
  4    2    1
  7    5    2   1
  12   10   6   2   1
First three polynomials v(n,x): 1, 2 + x, 4 + 2*x + x^2.

Cf. A210033, A208510.
Column k = 1 is A000071.
Row sums are A000225.
Column k = 2 is A001629.
Column k = 3 is A055243.
The version including k = 0 is A384893.
A034839 counts subsets by number of maximal runs, see also A202023, A202064.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
Cf. A053538, A116674, A119900, A268193, A384177, A384879, A384889, A384905.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, 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[%]    (* A210033 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210034 *)

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

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A004798 Convolution of Fibonacci numbers 1,2,3,5,... with themselves.

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A111006 Another version of Fibonacci-Pascal triangle A037027.

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A281205 T(n,k)=Number of nXk 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A006478 a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.

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A067330 Triangle read by rows of incomplete convolutions of Fibonacci numbers F(n+1) = A000045(n+1), n>=0.

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