A010049 -id:A010049 - cp's OEIS frontend

A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 8, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 8, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 8, 16, 13, 21, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18

Offset: 0

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Fibonacci(n+2).
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).
Also the number of sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference. The maximal case is A384883. For prime instead of binary indices we have A166469. - Gus Wiseman, Jul 05 2025

			From _Gus Wiseman_, Jul 05 2025: (Start)
The binary indices of 11 are {1,2,4}, with sparse subsets {{},{1},{2},{4},{1,4},{2,4}}, so a(11) = 6.
The maximal runs of binary indices of 11 are ((1,2),(4)), with lengths (2,1), so a(11) = F(2+2)*F(1+2) = 6.
The a(0) = 1 through a(12) = 3 sparse subsets are:
  0    1    2    3    4    5    6    7    8    9    10    11    12
  ------------------------------------------------------------------
  {}   {}   {}   {}   {}   {}   {}   {}   {}   {}    {}    {}    {}
       {1}  {2}  {1}  {3}  {1}  {2}  {1}  {4}  {1}   {2}   {1}   {3}
                 {2}       {3}  {3}  {2}       {4}   {4}   {2}   {4}
                           {1,3}     {3}       {1,4} {2,4} {4}
                                     {1,3}                 {1,4}
                                                           {2,4}
The greatest number whose set of binary indices is a member of column n above is A374356(n).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A245562, A000045, A001045, A071053, A245565, A246028.
A034839 counts subsets by number of maximal runs, strict partitions A116674.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A053538, A166469, A202023, A202064, A208342, A374356, A384883.

with(combinat); ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
                   od:
a:=mul(fibonacci(i+2), i in lis);
ans:=[op(ans),a];
od:
ans;

a[n_] := Sum[Mod[Binomial[3k, k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 29 2020, after Chai Wah Wu *)
spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[spars[bpe[n]]],{n,0,30}] (* Gus Wiseman, Jul 05 2025 *)

a(n)=my(s=1,k); while(n, n>>=valuation(n,2); k=valuation(n+1,2); s*=fibonacci(k+2); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

# use RLT function from A278159
from sympy import fibonacci
def A245564(n): return RLT(n,lambda m: fibonacci(m+2)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

A122931 Row sums of triangular array A122930.

Original entry on oeis.org

1, 2, 7, 18, 50, 132, 351, 924, 2431, 6380, 16732, 43848, 114869, 300846, 787815, 2062830, 5401054, 14140940, 37022755, 96928920, 253766591, 664375032, 1739365272, 4553731728, 11921847625, 31211839802, 81713718151, 213929389674

Offset: 1

Alford Arnold, Sep 20 2006

Also sums of the natural numbers with A000045 entries per row: for example, 1 2 3+4 5+6+7 8+9+10+11+12.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A000027, A000045, A001654, A003714, A010049, A035514, A122930.

Cf. A000217, A191797, A301809.

A000045 := proc(n) if n <= 1 then RETURN(n) ; else RETURN( A000045(n-1)+A000045(n-2)) ; fi ; end: A000071 := proc(n) RETURN(A000045(n)-1) ; end: A122931 := proc(n) local a45 ; a45 := A000045(n) ; RETURN (a45*(A000071(n+1)+(a45+1)/2)) ; end: for n from 1 to 30 do printf("%d,",A122931(n)) ; od ; # R. J. Mathar, Oct 07 2006

(#[[2]]^2-#[[1]]^2-#[[2]]+#[[1]])/2&/@Partition[Fibonacci[ Range[ 2,30]],2,1] (* or *) Module[{nn=30,fib},fib=Fibonacci[Range[nn]];Total/@ TakeList[ Range[Total[ fib]], fib]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 19 2018 *)

From R. J. Mathar, Oct 07 2006: (Start)
a(n) = Sum_{i=A000071(n+1)+1..A000071(n+2)} i.
a(n) = A000045(n)*floor(A000071(n+1) + (A000045(n)+1)/2). (End)
a(n) = Sum_{k=1..n} A000045(k)^2*A000045(n-k+1). - Gerald McGarvey, Nov 08 2007
a(n) = (F(n+2)^2 - F(n+1)^2 - F(n+2) + F(n+1))/2 where F(n)=Fibonacci(n). - Gary Detlefs, Mar 10 2011
G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)*(1-x-x^2)). - Colin Barker, Mar 12 2012
a(n) = F(n)*(F(n+3)-1)/2. - J. M. Bergot, Mar 16 2013
a(n) = (F(n+1) - 1)*(F(n+2) + 1)/2 + (n mod 2). - Greg Dresden, Sep 25 2021
a(n) = A301809(n+1) - A000045(n) = A191797(n+2) - A191797(n+1). - J.S. Seneschal, Jul 07 2025

More terms from R. J. Mathar, Oct 07 2006

A146005 a(n) = n*Lucas(n).

Original entry on oeis.org

0, 1, 6, 12, 28, 55, 108, 203, 376, 684, 1230, 2189, 3864, 6773, 11802, 20460, 35312, 60707, 104004, 177631, 302540, 513996, 871266, 1473817, 2488368, 4194025, 7057518, 11858508, 19898116, 33345679, 55814940, 93320819, 155867104

Offset: 0

R. J. Mathar, Oct 26 2008

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

I:=[0, 1, 6, 12]; [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, Dec 13 2012

Table[LucasL[n, 1]*n, {n, 0, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
CoefficientList[Series[x * (1 + 4*x - x^2)/(1 - x - x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
LinearRecurrence[{2,1,-2,-1},{0,1,6,12},40] (* Harvey P. Dale, Apr 03 2013 *)

a(n) = n*A000032(n).
G.f.: x(1+4x-x^2)/(1-x-x^2)^2.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4).
a(n) = A000045(n)-5*A000045(n+1)+5*A010049(n+1).
a(n) = A045925(n)+2*A099920(n-1).
E.g.f.: x*exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)). - G. C. Greubel, Jan 30 2016

A289207 a(n) = max(0, n-2).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 28 2017

This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).
The rows of this array are existing sequences of the OEIS, prepended with zeros:
row 0 is A118425,
row 1 is A006478,
row 2 is A001629,
row 3 is A010049,
row 4 is A006367,
row 5 is not in the OEIS.
It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.
For n = 1 to 8, rows with only one leading zero are also autosequences.

			Array of differences begin:
   0,   0,   0,   0,  0,   0,  0,  1,  4, 12, 30, 68, ...
   0,   0,   0,   0,  0,   0,  1,  3,  8, 18, 38, 76, ...
   0,   0,   0,   0,  0,   1,  2,  5, 10, 20, 38, 71, ...
   0,   0,   0,   0,  1,   1,  3,  5, 10, 18, 33, 59, ...
   0,   0,   0,   1,  0,   2,  2,  5,  8, 15, 26, 46, ...
   0,   0,   1,  -1,  2,   0,  3,  3,  7, 11, 20, 34, ...
   0,   1,  -2,   3, -2,   3,  0,  4,  4,  9, 14, 24, ...
   1,  -3,   5,  -5,  5,  -3,  4,  0,  5,  5, 10, 16, ...
  -4,   8, -10,  10, -8,   7, -4,  5,  0,  6,  6, 17, ...
  12, -18,  20, -18, 15, -11,  9, -5,  6,  0,  7,  7, ...
  ...

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Essentially the same as A023444. Cf. A001477, A118425, A006478, A001629, A010049, A006367, A199969.

a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]

G.f.: x^3 / (1-x)^2.

A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 3, 3, 4, 5, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2

Offset: 0

Gus Wiseman, Jul 02 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 27 are {1,2,4,5}, with maximal sparse subsets {{1,4},{1,5},{2,4},{2,5}}, so a(27) = 4.

For subsets of {1..n} we get A000931 (shifted), maximal case of A000045 (shifted).
This is the maximal case of A245564.
The greatest number whose binary indices are one of these subsets is A374356.
For prime instead of binary indices we have A385215, maximal case of A166469.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A202064 counts subsets containing n with k maximal runs.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A044813, A053538, A119900, A202023, A208342, A384177, A384890.

spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
maximize[sys_]:=Complement@@Prepend[Most[Subsets[#]]&/@sys,sys];
Table[Length[maximize[spars[bpe[n]]]],{n,0,100}]

A095353 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range [Fib(n+1),Fib(n+2)[ (where Fib = A000045).

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 7, 14, 23, 35, 56, 94, 155, 243, 402, 614, 1061, 1656, 2689, 4295, 6938, 11176, 18095, 29102, 46907, 75703, 122174, 197494, 317987, 514611, 829595, 1340861, 2166008, 3497040, 5645418, 9120129, 14733126, 23803219, 38460014

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratio a(n)/A095354(n) (i.e. average number of 1-fibits in Zeckendorf-expansions of primes p which Fib(n+1) <= p < Fib(n+2)) grows as: 1, 1, 1, 1.5, 2., 2.333333, 2.333333, 2.8, 3.285714, 3.181818, 3.5, 3.916667, 4.189189, 4.418182, 4.785714, 4.873016, 5.358586, 5.575758, 5.871179, 6.100852, 6.382705, 6.676225, 6.954266, 7.223132, 7.489542, 7.773978, 8.045173, 8.331323, 8.598659, 8.886546, 9.161734, 9.440489, 9.71936, 9.995484, 10.266207, 10.54327, 10.820602, 11.096084, 11.374267.

Ratio of that average compared to A010049(n)/A000045(n) (the expected value of that same sum computed for all integers in the same range) converges as: 1, 1, 0.666667, 0.9, 1, 1.037037, 0.919192, 0.99661, 1.063946, 0.945946, 0.96142, 1, 0.999059, 0.988519, 1.008389, 0.970278, 1.011305, 1.000122, 1.003368, 0.995592, 0.996635, 0.999338, 0.999601, 0.998575, 0.997298, 0.998427, 0.997837, 0.999078, 0.998056, 0.99941, 0.999296, 0.999567, 0.999834, 0.999811, 0.999265, 0.999347, 0.999451, 0.999382, 0.999555.

			a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100. a(4)=3, as in [5,8[ there are primes 5 and 7, whose Fibonacci-representations are 1000 and 1010 respectively and we have three 1-fibits in total. a(5)=2, as in [8,13[ there is only one prime 11, with Zeckendorf-representation 10100.

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095336, A095298 (similar sums and ratios computed in binary system).

a(2) corrected by Chai Wah Wu, Jan 16 2020

A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73

Offset: 1

Clark Kimberling, Mar 22 2012

Let T(n,k) denote the term in row n, column k.
T(n,n):  A000045 (Fibonacci numbers)
T(n,n-1): A010049 (second-order Fibonacci numbers)
T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...
T(n,2): 2,3,4,5,6,7,8,9,10,11,...
T(n,3): 3,5,7,9,11,13,15,17,19,...
T(n,4): A052905
Row sums: A000225
Alternating row sums: A094024 (signed)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.

Cf. A210553, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, 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[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

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

A102702 Expansion of (2-x-2*x^2-x^3)/(1-x-x^2)^2.

Original entry on oeis.org

2, 3, 6, 10, 18, 31, 54, 93, 160, 274, 468, 797, 1354, 2295, 3882, 6554, 11046, 18587, 31230, 52401, 87812, 146978, 245736, 410425, 684818, 1141611, 1901454, 3164458, 5262330, 8744599, 14521158, 24097797, 39965224, 66241330, 109731132

Offset: 0

Creighton Dement, Feb 04 2005

A floretion-generated sequence which results from a certain transform of the Fibonacci numbers. Specifically, (a(n)) is the (type 1B) tesfor-transform of the Fibonacci numbers (A000045) with respect to the floretion + .5'i + .5i' Note, for example, that the sequence A001629, appearing in the formula given, has the name "Fibonacci numbers convolved with themselves" and that this sequence arises in FAMP (see program code) under the name: the lesfor-transform (type 1B) of the Fibonacci numbers (A000045) with respect to the floretion + .5'i + .5i' . The denominator of the generating function has roots at the golden ratio phi and -(1+phi).
Floretion Algebra Multiplication Program. FAMP Code: (a(n)) = 2tesforseq[ + .5'i + .5i' ], 2lesforseq = A001629, jesforseq = A029907, vesforseq = A000045, ForType: 1B.
a(n) is the total number of parts not greater than 2 among all compositions of n+3 in which only the last part may be equal to 1. - Andrew Yezhou Wang, Jul 14 2019

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Chapter 15, page 187, "Hosoya's Triangle".
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
Mengmeng Liu and Andrew Yezhou Wang, The number of designated parts in compositions with restricted parts, Journal of Integer Sequences, 23 (2020), Article 20.1.8.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A001629, A029907.

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (2-x-2*x^2-x^3)/(1-x-x^2)^2)); // Marius A. Burtea, Dec 31 2019

CoefficientList[Series[(2-x-2x^2-x^3)/(x^4+2x^3-x^2-2x+1),{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{2,3,6,10},40] (* Harvey P. Dale, Apr 21 2014 *)

G.f.: (2-x-2*x^2-x^3)/(1-x-x^2)^2.
a(n) = 2*F(n+1) + A001629(n+3) - 2*A029907(n+1);
F(n+1) = a(n+2) - a(n+1) - a(n).
a(0)=2, a(1)=3, a(2)=6, a(3)=10, a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Harvey P. Dale, Apr 21 2014
a(n) = A010049(n+1) + A000045(n+2). - R. J. Mathar, May 21 2019
a(n) = ((2*n+10)*F(n+1)-(n-4)*F(n))/5. - Andrew Yezhou Wang, Jul 14 2019

Corrected by T. D. Noe, Nov 02 2006

A123603 Triangle T(n,k), 0<=k<=n, read by rows, with T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k-1) + T(n-2,k).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 3, 3, 5, 5, 9, 5, 5, 8, 10, 17, 17, 10, 8, 13, 18, 36, 35, 36, 18, 13, 21, 33, 69, 81, 81, 69, 33, 21, 34, 59, 133, 167, 199, 167, 133, 59, 34, 55, 105, 249, 345, 435, 435, 345, 249, 105, 55, 89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89

Offset: 0

Philippe Deléham, Nov 14 2006, Mar 14 2014

			Triangle begins:
1;
1, 1;
2, 1, 2;
3, 3, 3, 3;
5, 5, 9, 5, 5;
8, 10, 17, 17, 10, 8;
13, 18, 36, 35, 36, 18, 13;
21, 33, 69, 81, 81, 69, 33, 21;
34, 59, 133, 167, 199, 167, 133, 59, 34;
55, 105, 249, 345, 435, 435, 345, 249, 105, 55;
89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000045, A000129, A322239 (central terms).

CoefficientList[CoefficientList[Series[1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)
T[0, 0] := 1; T[n_, k_] := If[k < 0 || k > n, 0, T[n - 1, k - 1] + T[n - 1, k] + T[n - 2, k - 2] - T[n - 2, k - 1] + T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Oct 16 2017 *)

T(n,k) = T(n,n-k).
T(n,0) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A010049(n+1).
Sum_{k,0<=k<=n} T(n,k)*x^k = A000045(n+1), A000129(n+1), A030195(n+1), A015532(n+1) for x = 0, 1, 2, 3 respectively.
G.f.: 1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2).

A131913 Product of the square matrix in A065941 and the column vector (1, 2, 3, ...)'.

Original entry on oeis.org

1, 3, 6, 13, 25, 48, 89, 163, 294, 525, 929, 1632, 2849, 4947, 8550, 14717, 25241, 43152, 73561, 125075, 212166, 359133, 606721, 1023168, 1722625, 2895843, 4861254, 8149933, 13646809, 22825200, 38136089, 63653827, 106146534, 176849517, 294401825, 489706272

Offset: 0

Gary W. Adamson, Jul 27 2007

			a(4) = 25 = (1, 1, 3, 2, 1) dot (1, 2, 3, 4, 5) = (1 + 2 + 9 + 8 + 5), where (1, 1, 3, 2, 1) = row 4 of triangle A065941.
a(4) = 25 = A010049(4) + A001629(6) = 5 + 20.
a(5) = 48 = A055244(6) + A001629(4) = 43 + 5.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A065941, A010049, A001629, A055244.

a(n) = A010049(n) + A001629(n+2) = A055244(n+1) + A001629(n-1).
From Philippe Deléham, Dec 28 2013: (Start)
G.f.: (1+x-x^2)/(1-x-x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), a(0)=1, a(1)=3, a(2)=6, a(3)=13.
a(n) = a(n-1) + a(n-2) + 2*Fibonacci(n). (End)

cp's OEIS Frontend

A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

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A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

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