A064831 -id:A064831 - cp's OEIS frontend

A180666 Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.

0, 1, 2, 6, 15, 41, 106, 279, 729, 1911, 5001, 13095, 34281, 89752, 234971, 615165, 1610520, 4216400, 11038675, 28899630, 75660210, 198081006, 518582802, 1357667406, 3554419410, 9305590831, 24362353076, 63781468404

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n) are the Gi2 sums of the Golden Triangle A180662. See A180662 for information about these giraffe and other chess sums.

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

Cf. A064831, A180664, A180665, A115730, A180666.

nmax:=27: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n)*fibonacci(n+1) od: a(0):=0: a(1):=1: a(2):=2: a(3):=6: for n from 4 to nmax do a(n):=a(n-4)+A001654(n) od: seq(a(n),n=0..nmax);
A180666 := proc(n)
    option remember;
    if n <=3 then
        op(n+1,[0,1,2,6]) ;
    else
        procname(n-4)+A001654(n) ;
    end if;
end proc:
seq(A180666(n),n=0..100 ) ; # R. J. Mathar, Aug 18 2016

Take[Total@{#, PadLeft[Drop[#, -4], Length@ #]}, Length@ # - 4] &@ Table[Times @@ Fibonacci@ {n, n + 1}, {n, 0, 31}] (* or *)
CoefficientList[Series[(-x)/((x^2 - 3 x + 1) (x - 1) (x + 1)^2 (x^2 + 1)), {x, 0, 27}], x] (* Michael De Vlieger, Aug 18 2016 *)

a(n) = a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.
G.f.: (-x)/((x^2-3*x+1)*(x-1)*(x+1)^2*(x^2+1)).
a(n) = Sum_{k=0..floor(n/4)} A180662(n-3*k,n-4*k).
120*a(n) = 8*A001519(n) -10*A087960(n) -9*(-1)^n -15 -6*(n+1)*(-1)^n. - R. J. Mathar, Aug 18 2016

A202503 Fibonacci self-fission matrix, by antidiagonals.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 9, 8, 8, 8, 13, 14, 15, 13, 13, 13, 21, 23, 24, 24, 21, 21, 21, 34, 37, 39, 39, 39, 34, 34, 34, 55, 60, 63, 64, 63, 63, 55, 55, 55, 89, 97, 102, 103, 104, 102, 102, 89, 89, 89, 144, 157, 165, 167, 168, 168, 165, 165, 144, 144, 144

Offset: 1

Clark Kimberling, Dec 20 2011

The Fibonacci self-fission matrix, F, is the fission P^^Q, where P and Q are the matrices given at A202502 and A202451.  See A193842 for the definition of fission.
antidiagonal sums:  (1, 3, 8, 18, 38, ...), A064831
diagonal (1,  5, 14,  39, ...), A119996
diagonal (2,  8, 23,  63, ...), A180664
diagonal (2,  5, 15,  39, ...), A059840
diagonal (3,  8, 24,  63, ...), A080097
diagonal (5, 13, 39, 102, ...), A080143
diagonal (8, 21, 63, 165, ...), A080144
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

			Northwest corner:
1....1....2....3....5.....8....13...21
2....3....5....8...13....21....34...55
3....5....9...14...23....37....60...97
5....8...15...24...39....63...102...165
8...13...24...39...64...103...167...270

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A000045, A202451, A202453, A202502.

n = 14;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];
P = P1[[Range[2, n], Range[1, n]]];
F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)
TableForm[P]  (* A202502, modified lower triangular Fibonacci array *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[F]  (* A202503, Fibonacci fission array *)

A096979 Sum of the areas of the first n+1 Pell triangles.

Original entry on oeis.org

0, 1, 6, 36, 210, 1225, 7140, 41616, 242556, 1413721, 8239770, 48024900, 279909630, 1631432881, 9508687656, 55420693056, 323015470680, 1882672131025, 10973017315470, 63955431761796, 372759573255306, 2172602007770041

Offset: 0

Paul Barry, Jul 17 2004

Convolution of A059841(n) and A001109(n+1).

Partial sums of A084158.

S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
Index entries for linear recurrences with constant coefficients, signature (6,0,-6,1).

Cf. A096977, A064831, A096978.

Accumulate[LinearRecurrence[{5,5,-1},{0,1,5},30]] (* Harvey P. Dale, Sep 07 2011 *)
LinearRecurrence[{6, 0, -6, 1},{0, 1, 6, 36},22] (* Ray Chandler, Aug 03 2015 *)

G.f.: x/((1-x)*(1+x)*(1-6*x+x^2)).
a(n) = 6*a(n-1)-6*a(n-3)+a(n-4).
a(n) = (3-2*sqrt(2))^n*(3/32-sqrt(2)/16)+(3+2*sqrt(2))^n*(sqrt(2)/16+3/32)-(-1)^n/16-1/8.
a(n) = Sum_{k=0..n} (sqrt(2)*(sqrt(2)+1)^(2*k)/8-sqrt(2)*(sqrt(2)-1)^(2*k)/8)*(1+(-1)^(n-k))/2.
a(n) = Sum_{k=0..n} A000129(k)*A000129(k+1)/2. [corrected by Jason Yuen, Jan 14 2025]
a(n) = (A001333(n+1)^2 - 1)/8 = ((A000129(n) + A000129(n+1))^2 - 1)/8. - Richard R. Forberg, Aug 25 2013
a(n) = A002620(A000129(n+1)) = A000217(A048739(n-1)), n > 0. - Ivan N. Ianakiev, Jun 21 2014

A282464 a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.

Original entry on oeis.org

0, 1, 3, 15, 51, 176, 560, 1743, 5271, 15675, 45925, 133056, 381888, 1087645, 3077451, 8658951, 24245655, 67602608, 187789616, 519924075, 1435228575, 3951341811, 10852291273, 29740435200, 81340229376, 222058995001, 605201766675, 1646862596223, 4474969884411

Offset: 0

Bruno Berselli, Feb 16 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-10,10,4,-5,1).

Cf. A000045.
Partial sums of A169630.
Cf. A014286: partial sums of i*Fibonacci(i).
Cf. A064831: partial sums of (n+1-i)*Fibonacci(i)^2.

[&+[i*Fibonacci(i)^2: i in [0..n]]: n in [0..30]];

with(combinat): P:=proc(q) local a,n; a:=0; print(a); for n from 1 to q do
a:=a+n*fibonacci(n)^2; print(a); od; end: P(100); # Paolo P. Lava, Feb 17 2017

a[n_] := Sum[i*Fibonacci[i]^2, {i, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)
LinearRecurrence[{5,-4,-10,10,4,-5,1},{0,1,3,15,51,176,560},30] (* Harvey P. Dale, May 15 2021 *)

makelist(sum(i*fib(i)^2, i, 0, n), n, 0, 30);

a(n) = sum(i=0, n, i*fibonacci(i)^2) \\ Colin Barker, Feb 16 2017

[sum(i*fibonacci(i)^2 for i in [0..n]) for n in range(30)]

O.g.f.: x*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)/((1 - x)*(1 + x)^2*(1 - 3*x + x^2)^2).
a(n) = 5*a(n-1) - 4*a(n-2) - 10*a(n-3) + 10*a(n-4) + 4*a(n-5) - 5*a(n-6) + a(n-7).
a(n) = ((n-1)*Fibonacci(n) + n*Fibonacci(n-1))*Fibonacci(n) + (1 - (-1)^n)/2.

A080145 a(n) = Sum_{m=1..n} Sum_{i=1..m} F(i)*F(i+1) where F(n)=Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 4, 13, 37, 101, 269, 710, 1865, 4890, 12810, 33546, 87834, 229963, 602062, 1576231, 4126639, 10803695, 28284455, 74049680, 193864595, 507544116, 1328767764, 3478759188, 9107509812, 23843770261, 62423800984, 163427632705

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 31 2003

This is the 2-fold convolution of A001654 with the sequence 1,1,1,....
Equivalently, partial sums of A064831 which is the partial sums of A001654. - Joerg Arndt, Oct 01 2021
a(n) is the number of permutations p in Sn(321) such that p^(-1) has exactly one left peak. See Troyka and Zhuang. - Michel Marcus, Oct 01 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.
Justin M. Troyka and Yan Zhuang, Fibonacci numbers, consecutive patterns, and inverse peaks, arXiv:2109.14774 [math.CO], 2021.
Yan Zhuang, A lifting of the Goulden-Jackson cluster method to the Malvenuto-Reutenauer algebra, arXiv:2108.10309 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-3,4,-1).

Cf. A000032, A000045, A001654, A064831.

F:=Fibonacci;; List([0..30], n-> (4*F(n+1)*F(n+2)-2*n-3-(-1)^n)/4); # G. C. Greubel, Jul 23 2019

[(4*Lucas(2*n+3)+(-1)^(n+1)-10*n-15)/20: n in [0..30]]; // Vincenzo Librandi, Aug 22 2017

CoefficientList[Series[x/((1-2x-2x^2+x^3)(1-x)^2), {x, 0, 30}], x] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
With[{F=Fibonacci}, Table[(4*F[n+1]*F[n+2]-2*n-3-(-1)^n)/4, {n,0,30}]] (* G. C. Greubel, Jul 23 2019 *)

L(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=(4*L(2*n+3)-(-1)^n-10*n-15)/20 \\ Charles R Greathouse IV, Aug 26 2017

f=fibonacci; [(4*f(n+1)*f(n+2)-2*n-3-(-1)^n)/4 for n in (0..30)] # G. C. Greubel, Jul 23 2019

a(n) = F(n+1)*F(n+2) - floor((n+2)/2).
G.f.: x/((1 - 2*x - 2*x^2 + x^3)*(1-x)^2).
a(n) = (4*Lucas(2*n + 3) + (-1)^(n+1) - 10*n - 15)/20. - Ehren Metcalfe, Aug 21 2017
a(n) = (4*Fibonacci(n+1)*Fibonacci(n+2) - 2*n - 3 - (-1)^n)/4. - G. C. Greubel, Jul 23 2019
a(n) = Sum_{j=1..n} j*F(n+1-j)*F(n+2-j). - Michael A. Allen, Jan 07 2022

A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  (Fusion is defined at A193822; fission, at A193742; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A193917:
1
1...1
1...2...3
2...3...6...9
3...5...9...15...24
5...8...15..24...40...64
8...13..24..39...64...104..168
13..21..39..63...104..168..273..441
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
col 7:  A022355
col 8:  A022355
right edge, w(n,n):  A064831
w(n,n-1):  A001654
w(n,n-2):  A064831
w(n,n-3):  A059840
w(n,n-4):  A080097
w(n,n-5):  A080143
w(n,n-6):  A080144
Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

			First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Cf. A193722, A064831, A193918, A194000, A194001.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233

Offset: 0

Clark Kimberling, Aug 11 2011

See A193917 for the self-fusion of the same sequence of polynomials.  (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A194000:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
right edge, d(n,n):  A064831
d(n,n-1):  A059840
d(n,n-2):  A080097
d(n,n-3):  A080143
d(n,n-4):  A080144
...
Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

			First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).

Cf. A193842, A194001, A193917, A193918.

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

A230447 T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 5, 3, 3, 6, 9, 8, 6, 3, 9, 16, 17, 14, 9, 4, 12, 25, 33, 32, 23, 15, 4, 16, 38, 58, 65, 55, 39, 24, 5, 20, 54, 96, 124, 120, 94, 63, 40, 5, 25, 75, 150, 220, 244, 215, 157, 103, 64, 6, 30, 100, 225, 371, 464, 459, 372, 261, 167, 104

Offset: 0

Johannes W. Meijer, Oct 19 2013

The terms in the right hand columns of triangle T(n, k) and the terms in the rows of the square array Tsq(n, k) represent the Kn1p sums of the ‘Races with Ties’ triangle A035317.
For the definitions of the Kn1p sums see A180662. This sequence is related to A230448.
The first few row sums are: 1, 2, 6, 14, 32, 68, 144, 299, 616, 1258, 2559, 5185, 10478, … .

			The first few rows of triangle T(n, k) n >= 0 and 0 <= k <= n.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1
1|  1,  1
2|  2,  2,  2
3|  2,  4,  5,   3
4|  3,  6,  9,   8,   6
5|  3,  9, 16,  17,  14,    9
6|  4, 12, 25,  33,  32,   23,    15
7|  4, 16, 38,  58,  65,   55,    39,   24
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1,  1,  2,   3,   6,    9,   15,   24
1|  1,  2,  5,   8,  14,   23,   39,   63
2|  2,  4,  9,  17,  32,   55,   94,  157
3|  2,  6, 16,  33,  65,  120,  215,  372
4|  3,  9, 25,  58, 124,  244,  459,  831
5|  3, 12, 38,  96, 220,  464,  924, 1755
6|  4, 16, 54, 150, 371,  835, 1759, 3514
7|  4, 20, 75, 225, 596, 1431, 3191, 6705

Cf. (Triangle columns) A008619, A002620, A175287, A080239

Cf. A035317, A230448, A230449, A230135, A080239, A034851, A228570

T := proc(n, k): add(A035317(n-i, n-k+i), i=0..floor(k/2)) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
T := proc(n, k) option remember: if k=0 then return(A008619(n)) elif k=n then return(A080239(n+1)) else A230135(n, k) + procname(n-1, k) + procname(n-1, k-1) fi: end: A008619 := n -> floor(n/2) +1: A080239 := n -> add(combinat[fibonacci](n-4*k), k=0..floor((n-1)/4)): A230135 := proc(n, k): if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) then return(1) else return(0) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.
T(n, k) = sum(A035317(n-i, n-k+i), i = 0..floor(k/2)), n >= 0 and 0 <= k <= n.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(A035317(n+k-i, n+i), i=0..floor(k/2)), n >= 0 and k >= 0.
Tsq(n, k) = A080239(2*n+k+1) - sum(A035317(2*n+k-i, i), i=0..n-1).
The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
G.f.: a(n)/(4*(x-1)) + 1/(4*(x+1)) + (-1)^n*(x+2)/(10*(x^2+1)) - (A000032(2*n+3) + A000032(2*n+2)*x)/(5*(x^2+x-1)) + sum((-1)^(k+1) * A064831(n-k+1)/((x-1)^k), k= 2..n), n >= 0, with a(n) = A064831(n+1) + 2*A064831(n) - 2*A064831(n-1) + A064831(n-2).

A096978 Sum of the areas of the first n Jacobsthal rectangles.

Original entry on oeis.org

0, 1, 4, 19, 74, 305, 1208, 4863, 19398, 77709, 310612, 1242907, 4970722, 19884713, 79535216, 318148151, 1272578046, 5090341317, 20361307020, 81445344595, 325781145370, 1303125047521, 5212499258024, 20849998896239, 83399991856694

Offset: 0

Paul Barry, Jul 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,3,-14,8).

Cf. A096977, A064831.

[8*4^n/27-2*(-2)^n/27-(n+2)/9: n in [0..30]]; // Vincenzo Librandi, May 31 2011

LinearRecurrence[{4,3,-14,8},{0,1,4,19},30] (* or *) Table[(2^(2n+1)-3n - 3+(-2)^n)/27,{n,30}] (* Harvey P. Dale, Aug 08 2011 *)

G.f.: x/((1-x)^2*(1+2*x)*(1-4*x)).
a(n) = 8*4^n/27 - 2*(-2)^n/27 - (n+2)/9;
a(n) = Sum_{k=0..n} (2*4^k/3 + (-2)^k/3)*(n-k).
a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).
a(n) = Sum_{k=0..n} A001045(k)*A001045(k+1).
a(n-1) = Sum_{k=0..n} (-1)^(k+1)*A001045(k)*A001045(2*(n-k)). - Paul Barry, Aug 11 2009

A110033 A characteristic triangle for the Fibonacci numbers.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, 0, 3, -8, 1, 0, 0, 9, -21, 1, 0, 0, 0, 24, -55, 1, 0, 0, 0, 0, 64, -144, 1, 0, 0, 0, 0, 0, 168, -377, 1, 0, 0, 0, 0, 0, 0, 441, -987, 1, 0, 0, 0, 0, 0, 0, 0, 1155, -2584, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3025, -6765, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7920, -17711, 1

Offset: 0

Paul Barry, Jul 08 2005

			Rows begin
1;
-1,1;
1,-3,1;
0,3,-8,1;
0,0,9,-21,1;
0,0,0,24,-55,1;
0,0,0,0,64,-144,1;
0,0,0,0,0,168,-377,1;

Form the n X n Hankel matrices F(i+j-1), 1<=i, j<=n for the Fibonacci numbers and take the characteristic polynomials of these matrices. Triangle rows give coefficients of these characteristic polynomials. (Construction described by Michael Somos in A064831). Diagonal is (-1)^n*F(2n+2). Subdiagonal is A064831. Row sums are A110034. The unsigned matrix has row sums A110035.

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