A067330 -id:A067330 - cp's OEIS frontend

A067418 Triangle A067330 with rows read backwards.

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

A067988 Row sums of triangle A067330; also of triangle A067418.

Original entry on oeis.org

1, 3, 10, 25, 60, 133, 284, 585, 1175, 2310, 4464, 8502, 15995, 29775, 54920, 100487, 182556, 329555, 591550, 1056405, 1877821, 3323868, 5860800, 10297500, 18033925, 31487643, 54824854, 95211205, 164948700, 285121105, 491804144, 846631137, 1454746355, 2495275650

Offset: 0

Wolfdieter Lang, Feb 15 2002

a(n) is the sum of the positions of the 0's in all Fibonacci binary words of length n+1. A Fibonacci binary word is a binary word having no 00 subword. Example: a(3)=25 because the Fibonacci binary words of length 4 are 1110, 1111, 1101, 1010, 1011, 0110, 0111 and 0101, the positions of the 0's being 4, 3, 2, 4, 2, 1, 4, 1, 1 and 3 (their sum is 25). - Emeric Deutsch, Jan 04 2009

Cf. A000045, A001628, A067330, A067418.

a:=n->sum(binomial(n-j,j)*n*j/2,j=0..n): seq(a(n), n=2..30); # Zerinvary Lajos, Oct 19 2006

Table[((n+2)((3n+5)Fibonacci[n+1]+(n+1)Fibonacci[n]))/10,{n,0,30}] (* Harvey P. Dale, Feb 02 2020 *)

a(n) = (n+2)*((3*n+5)*F(n+1)+(n+1)*F(n))/10, with F(n) := A000045(n) (Fibonacci).
G.f.: (1+x^2)/(1-x-x^2)^3.
Sum_{j=0..n} binomial(n-j,j)*n*j/2. - Zerinvary Lajos, Oct 19 2006
E.g.f.: exp(x/2)*(5*(10 + 18*x + 7*x^2)*cosh(sqrt(5)*x/2) + sqrt(5)*(14 + 46*x + 15*x^2)*sinh(sqrt(5)*x/2))/50. - Stefano Spezia, Aug 30 2025

a(29)-a(33) from Stefano Spezia, Aug 30 2025

A067331 Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0.

Original entry on oeis.org

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, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675, 617363100

Offset: 0

Wolfdieter Lang, Feb 15 2002

Third diagonal of A067330. Third column of A067418.
From Emeric Deutsch, Jun 15 2010: (Start)
a(n) is the external path length of the Fibonacci tree of order n+3. A Fibonacci tree of order n (n >= 2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The external path length of a tree is the sum of the levels of its external nodes (i.e., leaves).
a(n) = Sum_{k>=0} k*A178524(n+2,k).
(End)
a(n) equals the penultimate immanant of the (n+3) X (n+3) tridiagonal matrix with ones along the main diagonal, the superdiagonal, and the subdiagonal. - John M. Campbell, Jan 01 2016
a(n) is the sum of the eccentricities of the vertices of the Fibonacci cube G(n+1). Example: a(1)=5; indeed, the Fibonacci cube G(2) is the path graph P(3), the vertices of which have eccentricities 2, 1, 2. - Emeric Deutsch, May 28 2017

			From _John M. Campbell_, Jan 03 2016: (Start)
Letting n=2, the external path length of the Fibonacci tree T(5) of order n+3=5 illustrated below is 12 = a(2) = F(1)*F(5) + F(2)*F(4) + F(3)*F(3).
     .
    / \
   /\ /\
  /\
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Robert Israel, Table of n, a(n) for n = 0..4720
Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the determinant Hosoya triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021.
J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, Electron. J. Combin. 21 (1) (2014), P1.7.
John M. Campbell, On the external path length of a Fibonacci tree.
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20(2) (1982), 168-178.
S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, HAL Id: hal-00836788, 2013.
S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, Ann. Comb. 18 (2014), 447-457.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Row sums of A108038.

Cf. A000045, A004070, A067330, A067418, A178523, A178524.

[((7*n+10)*Fibonacci(n+1)+4*(n+1)*Fibonacci(n))/5: n in [0..40]]; // Vincenzo Librandi, Jan 02 2016

f:= gfun:-rectoproc({a(n) = 2*a(n-1)+a(n-2) - 2*a(n-3)-a(n-4),a(0)=2,a(1)=5,a(2)=12,a(3)=25},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 06 2016

LinearRecurrence[{2, 1, -2, -1}, {2, 5, 12, 25}, 70] (* Vincenzo Librandi, Jan 02 2016 *)
Table[SeriesCoefficient[(2 + x)/(1 - x - x^2)^2, {x, 0, n}], {n, 0, 34}] (* Michael De Vlieger, Jan 02 2016 *)
Print[Table[Sum[Binomial[n + 3 - i, i]*(n + 2 - 2*i), {i, 0, Floor[(n + 3)/2]}], {n, 0, 100}]] (* John M. Campbell, Jan 04 2016 *)
Module[{nn=40,fibs},fibs=Fibonacci[Range[nn]];Table[ListConvolve[Take[ fibs,n],Take[fibs,{2,n+2}]],{n,nn-2}]][[All,2]] (* Harvey P. Dale, Aug 03 2019 *)

Vec((2+x)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 04 2016

a(n) = A067330(n+2, n) = A067418(n+2, 2) = Sum_{k=0..n} F(k+1)*F(n+3-k), n >= 0.
a(n) = ((7*n + 10)*F(n + 1) + 4*(n + 1)*F(n))/5, with F(n) = A000045(n) (Fibonacci).
G.f.: (2 + x)/(1 - x - x^2)^2.
a(n) = Sum_{i=0..floor((n+3)/2)} binomial(n+3-i, i)*(n + 2 - 2*i). - John M. Campbell, Jan 04 2016
E.g.f.: exp(x/2)*((50 + 55*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(18 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

Original entry on oeis.org

1, 3, 6, 4, 13, 17, 7, 19, 31, 38, 11, 32, 48, 69, 80, 18, 51, 79, 107, 140, 158, 29, 83, 127, 176, 220, 274, 303, 47, 134, 206, 283, 360, 432, 519, 566, 76, 217, 333, 459, 580, 706, 822, 963, 1039, 123, 351, 539, 742, 940, 1138, 1341, 1529, 1757, 1880, 199, 568, 872, 1201, 1520, 1844, 2163, 2492, 2796, 3165, 3364

Offset: 0

Wolfdieter Lang, Feb 15 2002

The diagonals d>=0 (d=0: main diagonal) give convolutions of Lucas numbers L(n+1) := A000204(n+1), n>=0, with those with d-shifted index: a(d+n,d) = Sum_{k=0..n} L(k+1)*L(d+n+1-k).
The diagonals give A004799(n-1), A067980-7 for d=n-m= 0..8, respectively. Row sums give A067989.
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+2*x)/(1-x-x^2) (g.f. Lucas L(n+1), n>=0).

			Triangle begins:
  {1};
  {3,6};
  {4,13,17};     p(2,x) = 4+13*x+17*x^2
  {7,19,31,38};
  ...

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

Cf. A067990 (triangle with rows read backwards).

Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 10}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

for(n=0,10, for(k=0,n, print1(sum(m=0,k,(fibonacci(m+2) + fibonacci(m))*(fibonacci(n-m+2) + fibonacci(n-m))), ", "))) \\ G. C. Greubel, Dec 17 2017

a(n, m) = Sum_{k=0..m} L(k+1)*L(n-k+1), n>=m>=0, else 0.
a(n, m) = (m+1)*L(n-m+1)*F(m) + ((m+1)*L(n-m+1) + m*L(n-m))*F(m+1), n>=m>=0, with F(n) := A000045(n) (Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for diagonals d= n-m>=0: (x^d)*(L(d+1)+L(d)*x)*(1-2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m-1) + m*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1) * F(n), hence a(n, m) = -5*A067330(n, m)+2*(m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067990 Triangle A067979 with rows read backwards.

Original entry on oeis.org

1, 6, 3, 17, 13, 4, 38, 31, 19, 7, 80, 69, 48, 32, 11, 158, 140, 107, 79, 51, 18, 303, 274, 220, 176, 127, 83, 29, 566, 519, 432, 360, 283, 206, 134, 47, 1039, 963, 822, 706, 580, 459, 333, 217, 76, 1880, 1757, 1529, 1341, 1138, 940, 742, 539, 351, 123, 3364, 3165, 2796, 2492, 2163, 1844, 1520, 1201

Offset: 0

Wolfdieter Lang, Feb 15 2002

The column m (without leading 0's) gives the convolution of Lucas numbers {L(n+1) := A000204(n+1)}, n>=0, with those with m-shifted index: a(n+m,m)=sum(L(k+1)*L(m+n+1-k),k=0..n), n>=0,m=0,1,...
The columns give A004799(n-1), A067980-7 for m= 0..8, respectively. Row sums give A067989.
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+2*x)/(1-x-x^2) (g.f. for Lucas {L(n+1)}).

			{1}; {6,3}; {17,13,4}; {38,31,19,7}; ...; p(2,x)=17+13*x+4*x^2.

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

Reverse /@ Table[Sum[LucasL[k + 1] LucasL[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)=(n-m+1)*L(m+1)*F(n-m)+((n-m+1)*L(m+1)+(n-m)*L(m))*F(n-m+1), n>=m>=0, else 0; with F(n) := A000045(n)(Fibonacci) and L(n) := A000032(n) (Lucas).
G.f. for column m=0, 1, ...: (x^m)*(L(m+1)+L(m)*x)*(1+2*x)/(1-x-x^2)^2.
a(n, m) = -(-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = -5*A067418(n, m)+2*(n-m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067430 Convolution of Fibonacci F(n+1), n>=0, with F(n+7), n>=0.

Original entry on oeis.org

13, 34, 81, 170, 340, 654, 1227, 2258, 4095, 7340, 13032, 22956, 40169, 69890, 121005, 208606, 358268, 613242, 1046535, 1781170, 3024123, 5123104, 8661456, 14616600, 24624325, 41419234, 69568137, 116690258

Offset: 0

Wolfdieter Lang, Feb 15 2002

Seventh diagonal of A067330. Seventh column of A067418.

a(n)= A067330(n+6, n) = A067418(n+6, 6)= sum(F(k+1)*F(n+7-k), k=0..n), n>=0.
a(n)= ((47*n+65)*F(n+1)+29*(n+1)*F(n))/5, F(n) := A000045(n) (Fibonacci).
G.f.: (13+8*x)/(1-x-x^2)^2.

A067977 Convolution of Fibonacci F(n+1), n>=0, with F(n+9), n>=0.

Original entry on oeis.org

34, 89, 212, 445, 890, 1712, 3212, 5911, 10720, 19215, 34116, 60096, 105158, 182965, 316780, 546113, 937918, 1605424, 2739760, 4662995, 7916984, 13412019, 22675272, 38265600, 64465450, 108433937

Offset: 0

Wolfdieter Lang, Feb 15 2002

Ninth diagonal of A067330. Ninth column of A067418.

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

Cf. A000045, A000204, A067330, A067418.

CoefficientList[Series[(34 + 21*x)/(1 - x - x^2)^2, {x, 0, 30}], x] (* Wesley Ivan Hurt, Feb 16 2017 *)
LinearRecurrence[{2,1,-2,-1},{34,89,212,445},30] (* Harvey P. Dale, Dec 22 2022 *)

a(n) = A067330(n+8, n) = A067418(n+8, 8) = Sum_{k=0..n} F(k+1)*F(n+9-k), n>=0.
G.f.: (34+21*x)/(1-x-x^2)^2.
a(n) = ((123*n+5*34)*F(n+1)+76*(n+1)*F(n))/5, F(n) := A000045(n) (Fibonacci); 34=F(9), 76=L(9), 123=L(10), L(n) := A000204(n) (Lucas).

A067332 Convolution of Fibonacci F(n+1), n>=0, with F(n+4), n>=0.

Original entry on oeis.org

3, 8, 19, 40, 80, 154, 289, 532, 965, 1730, 3072, 5412, 9471, 16480, 28535, 49196, 84496, 144638, 246845, 420140, 713353, 1208518, 2043264, 3448200, 5809275, 9771704, 16413019, 27530992, 46122320

Offset: 0

Wolfdieter Lang, Feb 15 2002

Fourth diagonal of A067330. Fourth column of A067418.

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

Table[((11n+15)Fibonacci[n+1]+7(n+1)Fibonacci[n])/5,{n,0,30}] (* or *) LinearRecurrence[{2,1,-2,-1},{3,8,19,40},30] (* Harvey P. Dale, Aug 25 2014 *)

a(n)= A067330(n+3, n) = A067418(n+3, 3) = sum(F(k+1)*F(n+4-k), k=0..n), n>=0.
a(n)= ((11*n+15)*F(n+1)+7*(n+1)*F(n))/5, with F(n) := A000045(n) (Fibonacci).
G.f.: (3+2*x)/(1-x-x^2)^2.
a(0)=3, a(1)=8, a(2)=19, a(3)=40, a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Harvey P. Dale, Aug 25 2014

A067333 Convolution of Fibonacci F(n+1), n>=0, with F(n+5), n>=0.

Original entry on oeis.org

5, 13, 31, 65, 130, 250, 469, 863, 1565, 2805, 4980, 8772, 15349, 26705, 46235, 79705, 136886, 234302, 399845, 680515, 1155385, 1957293, 3309096, 5584200, 9407525, 15823765, 26577559, 44579633, 74681770

Offset: 0

Wolfdieter Lang, Feb 15 2002

Fifth diagonal of A067330. Fifth column of A067418.

a(n)= A067330(n+4, n) = A067418(n+4, 4) = sum(F(k+1)*F(n+5-k), k=0..n), n>=0.
a(n)= ((18*n+25)*F(n+1)+11*(n+1)*F(n))/5, with F(n) := A000045(n) (Fibonacci).
G.f.: (5+3*x)/(1-x-x^2)^2.

A067334 Convolution of Fibonacci F(n+1), n>=0, with F(n+6), n>=0.

Original entry on oeis.org

8, 21, 50, 105, 210, 404, 758, 1395, 2530, 4535, 8052, 14184, 24820, 43185, 74770, 128901, 221382, 378940, 646690, 1100655, 1868738, 3165811, 5352360, 9032400, 15216800, 25595469, 42990578, 72110625, 120804090, 202142180, 337876622, 564176619, 941141410

Offset: 0

Wolfdieter Lang, Feb 15 2002

Sixth diagonal of A067330. Sixth column of A067418.

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

Cf. A067330, A067418.

CoefficientList[Series[(8+5x)/(1-x-x^2)^2,{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{8,21,50,105},40] (* Harvey P. Dale, Apr 07 2012 *)

a(n) = A067330(n+5, n) = A067418(n+5, 5) = Sum_{k=0..n} F(k+1)*F(n+6-k), n>=0.
a(n) = ((29*n+40)*F(n+1)+18*(n+1)*F(n))/5, with F(n) := A000045(n) (Fibonacci).
G.f.: (8+5*x)/(1-x-x^2)^2.
a(0)=8, a(1)=21, a(2)=50, a(3)=105, a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Harvey P. Dale, Apr 07 2012

More terms from Jason Yuen, Aug 27 2025

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A067418 Triangle A067330 with rows read backwards.

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A067988 Row sums of triangle A067330; also of triangle A067418.

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A067331 Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0.

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A067979 Triangle read by rows of incomplete convolutions of Lucas numbers L(n+1) = A000204(n+1), n>=0.

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A067990 Triangle A067979 with rows read backwards.

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A067430 Convolution of Fibonacci F(n+1), n>=0, with F(n+7), n>=0.

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A067977 Convolution of Fibonacci F(n+1), n>=0, with F(n+9), n>=0.

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A067332 Convolution of Fibonacci F(n+1), n>=0, with F(n+4), n>=0.

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