A191797 -id:A191797 - cp's OEIS frontend

A122931 Row sums of triangular array A122930.

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

A268317 Irregular triangle read by rows: T(n,k) gives the columns sum in the table Fib(n+1) X Fib(n), where k = 1..Fib(n), and 1's are assigned to cells on the longest diagonal path.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3

Offset: 0

Kival Ngaokrajang, Feb 01 2016

Inspired by sun flower spirals which come in Fib(i) and Fib(i+1) numbers in opposite directions. The present case of the Fib(n+1) X Fib(n) table has the following properties:
(i) Columns sum create the present irregular triangle.
(ii) Rows sum create the irregular triangle A268318.
(iii) The row sum of each of these irregular triangles is conjectured to be A000071.
(iv) The first differences of the sequence of half of the voids (0's) are conjectured to give A191797.
See illustrations in the links.

			Irregular triangle begins:
1
2
2  2
2  3  2
2  3  2  3  2
2  3  2  3  3  2  3  2
2  3  2  3  3  2  3  2  3  3  2  3  2
...

Kival Ngaokrajang, Illustration of initial terms, Sun flower spirals

Cf. A000071, A191797, A268318.

A246173 Triangle read by rows: T(n,k) is the number of vertex pairs at distance k of the Fibonacci cube Gamma(n) (1<=k<=n).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 10, 11, 6, 1, 20, 28, 21, 8, 1, 38, 64, 62, 35, 10, 1, 71, 140, 164, 120, 53, 12, 1, 130, 293, 402, 360, 210, 75, 14, 1, 235, 596, 935, 984, 708, 340, 101, 16, 1, 420, 1183, 2086, 2517, 2142, 1280, 518, 131, 18, 1, 744, 2304, 4507, 6120, 5991, 4260, 2164, 752, 165, 20, 1

Offset: 1

Emeric Deutsch, Aug 18 2014

The Fibonacci cube Gamma(n) is obtained from the n-cube Q(n) by removing all the vertices that contain two consecutive 1s.
The entries in row n are the coefficients of the Hosoya polynomial of the Fibonacci cube Gamma(n).
T(n,1) = A001629(n+1) = number of edges in Gamma(n).
Sum of entries in row n = A191797(n+2).
Sum(k*T(n,k), k>=1) = A238419(n) = the Wiener index of Gamma(n).

			Row 2 is 2,1. Indeed, Gamma(2) is the path-tree P(3) having vertex-pair distances 1,1, and 2.
Triangle starts:
1;
2,1;
5,4,1;
10,11,6,1;
20,28,21,8,1;

S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavzar, M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph

Cf. A001629, A191797, A238419, A246174

g := t*z/((1-z-z^2-t*z-t*z^2+t*z^3)*(1-z-z^2)): gserz := simplify(series(g, z = 0, 20)): for j to 18 do H[j] := sort(coeff(gserz, z, j)) end do: for j to 13 do seq(coeff(H[j], t, k), k = 1 .. j) end do; # yields sequence in triangular form

Rest /@ Rest[CoefficientList[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}, {t, 0, 5}], z], t]] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)
DeleteCases[CoefficientList[Series[t z/((1 - z - z^2 - t z - t z^2 + t z^3) (1 - z - z^2)), {z, 0, 10}], {z, t}], 0, {2}] // Flatten (* Eric W. Weisstein, Dec 11 2017 *)

G.f.: tz/((1-z-z^2-tz-tz^2+tz^3)(1-z-z^2)). Derived from Theorem 4.1 of the Klavzar-Mollard reference in which the g.f. of the ordered Hosoya polynomials is given.

A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Maghraoui Abdelkader, Aug 22 2015

Subsequence of A007318.

			1,
1,  1,
1,  1,
1,  2,  1,
1,  3,  3,   1,
1,  5, 10,  10,   5,    1,
1,  8, 28,  56,  70,   56,   28,    8,    1,
1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1

Jean-François Alcover, Table of n, a(n) for n = 0..388 (corrected by _N. J. A. Sloane_, Jan 18 2019)

Cf. A000045, A007318.

Cf. A036355, A037027, A228074, A162741, A034871, A034870, A191797.

Table[Binomial[Fibonacci[n], k], {n, 0, 8}, {k, 0, Fibonacci[n]}]//Flatten (* Jean-François Alcover, Nov 12 2015*)

v = vector(101,j,fibonacci(j)); i=0; n=0; while(n<100, for(k=0, n, print1(binomial(n, k), ", ","")); print(); i=i+1; n=v[i] ;)

T(n, k) = binomial(fibonacci(n), k).
T(n, 1) = fibonacci(n) = A000045(n).
T(n, 2) = A191797(n) for n>3.

A268318 Irregular triangle read by rows: T(n,k) gives the row sums in the table Fib(n+1) X Fib(n), where k = 1..Fib(n+1), and 1's are assigned to cells on the longest diagonal path.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2

Offset: 0

Kival Ngaokrajang, Feb 01 2016

Inspired by sun flower spirals which come in  Fib(i) and Fib(i+1) numbers in opposite directions. The present Fib(n+1) X Fib(n) table has the following properties:
(i) Columns sum create the irregular triangle A268317.
(ii) Rows sum create the present irregular triangle.
(iii) The row sums of each of these irregular triangles is conjectured to be A000071.
(iv) The first differences of the sequence of half of the  voids (0's) are conjectured to give A191797.
See illustrations in the links of A268317.

			Irregular triangle begins:
0
1
1 1
1 2 1
1 2 1 2 1
1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1
...

Cf. A000071, A191797, A268317.

A192017 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Fibonacci tree of order n (1<=k<=n; entries in row n are the coefficients of the corresponding Wiener polynomial).

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 7, 10, 9, 2, 12, 21, 27, 15, 3, 20, 40, 65, 57, 25, 3, 33, 72, 138, 163, 114, 37, 4, 54, 125, 270, 394, 378, 206, 54, 4, 88, 212, 500, 854, 1033, 796, 354, 74, 5, 143, 354, 891, 1716, 2479, 2463, 1571, 574, 100, 5, 232, 585, 1545, 3265, 5424, 6559, 5469, 2917, 896, 130, 6

Offset: 1

Emeric Deutsch, Jun 21 2011

The Fibonacci trees f(k) of order k are defined as follows: 1. f(-1) and f(0) each consist of a single node. 2. For k>=1, to the root of f(k-1), taken as the root of f(k), we attach with a rightmost edge the tree f(k-2). See the Iyer & Reddy references. These trees are not the same as the Fibonacci trees in A180566.

Sum of entries in row n is A191797(n+2).

			T(2,2)=1 because in the Fibonacci tree of order 2, namely /\, there is only 1 pair of nodes at distance 2 (the two leaves).
Triangle starts:
   1;
   2,  1;
   4,  4,  2;
   7, 10,  9,  2;
  12, 21, 27, 15,  3;
  20, 40, 65, 57, 25,  3;

K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.

Cf. A000071, A011973, A165910, A191797.

G := (1+t*z)/(1-z-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do r[n] := sort(coeff(Gser, z, n)) end do; w[0] := 0; w[1] := t; for n from 2 to 11 do w[n] := sort(expand(w[n-1]+w[n-2]+t*r[n-1]*r[n-2])) end do: for n from 1 to 11 do seq(coeff(w[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form

m = 11; Gser = Series[(1 + t*z)/(1 - z - t*z^2), {z, 0, m}]; Do[r[n] = Coefficient[Gser, z, n], {n, 0, m}]; w[0] = 0; w[1] = t; Do[w[n] = Expand[w[n - 1] + w[n - 2] + t*r[n - 1]*r[n - 2]] , {n, 2, m}]; Flatten[Table[Coefficient[w[n], t, j], {n, 1, m}, {j, 1, n}]] (* Jean-François Alcover, Sep 02 2011, after Maple *)

T(n,1) = A000071(n-2) (Fibonacci numbers minus 1).
Sum_{k=1..n} k*T(n,k) = A165910(n) (the Wiener indices).
The Wiener polynomial w(n,t) of the Fibonacci tree of order n satisfies the recurrence relation w(n,t) = w(n-1,t) + w(n-2,t) + t*r(n-1,t)*r(n-2,t), w(0,t)=w(1,t)=0, where r(n,t) is the generating polynomial of the nodes of the Fibonacci tree of order n with respect to the level of the nodes (for example, r(2,t) = 1 + 2t for the tree /\; see A011973 and the Maple program).

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A122931 Row sums of triangular array A122930.

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A268317 Irregular triangle read by rows: T(n,k) gives the columns sum in the table Fib(n+1) X Fib(n), where k = 1..Fib(n), and 1's are assigned to cells on the longest diagonal path.

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A246173 Triangle read by rows: T(n,k) is the number of vertex pairs at distance k of the Fibonacci cube Gamma(n) (1<=k<=n).

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A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

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A268318 Irregular triangle read by rows: T(n,k) gives the row sums in the table Fib(n+1) X Fib(n), where k = 1..Fib(n+1), and 1's are assigned to cells on the longest diagonal path.

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A192017 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Fibonacci tree of order n (1<=k<=n; entries in row n are the coefficients of the corresponding Wiener polynomial).

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