A001925 -id:A001925 - cp's OEIS frontend

A002940 Arrays of dumbbells.

1, 4, 11, 26, 56, 114, 223, 424, 789, 1444, 2608, 4660, 8253, 14508, 25343, 44030, 76136, 131110, 224955, 384720, 656041, 1115784, 1893216, 3205416, 5416441, 9136084, 15384563, 25866914, 43429784, 72821274, 121953943, 204002680, 340886973, 569047468, 949022608

Offset: 1

N. J. A. Sloane

Whitney transform of n. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005
a(n-1) is the permanent of the n X n 0-1 matrix with 1 in (i,j) position iff (i=1 and j1). For example, with n=5, a(4) = per([[1, 1, 1, 1, 0], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1]]) = 26. - David Callan, Jun 07 2006
a(n) is the internal path length of the Fibonacci tree of order n+2. 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 internal path length of a tree is the sum of the levels of all of its internal (i.e. non-leaf) nodes. - Emeric Deutsch, Jun 15 2010
Partial Sums of A023610 - John Molokach, Jul 03 2013

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 23-24.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A002941, A002889, A055608, A062123, A062124, A062125, A062126, A062127, A046741.

Cf. A000045, A001925, A023610, A054454, A006478, A067331, A178523.

a002940 n = a002940_list !! (n-1)
a002940_list = 1 : 4 : 11 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)
   (drop 5 a000045_list)
-- Reinhard Zumkeller, Jan 18 2014

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019

a[n_]:= a[n]= If[n<3, n^2, 2a[n-1] -a[n-3] +Fibonacci[n+1]]; Array[a, 32] (* Jean-François Alcover, Jul 31 2018 *)

my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019

((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

a(n) = 2*a(n-1) - a(n-3) + A000045(n+1).
G.f.: x*(1+x)/((1-x)*(1-x-x^2)^2).
a(n) = Sum_{k=0..n} ( Sum_{i=0..n} k*C(k, i-k) ). - Paul Barry, Feb 16 2005
E.g.f.: 2*exp(x) + exp(x/2)*((55*x - 50)*cosh(sqrt(5)*x/2) + sqrt(5)*(25*x - 22)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 03 2023

More terms from Henry Bottomley, Jun 02 2000

A202970 Symmetric matrix based on A001911, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 10, 6, 11, 21, 21, 11, 19, 39, 46, 39, 19, 32, 68, 87, 87, 68, 32, 53, 115, 153, 167, 153, 115, 53, 87, 191, 260, 296, 296, 260, 191, 87, 142, 314, 433, 505, 528, 505, 433, 314, 142, 231, 513, 713, 843, 904, 904, 843, 713, 513, 231, 375, 835

Offset: 1

Clark Kimberling, Dec 27 2011

Let s=A001911 (F(n+3)-2, where F(n)=A000045(n), the Fibonacci numbers), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202970 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202971 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...3...6....11...19
3...10..21...39...68
6...21..46...87...153
11..39..87...167..296
19..68..153..296..528

Cf. A202971, A202453, A202876.

s[k_] := -2 + Fibonacci[k + 3];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001891 *)
Table[m[1, j], {j, 1, 12}]     (* A001911 *)
Table[m[j, j], {j, 1, 12}]
Table[m[j, j + 1], {j, 1, 12}]
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A001925 *)

A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

Original entry on oeis.org

1, 1, 3, 1, 6, 7, 1, 9, 22, 14, 1, 12, 46, 64, 26, 1, 15, 79, 177, 162, 46, 1, 18, 121, 380, 571, 374, 79, 1, 21, 172, 700, 1496, 1632, 809, 133, 1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221, 1, 27, 301, 1799, 6271, 13013, 15754, 10446, 3316, 364

Offset: 0

N. J. A. Sloane, Jun 28 2015

See Riordan 1954 page 18 equation (9). - Michael Somos, Aug 26 2015

			Triangle T(n,k) begins:
1;
1,  3;
1,  6,  7;
1,  9,  22,   14;
1, 12,  46,   64,   26;
1, 15,  79,  177,  162,   46;
1, 18, 121,  380,  571,  374,   79;
1, 21, 172,  700, 1496, 1632,  809,  133;
1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221;
G.f. = 1 + (1 + 3*t)*u + (1 + 6*t + 7*t^2)*u^2 + ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See triangle on page 18)

Some diagonals: A001924, A001925, A001926.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      T(n-1, k) +2*T(n-1, k-1) +T(n-2, k-1)
     -T(n-3, k-3) +`if`(n=k, 1, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 02 2015

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k - 1] - T[n-3, k-3] + Boole[n == k]; T[, ] = 0; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2016 *)

{T(n, k) = polcoeff( polcoeff( 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)) + x * O(x^n), n), k)}; /* Michael Somos, Aug 26 2015 */

From Eq. (11) of Riordan (1954): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-3) + delta(n,k), where delta(n,k)=1 iff n=k, otherwise 0.

Sum_{n, k} T(n, k) * x^n*y^k = 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)). - Michael Somos, Aug 26 2015

More terms from Alois P. Heinz, Jul 02 2015

cp's OEIS Frontend

A002940 Arrays of dumbbells.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Sage

Formula

Extensions

A202970 Symmetric matrix based on A001911, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions