A000004 -id:A000004 - cp's OEIS frontend

A101386 Expansion of g.f.: (5 - 3x)/(1 - 6x + x^2).

5, 27, 157, 915, 5333, 31083, 181165, 1055907, 6154277, 35869755, 209064253, 1218515763, 7102030325, 41393666187, 241259966797, 1406166134595, 8195736840773, 47768254910043, 278413792619485, 1622714500806867, 9457873212221717, 55124524772523435, 321289275422918893

Offset: 0

Creighton Dement, Jan 23 2005

A floretion-generated sequence relating to NSW numbers and numbers n such that (n^2 - 8)/2 is a square. It is also possible to label this sequence as the "tesfor-transform of the zero-sequence" under the floretion given in the program code, below. This is because the sequence "vesseq" would normally have been A046184 (indices of octagonal numbers which are also a square) using the floretion given. This floretion, however, was purposely "altered" in such a way that the sequence "vesseq" would turn into A000004. As (a(n)) would not have occurred under "natural" circumstances, one could speak of it as the transform of A000004.
Floretion Algebra Multiplication Program FAMP code: - tesforseq[ + 3'i - 2'j + 'k + 3i' - 2j' + k' - 4'ii' - 3'jj' + 4'kk' - 'ij' - 'ji' + 3'jk' + 3'kj' + 4e], Note: vesforseq = A000004, lesforseq = A002315, jesforseq = A077445
From Wolfdieter Lang, Feb 05 2015: (Start)
All positive solutions x = a(n) of the (generalized) Pell equation x^2 - 2*y^2 = +7 based on the  fundamental solution (x2,y2) = (5,3) of the second class of (proper) solutions. The corresponding y solutions are given by y(n) = A253811(n).
All other positive solutions come from the first class of (proper) solutions based on the fundamental solution (x1,y1) = (3,1). These are given in A038762 and A038761.
All solutions of this Pell equation are found in A077443(n+1) and A077442(n), for n >= 0. See, e.g., the Nagell reference on how to find all solutions.
(End)

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 55, 56.
M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
Tanya Khovanova, Recursive Sequences
Morris Newman, Daniel Shanks, and H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981) 129-140. MR82b:20022.
Eric Weisstein's World of Mathematics, NSW Number.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000004, A000129, A001109, A002315, A038761, A038762, A046184, A054490, A164737.

Cf. A077442, A077443(n+1), A077445, A253811.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!((5 - 3*x)/(1-6*x+x^2))); // G. C. Greubel, Jul 26 2018

A101386:= (n) -> simplify(5*ChebyshevU(n, 3) - 3*ChebyshevU(n-1, 3)); seq( A101386(n), n = 0..30); # G. C. Greubel, Mar 17 2020

CoefficientList[ Series[(5-3x)/(1-6x+x^2), {x,0,30}], x] (* Robert G. Wilson v, Jan 29 2005 *)
LinearRecurrence[{6,-1},{5,27},30] (* Harvey P. Dale, Apr 23 2016 *)

Vec((5-3*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Feb 05 2015

[5*chebyshev_U(n,3) -3*chebyshev_U(n-1,3) for n in (0..30)] # G. C. Greubel, Mar 17 2020

a(n) = A002315(n) + A077445(n+1). Note: the offset of A077445 is 1.
a(n+1) - a(n) = 2*A054490(n+1).
a(n) = 6*a(n-1) - a(n-2), a(0)=5, a(1)=27. - Philippe Deléham, Nov 17 2008
From Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009: (Start)
a(n) = ((5+sqrt(18))*(3 + sqrt(8))^n + (5-sqrt(18))*(3 - sqrt(8))^n)/2.
Third binomial transform of A164737. (End)
a(n) = rational part of z(n), with z(n) = (5+3*sqrt(2))*(3+2*sqrt(2))^n, n >= 0, the general positive solutions of the second class of proper solutions. See the preceding formula. - Wolfdieter Lang, Feb 05 2015
a(n) = 5*A001109(n+1) - 3*A001109(n). - G. C. Greubel, Mar 17 2020
a(n) = Pell(2*n+2) + 3*Pell(2*n+1), where Pell(n) = A000129(n). - G. C. Greubel, Apr 17 2020
E.g.f.: exp(3*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Mar 16 2024

More terms from Robert G. Wilson v, Jan 29 2005

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 3, 4, 0, 0, 16, 4, 5, 0, 0, 50, 25, 40, 6, 0, 0, 120, 90, 288, 216, 7, 0, 0, 245, 245, 1176, 1764, 1603, 8, 0, 0, 448, 560, 3584, 8064, 14656, 13000, 9, 0, 0, 756, 1134, 9072, 27216, 74196, 131625, 118872, 10, 0, 0, 1200, 2100, 20160, 75600, 274800, 731250, 1320800, 1202880

Offset: 0

Luis Manuel Rivera Martínez, Dec 09 2013

For n >= 0, 0 <= k <= n, T(n, k) is the number of permutations of n symbols that k-commute with an n-cycle (we say that two permutations f and g k-commute if H(fg, gf) = k, where H(, ) denotes the Hamming distance between permutations).

Row sums give A000142.

			For n = 4 and k = 4, T(4, 4) = 4 because all the permutations of 4 symbols that 4-commute with permutation (1, 2, 3, 4) are (1, 3), (2, 4), (1, 2)(3, 4) and (1, 4)(2, 3).

Luis Manuel Rivera Martínez, Rows n = 0..30 of triangle, flattened
R. Moreno and L. M. Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306.5708 [math.CO], 2013-2014.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A007318, A000757.

T[n_, k_] := n Binomial[n, k] ((-1)^k+Sum[(-1)^j k!/(k-j)/j!, {j, 0, k-1}]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = n*C(n,k)*A000757(k), 0 <= k <= n.
Bivariate e.g.f.: G(z, u) = z*exp(z*(1-u))*(u/(1-z*u)+(1-log(1-z*u))*(1-u)).
T(n, 0)  = A001477(n), n>=0;
T(n, 1)  = A000004(n), n>=1;
T(n, 2)  = A000004(n), n>=2;
T(n, 3)  = A004320(n-2), n>=3;
T(n, 4)  = A027764(n-1), n>=4;
T(n, 5)  = A027765(n-1)*A000757(5), n>=5;
T(n, 6)  = A027766(n-1)*A000757(6), n>=6;
T(n, 7)  = A027767(n-1)*A000757(7), n>=7;
T(n, 8)  = A027768(n-1)*A000757(8), n>=8;
T(n, 9)  = A027769(n-1)*A000757(9), n>=9;
T(n, 10) = A027770(n-1)*A000757(10), n>=10;
T(n, 11) = A027771(n-1)*A000757(11), n>=11;
T(n, 12) = A027772(n-1)*A000757(12), n>=12;
T(n, 13) = A027773(n-1)*A000757(13), n>=13;
T(n, 14) = A027774(n-1)*A000757(14), n>=14;
T(n, 15) = A027775(n-1)*A000757(15), n>=15;
T(n, 16) = A027776(n-1)*A000757(16), n>=16. - Luis Manuel Rivera Martínez, Feb 08 2014
T(n, 0)+T(n, 3) = n*A050407(n+1), for n>=0. - Luis Manuel Rivera Martínez, Mar 06 2014

A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 4, 1, 0, 0, 3, 5, 2, 2, 0, 0, 3, 7, 6, 4, 2, 0, 0, 4, 9, 10, 11, 7, 1, 0, 0, 4, 11, 18, 21, 13, 7, 2, 0, 0, 4, 14, 26, 34, 31, 20, 7, 3, 0, 0, 4, 18, 37, 53, 59, 51, 32, 11, 2, 0, 0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0

Offset: 0

Alois P. Heinz, Feb 17 2017

			Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1,  1;
  0, 2,  2,  0;
  0, 2,  4,  1,  0;
  0, 3,  5,  2,  2,   0;
  0, 3,  7,  6,  4,   2,   0;
  0, 4,  9, 10, 11,   7,   1,  0;
  0, 4, 11, 18, 21,  13,   7,  2,  0;
  0, 4, 14, 26, 34,  31,  20,  7,  3,  0;
  0, 4, 18, 37, 53,  59,  51, 32, 11,  2, 0;
  0, 5, 21, 47, 82, 110, 117, 85, 35, 12, 2, 0;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000004, A000720, A071917, A320678, A320679, A320680, A320681, A320682, A320683, A320684, A320685.
Row sums give A127542.
Main diagonal gives A185012.
First lower diagonal gives A282518.
T(2n,n) gives A282517.

b:= proc(n, s) option remember; expand(`if`(n=0,
      `if`(isprime(s), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = Expand[If[n==0, If[PrimeQ[s], 1, 0], b[n-1, s] + x*b[n-1, s+n]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

A288387 Number T(n,k) of Dyck paths of semilength n such that the minimal number of peaks over all positive levels equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 8, 5, 0, 0, 1, 25, 13, 3, 0, 0, 1, 83, 35, 13, 0, 0, 0, 1, 282, 112, 30, 4, 0, 0, 0, 1, 971, 368, 61, 29, 0, 0, 0, 0, 1, 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1, 11940, 3992, 619, 188, 56, 0, 0, 0, 0, 0, 1, 42504, 13449, 2241, 345, 240, 6, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Jun 08 2017

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k<=n. T(n,k) = 0 if k>n.

T(0,0) = 1 by convention.

			. T(4,1) = 5:
.              /\      /\        /\/\    /\        /\/\
.         /\/\/  \  /\/  \/\  /\/    \  /  \/\/\  /    \/\ .
.
Triangle T(n,k) begins:
:    1;
:    0,    1;
:    1,    0,   1;
:    2,    2,   0,  1;
:    8,    5,   0,  0, 1;
:   25,   13,   3,  0, 0, 1;
:   83,   35,  13,  0, 0, 0, 1;
:  282,  112,  30,  4, 0, 0, 0, 1;
:  971,  368,  61, 29, 0, 0, 0, 0, 1;
: 3386, 1208, 172, 90, 5, 0, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Counting lattice paths

Columns k=0-10 give: A288539, A288540, A288541, A288542, A288543, A288544, A288545, A288546, A288547, A288548, A288549.
Row sums give A000108.
Main diagonal and first lower diagonal give: A000012, A000004.
Cf. A000007, A000027, A218152, A287822, A288386.

b:= proc(n, k, j) option remember; `if`(j=n, 1,
      add(add(binomial(i, m)*binomial(j-1, i-1-m),
      m=max(k, i-j)..i-1)*b(n-j, k, i), i=1..n-j))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n, k, j), j=k..n))
    end:
T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, k_, j_] := b[n, k, j] = If[j==n, 1, Sum[Sum[Binomial[i, m]*Binomial[ j-1, i-1-m], {m, Max[k, i - j], i - 1}]*b[n - j, k, i], {i, 1, n - j}]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[b[n, k, j], {j, k, n}]];
T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)

T(0,0) = 1, T(n,k) = A288386(n,k) - A288386(n,k+1).
T(2n,n-1) = A218152(n) for n>1.
T(2n,n) = A000007(n).
T(2n+1,n) = A000027(n+1) for n>0.

A318753 Number A(n,k) of rooted trees with n nodes such that no more than k subtrees extending from the same node have the same number of nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 3, 3, 0, 0, 1, 1, 2, 4, 7, 6, 0, 0, 1, 1, 2, 4, 8, 15, 12, 0, 0, 1, 1, 2, 4, 9, 18, 34, 25, 0, 0, 1, 1, 2, 4, 9, 19, 43, 79, 51, 0, 0, 1, 1, 2, 4, 9, 20, 46, 102, 190, 111, 0, 0, 1, 1, 2, 4, 9, 20, 47, 110, 250, 457, 240, 0

Offset: 0

Alois P. Heinz, Sep 02 2018

			Square array A(n,k) begins:
  0,  0,  0,   0,   0,   0,   0,   0,   0, ...
  1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  2,   2,   2,   2,   2,   2,   2, ...
  0,  2,  3,   4,   4,   4,   4,   4,   4, ...
  0,  3,  7,   8,   9,   9,   9,   9,   9, ...
  0,  6, 15,  18,  19,  20,  20,  20,  20, ...
  0, 12, 34,  43,  46,  47,  48,  48,  48, ...
  0, 25, 79, 102, 110, 113, 114, 115, 115, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-10 give: A063524, A032305, A213920, A318797, A318798, A318799, A318800, A318801, A318802, A318803, A318804.
Rows n=0-2 give: A000004, A000012, A057427.
Main diagonal gives A000081.
Cf. A318754.

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(A(i, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))
    end:
A:= (n, k)-> g(n-1$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];
A[n_, k_] := g[n - 1, n - 1, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k} A318754(n,j) for n > 0.

A(n,n+j) = A000081(n) for j >= -1.

A327085 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional simplex using up to k colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 21, 6, 0, 0, 10, 140, 405, 28, 0, 0, 20, 575, 7904, 17154, 252, 0, 0, 35, 1785, 76880, 1415648, 1920375, 4726, 0, 0, 56, 4606, 486522, 41453650, 855834880, 547375212, 150324, 0

Offset: 1

Robert A. Russell, Aug 19 2019

An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n-2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. The chiral colorings of its edges come in pairs, each the reflection of the other.

A(n,k) is also the number of chiral pairs of colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of chiral pairs of colorings of the vertices (0-dimensional simplices) of an equilateral triangle.

			Array begins with A(1,1):
  0 0   0    0     0      0       0       0        0        0         0 ...
  0 0   1    4    10     20      35      56       84      120       165 ...
  0 1  21  140   575   1785    4606   10416    21330    40425     71995 ...
  0 6 405 7904 76880 486522 2300305 8806336 28725192 82626270 214744629 ...
  ...
For A(2,3) = 1, the chiral pair is ABC-ACB.

Robert A. Russell, Table of n, a(n) for n = 1..325 First 25 antidiagonals.
Harald Fripertinger, The cycle type of the induced action on 2-subsets
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A327083 (oriented), A327084 (unoriented), A327086 (achiral), A327089 (exactly k colors), A325000(n,k-n) (vertices, facets), A337885 (faces, peaks), A337409 (orthotope edges, orthoplex ridges), A337413 (orthoplex edges, orthotope ridges).

Rows 1-4 are A000004, A000292(n-2), A337899, A331352.

CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]] == s[[i-1,1]], s[[i-1,2]] += s[[i,2]]; s = Delete[s,i], Null]]; s)
CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p, -1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p, -1]]]
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#,2]]], 1, -1] pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
array[n_, k_] := row[n] /. j -> k
Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
(* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
array[n_,k_] := Total[If[EvenQ[Total[1-Mod[#,2]]],1,-1] pc[#]k^ex[#] &/@ IntegerPartitions[n+1]]/(n+1)!
Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = Sum_{j=1..(n+1)*n/2} A327089(n,j) * binomial(k,j).
A(n,k) = A327083(n,k) - A327084(n,k) = (A327083(n,k) - A327086(n,k)) / 2 = A327084(n,k) - A327086(n,k).

A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2m-1,2m+1), where m = floor((n-2)/4).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 17, 28, 45, 72, 116, 188, 305, 494, 799, 1292, 2090, 3382, 5473, 8856, 14329, 23184, 37512, 60696, 98209, 158906, 257115, 416020, 673134, 1089154, 1762289, 2851444, 4613733, 7465176, 12078908, 19544084, 31622993, 51167078, 82790071

Offset: 2

Clark Kimberling

Essentially both the first difference sequence and partial sum of A005252, so its own shifted second difference and indeed virtually the same as A005252, so close to being its own shifted first difference.
From Paul Curtz, Jun 22 2011: (Start)
b(n) = 0,0,0,1,2,3,4,6, and differences are
   0,  0,  0,  1,  2,  3,  4,  6,
   0,  0,  1,  1,  1,  1,  2,  4,
   0,  1,  0,  0,  0,  1,  2,  3,
   1, -1,  0,  0,  1,  1,  1,  1,
  -2,  1,  0,  1,  0,  0,  0,  1,
   3, -1,  1  -1,  0,  0,  1,  1,
  -4,  2, -2,  1,  0,  1,  0,  0,
   6, -4,  3, -1,  1, -1,  0,  0;
b(n) is an autosequence (sequence identical to its inverse binomial transform signed) of first kind, i.e., its main diagonal is A000004.
Examples: A000045, A001045, A113405, A191754 (array). (End)
a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an odd number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 3; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, three of which have an odd number of ones. See the E. Munarini et al. reference, p. 323. - Emeric Deutsch, Jun 28 2015
a(n) is the number of odd permutations p of 1..n such that |p(i)-i| <= 1 for i=1..n. - Dmitry Efimov, Jan 08 2016

T. D. Noe, Table of n, a(n) for n = 2..502
Shishuo Fu and Dazhao Tang, Partitions with fixed largest hook length, arXiv:1604.04028 [math.CO], 2016.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 886
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
E. Munarini and N. Z. Salvi, Structural and enumerative properties of the Fibonacci cubes, Discrete Math., 255, 2002, 317-324.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A000004, A000045, A001045, A005252, A010892, A113405, A191754.

[n le 4 select n else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

LinearRecurrence[{2,-1,0,1},{1,2,3,4},39] (* Ray Chandler, Sep 23 2015 *)
CoefficientList[Series[1/((1-x-x^2)(1-x+x^2)), {x,0,40}], x] (* Vincenzo Librandi, Jan 09 2016 *)

Vec(x^2/((1-x-x^2)*(1-x+x^2)) + O(x^50)) \\ Michel Marcus, Feb 03 2016

def A024490(n): return (fibonacci(n+1) -chebyshev_U(n,1/2))/2
[A024490(n) for n in range(2,60)] # G. C. Greubel, Apr 10 2023

a(n) = A000045(n+1) - A005252(n).
a(n) = (A000045(n+1) - A010892(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
a(n) = Sum_{k=0..n} Fibonacci(k+1)*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3). - Paul Barry, May 18 2004
G.f.: x^2/((1-x-x^2)(1-x+x^2)). - Jon Perry, Jun 22 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1,k+1)*(1+(-1)^k)/2. - Paul Barry, Jul 05 2007
G.f.: (1 + Q(0)*x^4/2)/(1-x)^2, where Q(k) = 1 + 1/(1 - x*( 4*k + 2 - x + x^3)/( x*(4*k + 4 - x + x^3) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 07 2014
E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) - 5*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - Stefano Spezia, Aug 03 2022

Additional comments from Henry Bottomley, Apr 07 2000
Corrected by Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
Further corrections from Hugo van der Sanden, Oct 05 2006

A210472 Number A(n,k) of paths starting at {n}^k to a border position where one component equals 0 using steps that decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 33, 20, 1, 0, 1, 5, 196, 543, 70, 1, 0, 1, 6, 1305, 22096, 10497, 252, 1, 0, 1, 7, 9786, 1304045, 3323092, 220503, 924, 1, 0, 1, 8, 82201, 106478916, 1971644785, 574346824, 4870401, 3432, 1, 0

Offset: 0

Alois P. Heinz, Jan 22 2013

			A(0,3) = 1: [(0,0,0)].
A(1,1) = 1: [(1), (0)].
A(1,2) = 2: [(1,1), (0,1)], [(1,1), (1,0)].
A(1,3) = 3: [(1,1,1), (0,1,1)], [(1,1,1), (1,0,1)], [(1,1,1), (1,1,0)].
A(2,1) = 1: [(2), (1), (0)].
A(2,2) = 6: [(2,2), (1,2), (0,2)], [(2,2), (1,2), (1,1), (0,1)], [(2,2), (1,2), (1,1), (1,0)], [(2,2), (2,1), (1,1), (0,1)], [(2,2), (2,1), (1,1), (1,0)], [(2,2), (2,1), (2,0)].
Square array A(n,k) begins:
  0, 1,   1,      1,         1,             1, ...
  0, 1,   2,      3,         4,             5, ...
  0, 1,   6,     33,       196,          1305, ...
  0, 1,  20,    543,     22096,       1304045, ...
  0, 1,  70,  10497,   3323092,    1971644785, ...
  0, 1, 252, 220503, 574346824, 3617739047205, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened

Columns k=0-4 give: A000004, A000012, A000984, A209245, A209288.
Rows n=0-3 give: A057427, A001477, A093964, A210486.
Main diagonal gives A276490.
Cf. A089759 (unrestricted paths), A225094, A262809, A263159.

b:= proc() option remember; `if`(nargs=0, 0, `if`(args[1]=0, 1,
      add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs)))
    end:
A:= (n, k)-> b(n$k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[] = 0; b[args__] := b[args] = If[First[{args}] == 0, 1, Sum[b @@ Sort[ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]]; a[n_, k_] := b @@ Array[n&, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

A226545 Number A(n,k) of squares in all tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 25, 34, 25, 5, 0, 0, 6, 50, 98, 98, 50, 6, 0, 0, 7, 96, 256, 386, 256, 96, 7, 0, 0, 8, 180, 654, 1402, 1402, 654, 180, 8, 0, 0, 9, 331, 1625, 4938, 6940, 4938, 1625, 331, 9, 0

Offset: 0

Alois P. Heinz, Jun 10 2013

			A(3,3) = 1 + 6 + 6 + 6 + 6 + 9 = 34:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |     |  |   |_|  |_|   |  |_|_|_|  |_|_|_|  |_|_|_|
  |     |  |___|_|  |_|___|  |_|   |  |   |_|  |_|_|_|
  |_____|  |_|_|_|  |_|_|_|  |_|___|  |___|_|  |_|_|_|
Square array A(n,k) begins:
  0, 0,   0,    0,     0,      0,       0,        0, ...
  0, 1,   2,    3,     4,      5,       6,        7, ...
  0, 2,   5,   12,    25,     50,      96,      180, ...
  0, 3,  12,   34,    98,    256,     654,     1625, ...
  0, 4,  25,   98,   386,   1402,    4938,    16936, ...
  0, 5,  50,  256,  1402,   6940,   33502,   157279, ...
  0, 6,  96,  654,  4938,  33502,  221672,  1426734, ...
  0, 7, 180, 1625, 16936, 157279, 1426734, 12582472, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns (or rows) k=0-10 give: A000004, A001477, A067331(n-1) for n>0, A226546, A226547, A226548, A226549, A226550, A226551, A226552, A226553.
Main diagonal gives A226554.
Cf. A113881, A219924.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0,0] elif n=0 or l=[] then [1,0]
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
           (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n]))[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0}, Min[l] > 0, t=Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s={0, 0}; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h+{0, h[[1]]}][b[n, Join[l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]]]]] ]; s]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]][[2]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A229001 Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 12, 0, 1, 6, 18, 60, 0, 1, 10, 32, 96, 360, 0, 1, 18, 66, 186, 600, 2520, 0, 1, 34, 152, 426, 1222, 4320, 20160, 0, 1, 66, 378, 1110, 2964, 9086, 35280, 181440, 0, 1, 130, 992, 3186, 8254, 22818, 75882, 322560, 1814400

Offset: 0

Alois P. Heinz, Sep 10 2013

			A(3,2) = 32 = 9+5+5+5+5+3 = 3^2+4*(2^2+1^2)+3*1^2: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
Square array A(n,k) begins:
:    0,    0,    0,     0,     0,      0,      0, ...
:    1,    1,    1,     1,     1,      1,      1, ...
:    3,    4,    6,    10,    18,     34,     66, ...
:   12,   18,   32,    66,   152,    378,    992, ...
:   60,   96,  186,   426,  1110,   3186,   9846, ...
:  360,  600, 1222,  2964,  8254,  25620,  86782, ...
: 2520, 4320, 9086, 22818, 66050, 214410, 765506, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.
Rows n=0-2 give: A000004, A000012, A052548.
Main diagonal gives: A229002.

A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1
             -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := Sum[If[n == t, 1, n!/(t + 1)!*(t*(n - t + 1) + 1 - ((t + 1)*(n - t) + 1)/(t + 2))]* t^k, {t, 1, n}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A(n,k) = Sum_{t=1..n} t^k * A122843(n,t).

For fixed k, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. - Vaclav Kotesovec, Sep 12 2013

cp's OEIS Frontend

A101386 Expansion of g.f.: (5 - 3*x)/(1 - 6*x + x^2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A282516 Number T(n,k) of k-element subsets of [n] having a prime element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A288387 Number T(n,k) of Dyck paths of semilength n such that the minimal number of peaks over all positive levels equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A318753 Number A(n,k) of rooted trees with n nodes such that no more than k subtrees extending from the same node have the same number of nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A327085 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the edges of a regular n-dimensional simplex using up to k colors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2*m-1,2*m+1), where m = floor((n-2)/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A101386 Expansion of g.f.: (5 - 3x)/(1 - 6x + x^2).

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.

A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2m-1,2m+1), where m = floor((n-2)/4).