A001629 -id:A001629 - cp's OEIS frontend

A289207 a(n) = max(0, n-2).

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 28 2017

This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).
The rows of this array are existing sequences of the OEIS, prepended with zeros:
row 0 is A118425,
row 1 is A006478,
row 2 is A001629,
row 3 is A010049,
row 4 is A006367,
row 5 is not in the OEIS.
It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.
For n = 1 to 8, rows with only one leading zero are also autosequences.

			Array of differences begin:
   0,   0,   0,   0,  0,   0,  0,  1,  4, 12, 30, 68, ...
   0,   0,   0,   0,  0,   0,  1,  3,  8, 18, 38, 76, ...
   0,   0,   0,   0,  0,   1,  2,  5, 10, 20, 38, 71, ...
   0,   0,   0,   0,  1,   1,  3,  5, 10, 18, 33, 59, ...
   0,   0,   0,   1,  0,   2,  2,  5,  8, 15, 26, 46, ...
   0,   0,   1,  -1,  2,   0,  3,  3,  7, 11, 20, 34, ...
   0,   1,  -2,   3, -2,   3,  0,  4,  4,  9, 14, 24, ...
   1,  -3,   5,  -5,  5,  -3,  4,  0,  5,  5, 10, 16, ...
  -4,   8, -10,  10, -8,   7, -4,  5,  0,  6,  6, 17, ...
  12, -18,  20, -18, 15, -11,  9, -5,  6,  0,  7,  7, ...
  ...

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

Essentially the same as A023444. Cf. A001477, A118425, A006478, A001629, A010049, A006367, A199969.

a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]

G.f.: x^3 / (1-x)^2.

A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

Original entry on oeis.org

1, 2, 9, 44, 240, 1390, 8404, 52426, 334964, 2180928, 14418123, 96525656, 653077411, 4458529390, 30674865164, 212472058410, 1480446579602, 10369560147798, 72972217926122, 515674254743332, 3657933383804959, 26036659997517572, 185905008055923918

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Column k=2 of A378323.
Cf. A001629, A052705, A371577, A371578.
Cf. A270386, A364475.

a(n, r=2, s=1, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

a(n) = 2 * Sum_{k=0..n} binomial(3*k+2,k) * binomial(k,n-k)/(3*k+2).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364475.

A371578 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x) )^2.

Original entry on oeis.org

1, 2, 13, 102, 916, 8880, 90607, 958794, 10426089, 115798342, 1308035135, 14980661482, 173553196140, 2030265152576, 23948922940698, 284543368174220, 3402103050539715, 40903437537402792, 494215527894112099, 5997782678374854902, 73078635875447981850

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A001629, A052705, A371576, A371577.

Cf. A365184, A371580.

a(n, r=2, s=1, t=5, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

a(n) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(k,n-k)/(5*k+2).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365184.

A372102 Number of permutations of [n] whose non-fixed points are not neighbors.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 19, 45, 107, 278, 728, 2033, 5749, 17105, 51669, 162674, 520524, 1724329, 5807143, 20146861, 71048431, 257139686, 945626800, 3558489633, 13599579817, 53060155137, 210124405097, 847904374466, 3470756061140, 14453943647561, 61023690771451

Offset: 0

Alois P. Heinz, Apr 18 2024

nonn

			a(3) = 2: 123, 321.
a(4) = 4: 1234, 1432, 3214, 4231.
a(5) = 9: 12345, 12543, 14325, 15342, 32145, 32541, 42315, 52143, 52341.
a(6) = 19: 123456, 123654, 125436, 126453, 143256, 143652, 153426, 163254, 163452, 321456, 325416, 326451, 423156, 423651, 521436, 523416, 621453, 623154, 623451.

Alois P. Heinz, Table of n, a(n) for n = 0..892
Wikipedia, Permutation.

Cf. A000142, A000166, A001629, A011973, A098825, A131735, A162969, A165961, A371995.

a:= proc(n) option remember; `if`(n<4, [2$3, 4][n+1],
      3*a(n-1)+(n-2)*a(n-2)+(n-1)*(a(n-4)-a(n-3)))/2
    end:
seq(a(n), n=0..30);

a[n_] := Sum[Binomial[n + 1 - k, k] * Subfactorial[k], {k, 0, (n + 1)/2}];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Apr 24 2024 *)

a(n) = Sum_{j=0..floor((n+1)/2)} A000166(j)*A011973(n+1,j).
a(n) mod 2 = A131735(n+3).
Row sums of A371995(n+1), which are the antidiagonals of A098825. - Peter Luschny, Apr 24 2024
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 - 7/8) * n^(n/2 + 1) / 2^((n+3)/2). - Vaclav Kotesovec, Apr 25 2024

A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 3, 3, 4, 5, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2

Offset: 0

Gus Wiseman, Jul 02 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 27 are {1,2,4,5}, with maximal sparse subsets {{1,4},{1,5},{2,4},{2,5}}, so a(27) = 4.

For subsets of {1..n} we get A000931 (shifted), maximal case of A000045 (shifted).
This is the maximal case of A245564.
The greatest number whose binary indices are one of these subsets is A374356.
For prime instead of binary indices we have A385215, maximal case of A166469.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A202064 counts subsets containing n with k maximal runs.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A044813, A053538, A119900, A202023, A208342, A384177, A384890.

spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
maximize[sys_]:=Complement@@Prepend[Most[Subsets[#]]&/@sys,sys];
Table[Length[maximize[spars[bpe[n]]]],{n,0,100}]

A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 207, 394, 739, 1398, 2687, 5152, 9891, 19128, 37039, 71754, 139459, 271522, 528999, 1032308, 2017291, 3945186, 7723203, 15134440, 29679407, 58245068, 114389683, 224796210, 442021743, 869658304, 1711914351, 3371515306

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

Also the number of subsets of {1..n} with the same number of adjacent elements increasing by 1 as adjacent elements increasing by more than 1.

			The set {2,3,5,6,8} has maximal runs ((2,3),(5,6),(8)) and maximal anti-runs ((2),(3,5),(6,8)) so is counted under a(8).
The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}   {}       {}       {}
      {1}  {1}  {1}  {1}      {1}      {1}
           {2}  {2}  {2}      {2}      {2}
                {3}  {3}      {3}      {3}
                     {4}      {4}      {4}
                     {1,2,4}  {5}      {5}
                     {1,3,4}  {1,2,4}  {6}
                              {1,2,5}  {1,2,4}
                              {1,3,4}  {1,2,5}
                              {1,4,5}  {1,2,6}
                              {2,3,5}  {1,3,4}
                              {2,4,5}  {1,4,5}
                                       {1,5,6}
                                       {2,3,5}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}

Christian Sievers, Table of n, a(n) for n = 0..3328

The LHS is counted by A034839 (for partitions A384881, strict A116674), rank statistic A069010.
The case containing n + 1 is A217615.
The RHS is counted by A384893 or A210034 (for partitions A268193, strict A384905), rank statistic A384890.
Subsets of this type are ranked by A385575.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000071, A000079, A001629, A010027, A053538, A384177, A384879, A384889.

a:= proc(n) option remember; `if`(n<5, [1, 2, 3, 4, 7][n+1], ((3*n-4)*a(n-1)-
      (3*n-5)*a(n-2)+(5*n-12)*a(n-3)-2*(4*n-11)*a(n-4)+4*(n-3)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 06 2025

Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2==#1+1&]]==Length[Split[#,#2!=#1+1&]]&]],{n,0,10}]

a(n)=polcoef([1,1,1]*[x,0,0;x,x^2,1;0,x,x]^n*[1,0,0]~,n) \\ Christian Sievers, Jul 06 2025

Let M be the matrix [1,0,0; 1,x,1/x; 0,1,1]. Then a(n) is the sum of the constant terms of the entries in the left column of M^n. - Christian Sievers, Jul 06 2025

a(21) and beyond from Christian Sievers, Jul 06 2025

A054453 Triangle of partial row sums of triangle A054450(n,m), n >= m >= 0.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 15, 10, 6, 2, 1, 28, 20, 12, 7, 2, 1, 51, 38, 26, 14, 8, 2, 1, 92, 71, 50, 33, 16, 9, 2, 1, 164, 130, 97, 64, 41, 18, 10, 2, 1, 290, 235, 180, 130, 80, 50, 20, 11, 2, 1, 509, 420, 332, 244, 171, 98, 60, 22, 12, 2, 1

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z^2)*(Fib(z))^2)/(1-x*z/(1-z^2)) Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).
This is the second member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310(n,m) by repeated application of the partial row sums procedure.
The column sequences are A029907, A001629, A054454 for m=0..2.

			{1}; {2,1}; {4,2,1}; {8,5,2,1};...
Fourth row polynomial (n=3): p(3,x)= 8+5*x+2*x^2+x^3

Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, Double Dimers and Super Ptolemy Relations, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.

Cf. A049310, A054450, A000045, A029907, A001629. Row sums: A054455(n).

a(n, m)=sum(A054450(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j-1, m)*|A049310(n-j, 0)|, j=m..n) + A054450(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x^2)*(Fib(x))^2)*(x/(1-x^2))^m, m >= 0, with Fib(x) G.f. for A000045(n+1).

A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

Original entry on oeis.org

0, 1, 16, 2048, 1638400, 7247757312, 164995463643136, 18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, 271732164163901599116133024293512544256

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Discriminant of Chebyshev polynomial U_n (x) of second kind.
Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.
The absolute value of the discriminant of Pell polynomials is a(n-1).
Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - Rigoberto Florez, Sep 01 2018

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Eric Weisstein's World of Mathematics, Pell Polynomial
Index entries for sequences related to Chebyshev polynomials.

Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

[0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Join[{0},Table[(n+1)^(n-2) 2^n^2,{n,10}]] (* Harvey P. Dale, May 01 2015 *)

PARI
```
a(n)=if(n<1,0,(n+1)^(n-2)*2^(n^2))
```

a(n)=if(n<1,0,n++; poldisc(poltchebi(n)'/n))

a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.
a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.
a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

Formula and more terms from Vladeta Jovovic, Aug 07 2003

A089098 Sign twisted convoluted convolved Fibonacci numbers H_j^(2).

Original entry on oeis.org

1, 1, 3, 5, 11, 19, 37, 65, 120, 210, 376, 654, 1149, 1985, 3443, 5911, 10159, 17345, 29605, 50305, 85400, 144516, 244272, 411900, 693729, 1166209, 1958219, 3283145, 5498595, 9197455, 15369373, 25655489, 42787456, 71293590, 118695272, 197452746, 328227725

Offset: 1

N. J. A. Sloane, Dec 05 2003

Let "a" = the Fibonacci numbers, and "b" = the aerated Fibonacci numbers (1, 0, 1, 0, 2,...). Then A089098 = (1/2) * (a^2 + b), where a^2 = A001629, the Fibonacci numbers convolved with themselves: (1, 2, 5, 10, 20, 38,...).

Colin Barker, Table of n, a(n) for n = 1..1000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 2.
P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4,-1,0,0,2,1).

2nd column of A337009.

with(numtheory): f := z->-1/(1-z-z^2): m := proc(r,j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]),i=1..nops(d)): Wser := simplify(series(W,z=0,80)): coeff(Wser,z^j) end: seq(m(2,j),j=1..39);

(1-x-x^2+x^3)/((1-x-x^2)^2*(1-x^2-x^4)) + O[x]^40 // CoefficientList[#,x]& (* Jean-François Alcover, Jan 20 2018 *)

Vec(-x*(x-1)^2*(x+1)/((x^2+x-1)^2*(x^4+x^2-1)) + O(x^50)) \\ Colin Barker, Jul 23 2015

G.f.: (z/2)[1/(1-z-z^2)^2+1/(1-z^2-z^4)].

G.f.: -x*(x-1)^2*(x+1) / ((x^2+x-1)^2*(x^4+x^2-1)). - Colin Barker, Jul 23 2015

Edited by Emeric Deutsch, Mar 06 2004

A099924 Self-convolution of Lucas numbers.

Original entry on oeis.org

4, 4, 13, 22, 45, 82, 152, 274, 491, 870, 1531, 2676, 4652, 8048, 13865, 23798, 40713, 69446, 118144, 200510, 339559, 573894, 968183, 1630632, 2742100, 4604572, 7721797, 12933334, 21637221, 36159610, 60367976, 100687786

Offset: 0

Ralf Stephan, Nov 01 2004

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 57.

Michael De Vlieger, Table of n, a(n) for n = 0..4767
É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017). See remark 4.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 37.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A001629, A000032. Bisection: A203573 (even), 2*A203574 (odd).

Table[Sum[LucasL[k]LucasL[n-k],{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[ {2,1,-2,-1},{4,4,13,22},40] (* Harvey P. Dale, Mar 06 2012 *)

a(n) = (n+1)*L(n) + 2F(n+1) = Sum_{k=0..n} L(k)*L(n-k).
G.f.: (2-x)^2/(1-x-x^2)^2, corrected Aug 23 2022
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), a(0)=4, a(1)=4, a(2)=13, a(3)=22. - Harvey P. Dale, Mar 06 2012
a(n) = 2*A099920(n+1)-A099920(n). - R. J. Mathar, Aug 23 2022

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cp's OEIS Frontend

A289207 a(n) = max(0, n-2).

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A371576 G.f. satisfies A(x) = ( 1 + x*A(x)^(3/2) * (1 + x) )^2.

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A371578 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) * (1 + x) )^2.

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A372102 Number of permutations of [n] whose non-fixed points are not neighbors.

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A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

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A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

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A054453 Triangle of partial row sums of triangle A054450(n,m), n >= m >= 0.

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A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

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A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

A371578 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x) )^2.