A001891 -id:A001891 - cp's OEIS frontend

A054469 a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2, with a(0) = 1, a(1) = 7.

1, 7, 28, 85, 218, 499, 1053, 2092, 3970, 7272, 12958, 22596, 38739, 65535, 109714, 182185, 300620, 493635, 807555, 1317360, 2144396, 3485032, 5657028, 9174560, 14869613, 24088399, 39009168, 63156437, 102233030, 165466347, 267786673

Offset: 0

Barry E. Williams, Mar 31 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

Cf. A000045, A001891, A001911.

Right-hand column 11 of triangle A011794.

A054469:= func< n | Fibonacci(n+12) -(1/12)*(1716 +802*n +173*n^2 +20*n^3 +n^4) >;
[A054469(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024

RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[ n+3,3]/2},a,{n,30}] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^5*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)

a(n) = sum(i=1,(n+2)\2,binomial(n+5-i,n+2-2*i))+2*sum(i=1,(n+1)\2,binomial(n+5-i,n+1-2*i)) \\ Jason Yuen, Aug 13 2024

def A054469(n): return fibonacci(n+12) - (1716 + 802*n + 173*n^2 + 20*n^3 + n^4)//12
[A054469(n) for n in range(41)] # G. C. Greubel, Oct 21 2024

a(n) = a(n-1) + a(n-2) + (n+1)*(n+2)^2*(n+3)/12.
a(-n) = 0.
a(n) = (Sum_{i=1..floor((n+2)/2)} binomial(n+5-i, n+2-2*i)) + 2*(Sum_{i=1..floor((n+1)/2)} binomial(n+5-i, n+1-2*i)).
G.f.: (1+x) / ((1-x)^5*(1-x-x^2)). - Colin Barker, Jun 11 2013
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+12) - Sum_{j=0..4} Fibonacci(11-2*j) * binomial(n+j, j).
a(n) = Fibonacci(n+12) - (1/12)*(1716 + 802*n + 173*n^2 + 20*n^3 + n^4). (End)

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 *)

A356619 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 3 for i = 2,...,k.

Original entry on oeis.org

0, 1, 4, 11, 25, 52, 103, 198, 374, 699, 1298, 2401, 4431, 8166, 15037, 27676, 50924, 93685, 172336, 316999, 583077, 1072472, 1972611, 3628226, 6673378, 12274287, 22575966, 41523709, 76374043, 140473802, 258371641, 475219576, 874065112, 1607656425

Offset: 0

Clark Kimberling, Aug 24 2022

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A001891, A062544, A221949, A356620, A356621.

maxDiff = 3;
t = Map[Length[Select[Map[{#, Max[Differences[#]]} &,
     Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &, Range[16]]
FindGeneratingFunction[%, x]
FindLinearRecurrence[t]
LinearRecurrence[{3, -2, 0, -1, 1}, {0, 1, 4, 11, 25}, 45]

G.f.: x*(1 + x + x^2)/((-1 + x)^2*(1 - x - x^2 - x^3)).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-4) + a(n-5).
a(n) = A221949(n+2)-1 for n >= 0.

A356620 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 4 for i = 2,...,k.

Original entry on oeis.org

0, 1, 4, 11, 26, 56, 115, 230, 453, 884, 1716, 3321, 6416, 12383, 23886, 46060, 88803, 171194, 330009, 636136, 1226216, 2363633, 4556076, 8782147, 16928162, 32630112, 62896595, 121237118, 233692093, 450456028, 868281948, 1673667305, 3226097496, 6218502903

Offset: 0

Clark Kimberling, Sep 04 2022

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,0,-1,1).

Cf. A001891, A356619, A356621.

maxDiff = 4; t = Map[Length[Select[Map[{#, Max[Differences[#]]} &,
      Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &, Range[18]]

G.f.: (x (-1 - x - x^2 - x^3))/((-1 + x)^2 (-1 + x + x^2 + x^3 + x^4)).

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-5) + a(n-6).

A356621 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 5 for i = 2,...,k.

Original entry on oeis.org

0, 1, 4, 11, 26, 57, 119, 242, 485, 964, 1907, 3762, 7410, 14583, 28686, 56413, 110924, 218091, 428777, 842976, 1657271, 3258134, 6405349, 12592612, 24756452, 48669933, 95682600, 188107071, 369808798, 727024989, 1429293531, 2809917134, 5524151673

Offset: 0

Clark Kimberling, Sep 04 2022

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,0,0,-1,1).

Cf. A001891, A356619, A356620.

maxDiff = 5; t = Map[Length[Select[Map[{#, Max[Differences[#]]} &,
      Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &, Range[20]]

G.f.: (x (-1 - x - x^2 - x^3 - x^4))/((-1 + x)^2 (-1 + x + x^2 + x^3 + x^4 + x^5)).

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-6) + a(n-7).

A001884 Hit polynomials.

Original entry on oeis.org

1, 0, 1, 2, 20, 104, 775, 6140, 55427, 553802, 6087992, 72994152, 948103477, 13262133736, 198769630061, 3177862894922, 53984653965996, 971068821144112, 18438722595913195, 368558842844143268, 7735520783692157215, 170095060428041137778, 3910332719957508452016, 93806427360751009531632

Offset: 1

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. See A001883.
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).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Second column of A080018.

Cf. A001883-A001891.

More terms from Vladeta Jovovic and Vladimir Baltic, Jan 20 2003

a(21)-a(24) from Vaclav Kotesovec, Oct 10 2017

A001890 Hit polynomials.

Original entry on oeis.org

1, 10, 34, 206, 1351, 10543, 92708, 912884, 9917445, 117838808, 1519483258, 21128310078, 315093762147, 5016410089130, 84909414423784, 1522548805068310, 28830824064870329, 574880701875755325, 12039866150973004846, 264230694283295736788, 6063848537910027941323

Offset: 3

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
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).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Fourth column of A080061.

Cf. A001883-A001891.

(* Program not suitable to compute more than a dozen terms *)
M[n_] := Table[If[0 <= i - j <= 2, x, 1], {i, 1, n}, {j, 1, n}];
a[n_] := Coefficient[Permanent[M[n]], x, 3];
Table[an = a[n]; Print[n, " ", an]; an, {n, 3, 15}] (* Jean-François Alcover, Jan 12 2018 *)

More terms from Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

a(21)-a(23) from Vaclav Kotesovec, Oct 10 2017

A054470 Partial sums of A054469.

Original entry on oeis.org

1, 8, 36, 121, 339, 838, 1891, 3983, 7953, 15225, 28183, 50779, 89518, 155053, 264767, 446952, 747572, 1241207, 2048762, 3366122, 5510518, 8995550, 14652578, 23827138, 38696751, 62785150, 101794318, 164950755, 267183785, 432650132

Offset: 0

Barry E. Williams, Mar 31 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (7,-20,29,-20,1,8,-5,1).

Cf. A000045, A001891, A001911, A054469.

Right-hand column 13 of triangle A011794.

A054470:= func< n | Fibonacci(n+14) - (45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)/120 >;
[A054470(n): n in [0..40]]; // G. C. Greubel, Oct 21 2024

Accumulate[RecurrenceTable[{a[0]==1,a[1]==7,a[n]==a[n-1]+a[n-2]+(n+2) Binomial[n+3,3]/2},a,{n,40}]] (* Harvey P. Dale, Sep 22 2013 *)
CoefficientList[Series[(1+x)/((1-x)^6*(1-x-x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)

def A054470(n): return fibonacci(n+14) -(45120 +21458*n +4925*n^2 +680*n^3 +55*n^4 +2*n^5)//120
[A054470(n) for n in range(41)] # G. C. Greubel, Oct 21 2024

a(n) = a(n-1) + a(n-2) + (2*n+5)*C(n+4, 4)/5, with a(-n) = 0.
a(n) = Sum_{j=1..[(n+2)/2]} binomial(n+6-j, n+2-2*j) + 2*Sum_{j=1..[(n+1)/2]} binomial(n+6-j, n+1-2*j), where [x]=greatest integer in x.
G.f.: (1+x) / ((1-x)^6*(1-x-x^2)). - Colin Barker, Jun 11 2013
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+14) - Sum_{j=0..5} Fibonacci(13-2*j)*binomial(n+j,j).
a(n) = Fibonacci(n+14) - (1/120)*(45120 + 21458*n + 4925*n^2 + 680*n^3 + 55*n^4 + 2*n^5). (End)

A163250 a(n) = A000045(n+6) - (n^2 + 4*n + 8).

Original entry on oeis.org

0, 0, 1, 5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, 24156724, 39087009, 63244757, 102332855

Offset: 0

Jonathan Vos Post, Jul 23 2009

Given on p. 2 of Freixas, and proved as Theorem 3.2.
Partial sums of A001891. - Bill McEachen, Jan 20 2023
Original name was: The number of nonisomorphic complete simple games with n voters of two different types. - Charles R Greathouse IV, Jan 22 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Josep Freixas, Xavier Molinero, and Salvador Roura, A Fibonacci sequence for linear structures with two types of components, arXiv:0907.3853 [math.CO], Jul 22 2009.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A053808, A001891.

List([0..35],n->Fibonacci(n+6)-(n^2+4*n+8)); # Muniru A Asiru, Oct 28 2018

[Fibonacci(n+6)-(n^2+4*n+8): n in [0..40]]; // Vincenzo Librandi, Sep 22 2017

with(numtheory): seq(coeff(series(x^2*(1+x)/((x^2+x-1)*(x-1)^3),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 28 2018

LinearRecurrence[{4,-5,1,2,-1}, {0,0,1,5,15}, 40] (* or *) Table[ Fibonacci[n+6] -(n^2+4*n+8), {n,0,40}] (* G. C. Greubel, Dec 12 2016 *)

concat([0,0], Vec(x^2*(1+x)/((1-x-x^2)*(1-x)^3) + O(x^40))) \\ G. C. Greubel, Dec 12 2016

f=fibonacci; [f(n+6) -(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 06 2019

a(n) = F(n+6) - (n^2 + 4*n + 8), where F(n) are the Fibonacci numbers.
From R. J. Mathar, Jul 27 2009: (Start)
a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5).
G.f.: x^2*(1+x)/((1-x-x^2)*(1-x)^3). (End)
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i^2 * C(n-k-1,k-i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A053808(n-2) for n >= 2. - Georg Fischer, Oct 28 2018
a(n) = (n-1)^2 + a(n-1) + a(n-2), n>2 (conjectured). - Bill McEachen, Jan 20 2023

More terms from R. J. Mathar, Jul 27 2009

New name using given formula from Joerg Arndt, Jan 21 2023

A001885 Hit polynomials.

Original entry on oeis.org

2, 2, 10, 28, 207, 1288, 10366, 91296, 903037, 9832848, 117032570, 1510932116, 21028774738, 313832463386, 4999133311044, 84655108256252, 1518546437350265, 28763765236019284, 573689119174695326, 12017485839703597024, 263787711208968183879, 6054632852404055079936

Offset: 2

N. J. A. Sloane

nonn

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883).
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).

N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)

Third column of A080018.

Cf. A001883-A001891.

More terms from Vladeta Jovovic and Vladimir Baltic, Jan 20 2003

a(21)-a(23) from Vaclav Kotesovec, Oct 10 2017

cp's OEIS Frontend

A054469 a(n) = a(n-1) + a(n-2) + (n+2)*binomial(n+3, 3)/2, with a(0) = 1, a(1) = 7.

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A202970 Symmetric matrix based on A001911, by antidiagonals.

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A356619 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 3 for i = 2,...,k.

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A356620 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 4 for i = 2,...,k.

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A356621 a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 5 for i = 2,...,k.

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A001884 Hit polynomials.

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A001890 Hit polynomials.

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A054470 Partial sums of A054469.

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