Louis Rogliano - cp's OEIS frontend

User: Louis Rogliano

Louis Rogliano has authored 3 sequences.

A306262 Difference between maximum and minimum sum of products of successive pairs in permutations of [n].

0, 0, 0, 4, 11, 24, 42, 68, 101, 144, 196, 260, 335, 424, 526, 644, 777, 928, 1096, 1284, 1491, 1720, 1970, 2244, 2541, 2864, 3212, 3588, 3991, 4424, 4886, 5380, 5905, 6464, 7056, 7684, 8347, 9048, 9786, 10564, 11381, 12240, 13140, 14084, 15071, 16104, 17182

Offset: 0

Louis Rogliano, Feb 01 2019

nonn

			a(4) = 11 = 23 - 12. 1342 and 2431 have sums 23, 3214 and 4123 have sums 12.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A026035, A101986.

a:= n-> `if`(n=0, 0, (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
    <0|0|0|0|1>, <-1|3|-2|-2|3>>^n. <<1, 0, 0, 4, 11>>)[1, 1]):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 02 2019

a[n_] := Module[
  {min, max, perm, g, mperm},
  perm = Permutations[Range[n]];
  g[x_] := Sum[x[[i]] x[[i + 1]], {i, 1, Length[x] - 1}];
  mperm = Map[g, perm];
  min = Min[mperm];
  max = Max[mperm];
  Return[max - min]]
LinearRecurrence[{3,-2,-2,3,-1},{0,0,0,4,11,24},60] (* Harvey P. Dale, Aug 05 2020 *)

concat([0,0,0], Vec(x^3*(4 - x - x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Feb 05 2019

a(n+1) = a(n) + 1/4*((-1+(-1)^(n-1))^2+2*(n-1)*(n+4)) with a(n) = 0 for n <= 2.
From Alois P. Heinz, Feb 01 2019: (Start)
G.f.: -(x^2+x-4)*x^3/((x+1)*(x-1)^4).
a(n) = (2*n^3+6*n^2-26*n+15-3*(-1)^n)/12 for n > 0.
a(n) = A101986(n-1) - A026035(n) for n > 0. (End)
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Wesley Ivan Hurt, May 28 2021
a(n) = A110610(n+1) - A110611(n+1). - Talmon Silver, Sep 24 2025

More terms from Alois P. Heinz, Feb 01 2019

A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 11, 4, 1, 14, 26, 19, 5, 1, 20, 50, 55, 29, 6, 1, 27, 85, 125, 99, 41, 7, 1, 35, 133, 245, 259, 161, 55, 8, 1, 44, 196, 434, 574, 476, 244, 71, 9, 1, 54, 276, 714, 1134, 1176, 804, 351, 89, 10, 1, 65, 375, 1110, 2058, 2562, 2190, 1275, 485, 109, 11

Offset: 1

Louis Rogliano, Nov 26 2015

By empirical observation: Sum of rows is A002064.

			Triangle begins:
  1
  1    2
  1    5    3
  1    9   11    4
  1   14   26   19    5
  1   20   50   55   29    6
  1   27   85  125   99   41    7
  1   35  133  245  259  161   55    8
  1   44  196  434  574  476  244   71    9
  1   54  276  714 1134 1176  804  351   89   10
  1   65  375 1110 2058 2562 2190 1275  485  109   11

Cyann Donnot, Antoine Genitrini, Yassine Herida, Unranking Combinations Lexicographically: an efficient new strategy compared with others, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
Antoine Genitrini and Martin Pépin, Lexicographic unranking of combinations revisited, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.

Columns are: A000012 (k=1), A000096 (k=2), A051925 (k=3), A215862 (k=4), A264750 (k=5).
Cf. A007318 (binomial(n-1,k-1) = number of sequences of k throws of an n-sided die in which the sum of the throws equals n).
See also A002064.

T[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
(* The rows are obtained by: *)
g[n_] := Table[T[n,k], {k,1,n}]
(* And the triangle is obtained by: *)
Table[g[n],{n,1,number_of_rows_wanted}]

Sum_{k = 1..n} T(n,k)*k/n^k = ((n+1)/n)^(n-1) = expected value of k.
Lim_{n->infinity} (expected value of k) = e = 2.71828182845... - Jon E. Schoenfield, Nov 26 2015
T(n,k) = Sum_{i=k..n} i*binomial(i-2,k-2). - Danny Rorabaugh, Mar 04 2016
T(n,n-1) = 2*T(n-1,n-1) + T(n-1,n-2).
By empirical observation, g.f. for column k: (x-k)/(x-1)^(k+1).

A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

Original entry on oeis.org

5, 29, 99, 259, 574, 1134, 2058, 3498, 5643, 8723, 13013, 18837, 26572, 36652, 49572, 65892, 86241, 111321, 141911, 178871, 223146, 275770, 337870, 410670, 495495, 593775, 707049, 836969, 985304, 1153944, 1344904, 1560328, 1802493, 2073813, 2376843, 2714283

Offset: 5

Louis Rogliano, Nov 23 2015

Sequence gives the second column of A185508. [Bruno Berselli, Nov 24 2015]

Number of 5-tuples (t_1, ..., t_5) with 1 <= t_j <= n, Sum_{j <= 4} t_j < n and Sum_{j<=5} t_j >= n. - Robert Israel, Nov 25 2015

			From _Jon E. Schoenfield_, Nov 26 2015: (Start)
For n=5, the a(5) = 5 sequences (i.e., quintuples or 5-tuples) are {1,1,1,1,1}, {1,1,1,1,2}, {1,1,1,1,3}, {1,1,1,1,4} and {1,1,1,1,5}. (Each of the first four throws must be a 1; otherwise, the sum of the throws would reach or exceed 5 before the 5th throw.)
For n=6, each of the quintuples must have four throws whose sum is less than 6, followed by a fifth throw that brings the sum to at least 6, so the a(6) = 29 quintuples are the 5 quintuples {1,1,1,1,t_5} where t_5 is any value in 2..6 and the four sets of 6 quintuples {1,1,1,2,t_5}, {1,1,2,1,t_5}, {1,2,1,1,t_5} and {2,1,1,1,t_5} where t_5 is any value in 1..6. (End)

Colin Barker, Table of n, a(n) for n = 5..1000
Louis Rogliano, Sequence A264750
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000096 (k=2), A051925 (k=3), A215862 (k=4).

Cf. A185508.

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: n in [5..40]]; // Vincenzo Librandi, Nov 24 2015

A264750:=n->(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: seq(A264750(n), n=5..50); # Wesley Ivan Hurt, Nov 24 2015

f[n_, k_] := Module[
{i, total = 0, part, perm},
part = IntegerPartitions[n, {k}];
perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}],      1];
For[i = 1, i <= Length[perm], i++,    total += n + 1 - perm[[i, k]]    ];
Return[total];   ]
And the sequences are obtained by:
h[k_] := Table[f[i, k], {i, k, number_of_terms_wanted}]
Table[(n - 4) (n - 3) (n - 2) (n - 1) (4 n + 5)/120, {n, 5, 40}] (* Bruno Berselli, Nov 24 2015 *)

Vec(x^5*(5-x)/(1-x)^6 + O(x^100)) \\ Colin Barker, Nov 23 2015

for(n=5, 40, print1((n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120", ")); \\ Bruno Berselli, Nov 24 2015

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120 for n in (5..40)] # Bruno Berselli, Nov 24 2015

From Colin Barker, Nov 23 2015: (Start)
a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*(4*n + 5)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>10.
G.f.: x^5*(5 - x) / (1 - x)^6. (End)

Offset changed by Robert Israel, Nov 25 2015

Formulae, b-file adapted to the new offset and definition rephrased by the Editors of the OEIS, Nov 26 2015

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User: Louis Rogliano

A306262 Difference between maximum and minimum sum of products of successive pairs in permutations of [n].

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A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

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A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

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Magma

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