A004120 Expansion of (1 + x - x^5) / (1 - x)^3. (Formerly M3354)

1, 4, 9, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

D. R. Breach, Solution to Problem 68-16, SIAM Rev. 12 (1970), 294-297.

D. R. Breach, Letter to N. J. A. Sloane, Jun 1980

Philippe Flajolet, Balls and urns, etc. A problem in submarine detection (solution to 68-16)

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 109-111.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

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

Formula

a(n) = n*(n + 11)/2 - 5, n>=3. - R. J. Mathar, Mar 15 2011

a(n) = A302537(n-1), n>=3. - R. J. Mathar, Apr 24 2024

Maple

A004120:=(-1-z+z**5)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

i=7; s=1; lst={s}; Do[s+=n+i; If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 30 2008 *)
CoefficientList[Series[(1+x-x^5)/(1-x)^3, {x, 0, 50}], x] (* or *) Join[ {1, 4, 9}, LinearRecurrence[{3, -3, 1}, {16, 25, 35}, 50]] (* Harvey P. Dale, Oct 11 2011 *)

Prog

(Magma) [1, 4, 9], [n*(n+11)/2-5: n in [3..30]]; // Vincenzo Librandi, Oct 08 2011
(PARI) a(n)=if(n>2, (n^2+11*n)/2-5, (n+1)^2) \\ Charles R Greathouse IV, Sep 30 2015

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved

A331220 Positive numbers of the form u * v where the ternary representations of u and of v have the same number of digits d for d = 0..2.

1, 4, 9, 16, 25, 35, 36, 49, 64, 81, 100, 120, 121, 144, 165, 169, 196, 209, 224, 225, 231, 256, 285, 289, 308, 315, 324, 352, 361, 391, 399, 400, 425, 441, 480, 484, 529, 575, 576, 625, 676, 729, 784, 840, 841, 900, 957, 961, 1008, 1024, 1080, 1088, 1089

Offset

1,2

Comments

In other words, the terms of this sequence are squares or the products of two ternary anagrams.

Leading zeros are ignored.

If m belongs to the sequence, then 9*m also belongs to the sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Example

The ternary representations of 5 and 7 are "12" and "21", respectively.
So 5 and 7 are ternary anagrams, and 35 = 7*5 belongs to the sequence.

Prog

(PARI) is(n, base=3) = fordiv (n, d, if (vecsort(digits(d, base))==vecsort(digits(n/d, base)), return (1))); return (0)

Crossrefs

Cf. A331219 (binary analog), A331221 (decimal analog), A331268, A331269.

Keyword

nonn,base

Author

Rémy Sigrist, Jan 12 2020

Status

approved

A052118 Number of nonnegative integer pairs (i,j) with binomial(i+r,r) + binomial(j+r,r) <= binomial(n+r,r), r=5.

0, 1, 4, 9, 16, 25, 35, 48, 63, 78, 97, 118, 139, 163, 190, 217, 248, 277, 312, 349, 384, 424, 465, 508, 553, 600, 649, 700, 752, 805, 864, 921, 980, 1045, 1106, 1175, 1241, 1310, 1383, 1456, 1529, 1610, 1687, 1770, 1850, 1937, 2024, 2113, 2204, 2295, 2390

Offset

0,3

Links

Table of n, a(n) for n=0..50.

Crossrefs

Cf. A052115, A052116, A052117.

Keyword

nonn

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 21 2000

Status

approved

A180174 Triangle read by rows of the numbers C(n,k) of k-subsets of a quadratically populated n-multiset M.

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 5, 7, 9, 10, 10, 10, 10, 10, 9, 7, 5, 3, 1, 1, 4, 9, 16, 25, 35, 45, 55, 65, 75, 84, 91, 96, 99, 100, 100, 100, 99, 96, 91, 84, 75, 65, 55, 45, 35, 25, 16, 9, 4, 1, 1, 5, 14, 30, 55, 90, 135, 190, 255, 330, 414, 505, 601, 700, 800, 900, 1000, 1099

Offset

0,5

Comments

The multiplicity m(i) of the i-th element with 1 <= i <= n is m(i)=i^2.

Thus M=[1,2,2,2,2,...,i^2 x i,...,n^2 x n].

Row sum is equal to A028361.

Column for k=2 is equal to AA000096.

Column for k=3 is equal to AA005581.

Column for k=4 is equal to AA005582.

The number of coefficients C(n,k) for given n is equal to A056520.

Links

Table of n, a(n) for n=0..72.

Formula

C(0,0) = 0.

C(n,k) = sum_{j=(k-LS+1)}^{k} C(n-1,j).

for n > 0 and k=1,...,LR with LS = n^2+1 and LR = n*(n+1)*(2*n+1)/6.

C(n,k) = C(n,LR-k).

Example

For n=4 one has M=[1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4].
For k=7 we have 55 subsets from M:
[1, 2, 2, 3, 3, 4, 4], [1, 2, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 4, 4],
[1, 2, 2, 3, 4, 4, 4], [1, 2, 2, 3, 3, 3, 4], [1, 2, 2, 2, 3, 4, 4],
[1, 2, 2, 2, 3, 3, 4], [2, 2, 3, 3, 4, 4, 4], [2, 2, 3, 3, 3, 4, 4],
[2, 2, 2, 3, 3, 4, 4], [1, 2, 2, 2, 3, 3, 3], [1, 2, 2, 2, 4, 4, 4],
[1, 3, 3, 3, 4, 4, 4], [2, 3, 3, 3, 4, 4, 4], [2, 2, 2, 3, 4, 4, 4],
[2, 2, 2, 3, 3, 3, 4], [1, 2, 3, 4, 4, 4, 4], [1, 2, 3, 3, 3, 3, 4],
[1, 2, 2, 2, 2, 3, 4], [1, 2, 2, 3, 3, 3, 3], [1, 2, 2, 2, 2, 3, 3],
[1, 2, 2, 4, 4, 4, 4], [1, 2, 2, 2, 2, 4, 4], [1, 3, 3, 4, 4, 4, 4],
[1, 3, 3, 3, 3, 4, 4], [2, 3, 3, 4, 4, 4, 4], [2, 3, 3, 3, 3, 4, 4],
[2, 2, 3, 4, 4, 4, 4], [2, 2, 3, 3, 3, 3, 4], [2, 2, 2, 2, 3, 4, 4],
[2, 2, 2, 2, 3, 3, 4], [2, 2, 2, 3, 3, 3, 3], [2, 2, 2, 2, 3, 3, 3],
[2, 2, 2, 4, 4, 4, 4], [2, 2, 2, 2, 4, 4, 4], [3, 3, 3, 4, 4, 4, 4],
[3, 3, 3, 3, 4, 4, 4], [1, 2, 3, 3, 3, 3, 3], [1, 2, 4, 4, 4, 4, 4],
[1, 3, 4, 4, 4, 4, 4], [1, 3, 3, 3, 3, 3, 4], [2, 3, 4, 4, 4, 4, 4],
[2, 3, 3, 3, 3, 3, 4], [2, 2, 3, 3, 3, 3, 3], [2, 2, 4, 4, 4, 4, 4],
[3, 3, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 4, 4], [1, 3, 3, 3, 3, 3, 3],
[1, 4, 4, 4, 4, 4, 4], [2, 3, 3, 3, 3, 3, 3], [2, 4, 4, 4, 4, 4, 4],
[3, 4, 4, 4, 4, 4, 4], [3, 3, 3, 3, 3, 3, 4], [3, 3, 3, 3, 3, 3, 3],
[4, 4, 4, 4, 4, 4, 4].

Maple

with(combinat)
kend := 4;
Liste := NULL;
for k from 0 to kend do
Liste := Liste, `$`(k, k^2)
end do;
Liste := [Liste];
for k from 0 to 2^(kend+1)-1 do
Teilergebnis[k] := choose(Liste, k)
end do;
seq(nops(Teilergebnis[k]), k = 0 .. 2^(kend+1)-1)
' Excel VBA
Sub A180174()
Dim n As Long, nend As Long, k As Long, kk As Long, length_row As Long, length_sum As Long
Dim ATable(10, -1000 To 1000) As Double, Summe As Double
Dim offset_row As Integer, offset_column As Integer
Worksheets("Tabelle2").Select
Cells.Select
Selection.ClearContents
Range("A1").Select
offset_row = 1
offset_column = 1
nend = 7
ATable(0, 0) = 1
Cells(0 + offset_row, 0 + offset_column) = 1
For n = 1 To nend
length_row = n * (n + 1) * (2 * n + 1) / 6
length_sum = n ^ 2 + 1
For k = 0 To length_row / 2
Summe = 0
For kk = k - length_sum + 1 To k
Summe = Summe + ATable(n - 1, kk)
Next kk
ATable(n, k) = Summe
Cells(n + offset_row, k + offset_column) = ATable(n, k)
ATable(n, length_row - k) = Summe
Cells(n + offset_row, length_row - k + 0 + offset_column) = ATable(n, k)
Next k
Next n
End Sub

Crossrefs

Cf. A007318, A008302, A028361, A056520, A000096, A005581, A005582.

Keyword

nonn,tabf

Author

Thomas Wieder, Aug 15 2010

Status

approved