cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-37 of 37 results.

A080041 a(n) = floor((1+sqrt(3))^n).

Original entry on oeis.org

1, 2, 7, 20, 55, 152, 415, 1136, 3103, 8480, 23167, 63296, 172927, 472448, 1290751, 3526400, 9634303, 26321408, 71911423, 196465664, 536754175, 1466439680, 4006387711, 10945654784, 29904084991, 81699479552, 223207129087
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003

Keywords

Links

Crossrefs

Cf. A080040.

Programs

  • Mathematica
    CoefficientList[Series[(1+2t^3)/(1-2t-3t^2+2t^3+2t^4), {t, 0, 30}], t]
Floor[(1+Sqrt[3])^Range[0,30]] (* Harvey P. Dale, Sep 14 2012 *)

Formula

a(n) = A080040(n) - (1+(-1)^n)/2.
G.f.: (1+2t^3)/( (t-1)*(1+t)*(2*t^2+2*t-1)).

A172012 Expansion of (2-3*x)/(1-3*x-3*x^2) .

Original entry on oeis.org

2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079, 77782556128482, 294896059795683
Offset: 0

Views

Author

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

Keywords

Comments

The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.
The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.

Links

Programs

  • Mathematica
    CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,3},{2,3},31] (* Harvey P. Dale, Aug 24 2011 *)

Formula

a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).
L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).
a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [Harvey P. Dale, Aug 24 2011]
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

Extensions

Edited and extended by R. J. Mathar, Jan 23 2010

A270444 Expansion of 2*(1+2*x) / (1-8*x+4*x^2).

Original entry on oeis.org

2, 20, 152, 1136, 8480, 63296, 472448, 3526400, 26321408, 196465664, 1466439680, 10945654784, 81699479552, 609813217280, 4551707820032, 33974409691136, 253588446248960, 1892809931227136, 14128125664821248, 105453765593661440
Offset: 1

Views

Author

Altug Alkan, Mar 17 2016

Keywords

Comments

If p is an odd prime, a((p+1)/2) == 2 mod p. In other words, a((p+1)/2) - 2^p is divisible by p where p is an odd prime.

Examples

			a(2) = 20 because (1 + sqrt(3))^3 + (1 - sqrt(3))^3 = 20.

Links

Crossrefs

Cf. A077444, A080040, A080041, A107903.

Programs

  • Mathematica
    CoefficientList[Series[2(1+2x)/(1-8x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-4},{2,20},30] (* Harvey P. Dale, Jun 09 2020 *)
  • PARI
    Vec(2*(1+2*x)/(1-8*x+4*x^2) + O(x^100))

Formula

G.f.: 2*(1+2*x)/(1-8*x+4*x^2).
a(n) = (1+sqrt(3))^(2*n-1) + (1-sqrt(3))^(2*n-1).
a(n) = 2 * A107903(n-1).

A384147 Array A(n,k) = n*(A(n-1,k)+A(n-2,k)+...+A(n-k,k)), where A(n,k) = n if n <= k, read by antidiagonals with n >= 1 and k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 8, 3, 4, 1, 20, 3, 4, 5, 1, 56, 27, 4, 5, 6, 1, 152, 99, 4, 5, 6, 7, 1, 416, 387, 64, 5, 6, 7, 8, 1, 1136, 1539, 304, 5, 6, 7, 8, 9, 1, 3104, 6075, 1504, 125, 6, 7, 8, 9, 10, 1, 8480, 24003, 7504, 725, 6, 7, 8, 9, 10, 11, 1, 23168, 94851, 37504, 4325, 216, 7, 8, 9, 10, 11, 12
Offset: 1

Views

Author

Jason Bard, May 25 2025

Keywords

Comments

Taking only the triangle where 1<=n<=k and reading by rows yields A002024.

Examples

			Top left corner of the array:
   1  1  1  1  1   1    1    1     1      1      1
   2  2  8 20 56 152  416 1136  3104   8480  23168
   3  3  3 27 99 387 1539 6075 24003  94851 374787
   4  4  4  4 64 304 1504 7504 37504 187264 935104
   5  5  5  5  5 125  725 4325 25925 155525 933125
   6  6  6  6  6   6  216 1476 10296  72036 504216
   7  7  7  7  7   7    7  343  2695  21511 172039
   8  8  8  8  8   8    8    8   512   4544  40832
   9  9  9  9  9   9    9    9     9    729   7209
  10 10 10 10 10  10   10   10    10     10   1000
  ...

Links

Crossrefs

Cf. A000012 (row 1), A080040 (row 2).
Cf. A000578, A002024, A002260.

Programs

  • Mathematica
    nmax = 100; AntiDiagonalFlatten[matrix_] := Module[{n = Length@matrix}, Flatten[Table[matrix[[i, s - i]], {s, 2, 2 n}, {i, Max[1, s - n], Min[n, s - 1]}], 1]]; A384147 = AntiDiagonalFlatten[Table[LinearRecurrence[ConstantArray[n, n], ConstantArray[n, n], {1, nmax}], {n, 1, nmax}]][[;; nmax*(nmax + 1)/2]]

Formula

A(m,m+1) = m^3 for all m >= 1.
A(m,m+2) = m^4 + m^3 - m^2 for all m >= 1.
A(m,m+3) = m^5 + 2m^4 - 2m^2 for all m >= 1.
A(m,m+4) = m^6 + 3m^5 + 2m^4 - 2m^3 - 3m^2 for all m >= 3.
A(m,m+5) = m^7 + 4m^6 + 5m^5 - 5m^3 - 4m^2 for all m >= 4.
...
A(m,m+k) ~ O(m^(k+2)) for all m >= k-1 may be derived similarly.

A093331 Number of ternary necklaces of length n with no subsequence 00.

Original entry on oeis.org

2, 5, 8, 17, 32, 76, 164, 398, 948, 2336, 5756, 14460, 36344, 92282, 235120, 602348, 1548320, 3995602, 10340300, 26838896, 69830576, 182111414, 475898036, 1246011050, 3267979208, 8584907756, 22585675348, 59501499506, 156955442072
Offset: 1

Views

Author

Philippe Deléham, Apr 25 2004

Keywords

Formula

a(n) = (1/n)*sum_{d divides n } totient(n/d)*A080040(d).

A287594 Number of independent vertex sets in the n-helm graph.

Original entry on oeis.org

3, 4, 12, 28, 72, 184, 480, 1264, 3360, 8992, 24192, 65344, 177024, 480640, 1307136, 3559168, 9699840, 26452480, 72173568, 196989952, 537802752, 1468536832, 4010582016, 10954043392, 29920862208, 81733033984, 223274237952, 609947435008, 1666309128192
Offset: 0

Views

Author

Eric W. Weisstein, May 27 2017

Keywords

Comments

Extended to a(0)-a(2) using the formula.

Links

Crossrefs

Cf. A080040.

Programs

  • Mathematica
    Table[2^n + (1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand
Table[2^n + 2^(n/2) LucasL[n, Sqrt[2]], {n, 0, 20}] // Round
LinearRecurrence[{4, -2, -4}, {4, 12, 28}, {0, 20}]
CoefficientList[Series[(3 - 8 x + 2 x^2)/(1 - 4 x + 2 x^2 + 4 x^3), {x, 0, 20}], x]

Formula

a(n) = 2^n+A080040(n).
a(n) = 2^n+(1-sqrt(3))^n+(1+sqrt(3))^n.
a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3).
G.f.: (3-8*x+2*x^2)/((1-2*x)*(1-2*x-2*x^2)).

A328286 Expansion of e.g.f. -log(1 - x - x^2/2).

Original entry on oeis.org

1, 2, 5, 21, 114, 780, 6390, 61110, 667800, 8210160, 112152600, 1685237400, 27624920400, 490572482400, 9381882510000, 192238348302000, 4201639474032000, 97572286427616000, 2399151995223984000, 62268748888378032000, 1701213856860117600000
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 11 2019

Keywords

Crossrefs

Cf. A009014, A039647, A080040, A080599 (exponential transform).

Programs

  • Maple
    b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
      binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 11 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ Michel Marcus, Oct 11 2019

Formula

a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.
D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
Previous Showing 31-37 of 37 results.

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