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.

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A244352 a(n) = Pell(n)^3 - Pell(n)^2, where Pell(n) is the n-th Pell number (A000129).

Original entry on oeis.org

0, 0, 4, 100, 1584, 23548, 338100, 4798248, 67750848, 954701400, 13441659268, 189185124940, 2662308356400, 37463104912660, 527155118240244, 7417689205890000, 104375121328998144, 1468671237346368048, 20665783224031936900, 290789699203441908148
Offset: 0

Views

Author

Colin Barker, Jun 26 2014

Keywords

Examples

			a(3) = Pell(3)^3 - Pell(3)^2 = 5^3 - 5^2 = 100.

Links

Crossrefs

Cf. A000129, A079291, A110272.

Programs

  • Magma
    Pell:= func< n | n eq 0 select 0 else Evaluate(DicksonSecond(n-1,-1),2) >;
[Pell(n)^3 - Pell(n)^2: n in [0..40]]; // G. C. Greubel, Aug 20 2022
  • Mathematica
    CoefficientList[Series[4*x^2*(3*x^3-4*x^2+8*x+1) / ((x+1)*(x^2-6*x+1)*(x^2-2*x-1)*(x^2+14*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 26 2014 *)
  • PARI
    pell(n) = round(((1+sqrt(2))^n-(1-sqrt(2))^n)/(2*sqrt(2)))
vector(50, n, pell(n-1)^3-pell(n-1)^2)
  • SageMath
    def Pell(n): return lucas_number1(n,2,-1)
[Pell(n)^3 -Pell(n)^2 for n in (0..40)] # G. C. Greubel, Aug 20 2022

Formula

a(n) = A110272(n) - A079291(n).
G.f.: 4*x^2*(1+8*x-4*x^2+3*x^3) / ((1+x)*(1-6*x+x^2)*(1+2*x-x^2)*(1-14*x-x^2)).
a(n) = A045991(A000129(n)). - Michel Marcus, Jun 26 2014

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Original entry on oeis.org

-2, 3, 20, 55, 114, 203, 328, 495, 710, 979, 1308, 1703, 2170, 2715, 3344, 4063, 4878, 5795, 6820, 7959, 9218, 10603, 12120, 13775, 15574, 17523, 19628, 21895, 24330, 26939, 29728, 32703, 35870, 39235, 42804, 46583, 50578, 54795, 59240, 63919, 68838, 74003, 79420, 85095
Offset: 0

Views

Author

Bruno Berselli, Sep 04 2018

Keywords

Comments

First differences are in A004538.
a(n) is divisible by 11 for n = 3, 7, 9, 14, 18, 20, 25, 29, 31, 36, 40, ... with formula (1/3)*(11*m + (1 + (m mod 3))*(-1)^((m-1) mod 3) + 8), m >= 0.

Links

Crossrefs

Cf. A004538.
Subsequence of A047216.
Similar sequences (see Table in Links section): A011379, A027444, A033445, A034262, A045991, A069778.

Programs

  • GAP
    List([0..50], n -> (n+2)*(n^2+n-1));
  • Julia
    [(n+2)*(n^2+n-1) for n in 0:50] |> println
  • Magma
    [(n+2)*(n^2+n-1): n in [0..50]];
  • Maple
    seq((n+2)*(n^2+n-1),n=0..43); # Paolo P. Lava, Sep 04 2018
  • Mathematica
    Table[(n + 2) (n^2 + n - 1), {n, 0, 50}]
  • Maxima
    makelist((n+2)*(n^2+n-1), n, 0, 50);
  • PARI
    vector(50, n, n--; (n+2)*(n^2+n-1))
  • Python
    [(n+2)*(n**2+n-1) for n in range(50)]
  • Sage
    [(n+2)*(n^2+n-1) for n in (0..50)]

Formula

O.g.f.: (-2 + 11*x - 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-2 + 5*x + 6*x^2 + x^3)*exp(x).
a(n) = -A033445(-n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

A384125 Array read by antidiagonals: T(n,m) is the number of edges in the n X m rook graph K_n X K_m.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 6, 9, 9, 6, 10, 16, 18, 16, 10, 15, 25, 30, 30, 25, 15, 21, 36, 45, 48, 45, 36, 21, 28, 49, 63, 70, 70, 63, 49, 28, 36, 64, 84, 96, 100, 96, 84, 64, 36, 45, 81, 108, 126, 135, 135, 126, 108, 81, 45, 55, 100, 135, 160, 175, 180, 175, 160, 135, 100, 55
Offset: 1

Views

Author

Andrew Howroyd, May 20 2025

Keywords

Examples

			Array begins:
=======================================
n\m |  1  2   3   4   5   6   7   8 ...
----+----------------------------------
  1 |  0  1   3   6  10  15  21  28 ...
  2 |  1  4   9  16  25  36  49  64 ...
  3 |  3  9  18  30  45  63  84 108 ...
  4 |  6 16  30  48  70  96 126 160 ...
  5 | 10 25  45  70 100 135 175 220 ...
  6 | 15 36  63  96 135 180 231 288 ...
  7 | 21 49  84 126 175 231 294 364 ...
  8 | 28 64 108 160 220 288 364 448 ...
  ...

Links

Crossrefs

Main diagonal is A045991.
Columns 1..6 are A000217(n-1), A000290, A045943, A054000, A269457(n-1), A067707.
Cf. A003991 (number of vertices), A360855 (triangles), A384120 (all cliques).

Programs

  • Mathematica
    Table[#*Binomial[m, 2] + m*Binomial[#, 2] &[n - m + 1], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, May 22 2025 *)
  • PARI
    T(n,m) = n*binomial(m,2) + m*binomial(n,2)

Formula

T(n,m) = n*binomial(m,2) + m*binomial(n,2).
T(n,m) = binomial(n*m,2) - 2*binomial(n,2)*binomial(m,2).
T(n,m) = T(m,n).

A133062 a(n) = 5*p^5 - 3*p^3 + 2*p^2 with p = prime(n).

Original entry on oeis.org

144, 1152, 15300, 83104, 801504, 1850212, 7085124, 12360640, 32146272, 102484260, 143058304, 346570564, 579077604, 734807392, 1146417984, 2090536452, 3574012320, 4222308004, 6749732224, 9020083104, 10364201572, 15383815360, 19693501632, 27918198180, 42933982084, 52547432004
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=83104 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035 - 1029 + 98 = 83104.

Crossrefs

Cf. A045991, A133071.

Programs

  • Magma
    [5*p^5-3*p^3+2*p^2: p in PrimesUpTo(200)] // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    5#^5-3#^3+2#^2&/@Prime[Range[30]] (* Harvey P. Dale, Mar 07 2017 *)

Formula

a(n) = 5*A050997(n) - 3*A030078(n) + 2*A001248(n) = A134631(A000040(n)).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133064 a(n) = 5*p^5 + 3*p^3 + 2*p^2, where p = prime(n).

Original entry on oeis.org

192, 1314, 16050, 85162, 809490, 1863394, 7114602, 12401794, 32219274, 102630594, 143237050, 346874482, 579491130, 735284434, 1147040922, 2091429714, 3575244594, 4223669890, 6751536802, 9022230570, 10366535674, 15386773594, 19696932354, 27922427994, 42939458122, 52553613810
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=85162 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035 + 1029 + 98 = 85162.

Crossrefs

Cf. A000290, A000578, A000584, A045991, A133073, A000040 (prime numbers).

Programs

  • Magma
    [5*p^5+3*p^3+2*p^2: p in PrimesUpTo(200)] // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    5#^5+3#^3+2#^2&/@Prime[Range[30]] (* Harvey P. Dale, Dec 17 2011 *)
  • PARI
    a(n) = my(p=prime(n)); 5*p^5 + 3*p^3 + 2*p^2; \\ Michel Marcus, Mar 11 2022

Formula

a(n) = 5*prime(n)^5 + 3*prime(n)^3 + 2*prime(n)^2, where prime(n)= A000040(n).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010
Previous Showing 61-65 of 65 results.

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