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.

Showing 1-5 of 5 results.

A107732 Column 1 of the array in A107735.

Original entry on oeis.org

1, 2, 5, 4, 21, 8, 85, 16, 341, 32, 1365, 64, 5461, 128, 21845, 256, 87381, 512, 349525, 1024, 1398101, 2048, 5592405, 4096, 22369621, 8192, 89478485, 16384, 357913941, 32768, 1431655765, 65536, 5726623061, 131072, 22906492245, 262144, 91625968981, 524288, 366503875925
Offset: 3

Views

Author

N. J. A. Sloane, Jun 10 2005

Keywords

References

  • S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

Links

Crossrefs

Cf. A000079, A001045, A002450, A107735.

Programs

  • Mathematica
    Table[(3 (1 + (-1)^n) 2^(n/2) - (1 - (-1)^n) (2 - 2^n))/12, {n, 3, 50}] (* Bruno Berselli, Mar 26 2019 *)
  • PARI
    Vec(x^3*(1 + 2*x - 2*x^2 - 10*x^3 + 8*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)) + O(x^40)) \\ Colin Barker, Mar 26 2019
  • SageMath
    def a(n): return (2^n-2)//6 if is_odd(n) else 2^(n//2-1)
print([a(n) for n in (3..41)]) # Peter Luschny, Mar 26 2019

Formula

a(2*k+2) = 2^k = A000079(k), a(2*k+1) = (4^k-1)/3 = A002450(k) = A001045(2*k).
a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n > 8. - Chai Wah Wu, Jun 19 2016
G.f.: x^3*(1 + 2*x - 2*x^2 - 10*x^3 + 8*x^5)/(1 - 7*x^2 + 14*x^4 - 8*x^6). - Chai Wah Wu, Jun 19 2016
a(n) = (3*(1 + (-1)^n)*2^(n/2) - (1 - (-1)^n)*(2 - 2^n))/12. - Colin Barker, Mar 26 2019
a(n) = (2^n - 2)/6 if n is odd else 2^(n/2 - 1). - Peter Luschny, Mar 26 2019

Extensions

More terms from Chai Wah Wu, Jun 19 2016

A107731 Row 7 of the array in A107735.

Original entry on oeis.org

1, 21, 141, 521, 1401, 3101, 6021, 10641, 17521, 27301, 40701, 58521, 81641, 111021, 147701, 192801, 247521, 313141, 391021, 482601, 589401, 713021, 855141, 1017521, 1202001, 1410501, 1645021, 1907641, 2200521, 2525901, 2886101, 3283521
Offset: 0

Views

Author

N. J. A. Sloane, Jun 10 2005

Keywords

References

  • S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

Programs

  • Mathematica
    nn:=31; CoefficientList[Series[(1 + 16*x + 46*x^2 + 16*x^3 + x^4)/(1-x
)^5, {x,0,nn}], x] (* Georg Fischer, Apr 17 2020 *)
  • PARI
    {a(n)= 1+ (n^2+n)* (n^2+n+1)* 10/3} /* Michael Somos, Mar 20 2007 */

Formula

a(n) = 1 + (n^2+n)*(n^2+n+1)*10/3. a(-1-n)= a(n). - Michael Somos, Mar 20 2007
G.f.: (1 + 16*x + 46*x^2 + 16*x^3 + x^4)/(1-x)^5. - Michael Somos, Mar 20 2007; corrected by Georg Fischer, Apr 17 2020

Extensions

More terms from Michael Somos, Mar 20 2007

A107733 Column 2 of the array in A107735.

Original entry on oeis.org

1, 3, 13, 11, 141, 43, 1485, 171, 15565, 683, 163021, 2731, 1707213, 10923, 17878221, 43691, 187223245, 174763, 1960627405, 699051, 20531956941, 2796203, 215013444813, 11184811, 2251650026701, 44739243, 23579585203405, 178956971, 246928622013645, 715827883, 2585870100909261, 2863311531
Offset: 3

Views

Author

N. J. A. Sloane, Jun 10 2005

Keywords

Comments

The second bisection [3, 11, 43, 171, 683, ...] is A007583. - Jean-François Alcover, Oct 22 2019 [noticed by Paul Curtz in a private e-mail].

References

  • S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

Links

Crossrefs

Cf. A007583, A107735.

Programs

  • Mathematica
    LinearRecurrence[{0, 17, 0, -80, 0, 128, 0, -64}, {1, 3, 13, 11, 141, 43, 1485, 171}, 32] (* Jean-François Alcover, Oct 22 2019 *)

Formula

a(n) = 1 + Sum_{j=1..g} 2^(2j-1) if n = 2g+2, = 1 + 4 Sum_{j=1..g} C(2g+1, 2j) 5^(j-1) if n = 2g+1.
From Chai Wah Wu, Jun 19 2016: (Start)
a(n) = 17*a(n-2) - 80*a(n-4) + 128*a(n-6) - 64*a(n-8) for n > 10.
G.f.: x^3*(-64*x^7 + 96*x^5 - 40*x^3 - 4*x^2 + 3*x + 1)/(64*x^8 - 128*x^6 + 80*x^4 - 17*x^2 + 1). (End)

Extensions

More terms from Emeric Deutsch, Jun 22 2005

A107734 Column 3 of the array in A107735.

Original entry on oeis.org

1, 4, 25, 24, 521, 160, 10569, 1088, 213577, 7424, 4313673, 50688, 87118409, 346112, 1759416905, 2363392, 35532608073, 16138240, 717604819529, 110198784, 14492509704777, 752484352, 292685934269001, 5138284544, 5910988354523721, 35086401536, 119376366388154953
Offset: 3

Views

Author

N. J. A. Sloane, Jun 10 2005

Keywords

References

  • Shigeru Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

Crossrefs

Cf. A107732, A107733, A107735.

Extensions

More terms from Amiram Eldar, May 03 2024

A006527 a(n) = (n^3 + 2*n)/3.

Original entry on oeis.org

0, 1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531, 28424, 30405
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of ways to color vertices (or edges) of a triangle using <= n colors, allowing only rotations.
Also: dot_product (1,2,...,n)*(2,3,...,n,1), n >= 0. - Clark Kimberling
Start from triacid and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - Robert G. Wilson v, Aug 02 2002
Starting with offset 1 = row sums of triangle A158822 and binomial transform of (1, 3, 4, 2, 0, 0, 0, ...). - Gary W. Adamson, Mar 28 2009
One-ninth of sum of three consecutive cubes: a(n) = ((n-1)^3 + n^3 + (n+1)^3)/9. - Zak Seidov, Jul 22 2013
For n > 2, number of different cubes, formed after splitting a cube in color C_1, by parallel planes in the colors C_2, C_3, ..., C_n in three spatial dimensions (in the order of the colors from a fixed vertex). Generally, in a large hypercube n^d is f(n,d) = C(n+d-1, d) + C(n, d) different small hypercubes. See below for my formula a(n) = f(n,3). - Thomas Ordowski, Jun 15 2014
a(n) is a square for n = 1, 2 & 24; and for no other values up to 10^7 (see M. Gardner). - Michel Marcus, Sep 06 2015
Number of unit tetrahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb. - Jason Pruski, Aug 23 2017

References

  • M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
  • S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Column 1 of triangle A094414. Row 6 of the array in A107735.
Cf. A002264, A005843, A054602, A059100, A135184, A158822, A177342.
Cf. A000292 (unoriented), A000292(n-2) (chiral), A000290 (achiral) triangle colorings.
Row 2 of A324999 (simplex vertices and facets) and A327083 (simplex edges and ridges).

Programs

  • Haskell
    a006527 n = n * (n ^ 2 + 2) `div` 3  -- Reinhard Zumkeller, Jan 06 2014
  • Magma
    [(n^3 + 2*n)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011
  • Maple
    A006527:=z*(1+z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat):seq(lcm(fibonacci(4,n),fibonacci(2,n))/3,n=0..42); # Zerinvary Lajos, Apr 20 2008
  • Mathematica
    Table[ (n^3 + 2*n)/3, {n, 0, 45} ]
LinearRecurrence[{4,-6,4,-1},{0,1,4,11},46] (* or *) CoefficientList[ Series[(x+x^3)/(x-1)^4,{x,0,49}],x] (* Harvey P. Dale, Jun 13 2011 *)
  • PARI
    a(n)=n*(n^2+2)/3 \\ Charles R Greathouse IV, Jul 25 2011

Formula

a(0)=0, a(1)=1, a(2)=4, a(3)=11; for n > 3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jun 13 2011
From Paul Barry, Mar 13 2003: (Start)
a(n) = 2*binomial(n+1, 3) + binomial(n, 1).
G.f.: x*(1+x^2)/(1-x)^4. (End)
a(n) = A000292(n) + A000292(n-2). - Alexander Adamchuk, May 20 2006
a(n) = n*A059100(n)/3. - Lekraj Beedassy, Feb 06 2007
a(n) = A054602(n)/3. - Zerinvary Lajos, Apr 20 2008
a(n) = ( n + Sum_{i=1..n} A177342(i) )/(n+1), with n > 0. - Bruno Berselli, May 19 2010
a(n) = A002264(A000578(n) + A005843(n)). - Reinhard Zumkeller, Jun 16 2011
a(n) = binomial(n+2, 3) + binomial(n, 3). - Thomas Ordowski, Jun 15 2014
a(n) = A000292(n) - A000292(-n). - Bruno Berselli, Sep 22 2016
E.g.f.: (x/3)*(3 + 3*x + x^2)*exp(x). - G. C. Greubel, Sep 01 2017
From Robert A. Russell, Oct 20 2020: (Start)
a(n) = 1*C(n,1) + 2*C(n,2) + 2*C(n,3), where the coefficient of C(n,k) is the number of oriented triangle colorings using exactly k colors.
a(n) = 2*A000292(n) - A000290(n) = 2*A000292(n-2) + A000290(n). (End)
Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i*sqrt(2)) + polygamma(0, 1+i*sqrt(2)))/4 = 1.45245201414472469745354677573358867... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

Extensions

More terms from Alexander Adamchuk, May 20 2006
Corrected and replaced 5th formula from Harvey P. Dale, Jun 13 2011
Deleted an erroneous comment. - N. J. A. Sloane, Dec 10 2018
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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