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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A005685 Number of Twopins positions.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 16, 26, 40, 65, 101, 163, 257, 416, 663, 1073, 1719, 2781, 4472, 7236, 11664, 18873, 30465, 49293, 79641, 128862, 208315, 337061, 545071, 881943, 1426520, 2308158, 3733880, 6041545, 9774133
Offset: 4

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Author

N. J. A. Sloane

Keywords

Comments

The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0) = 0, a(1) = 1, a(2) = 1 and a(3) = 1. The formula for a(n) confirms these values. - Johannes W. Meijer, Aug 24 2013

References

  • R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • Maple
    A005685 := -(-1-z**3+2*z**4+z**2+z**5+z**6+z**7)/(z**2-z+1)/(z**2+z-1)/(z**4+z**2-1);
  • PARI
    a(n)=(2*fibonacci(floor((n+1)/2))+fibonacci(n)+[0,1,1,0,-1,-1][(n%6)+1])/4; /* Ralf Stephan, Aug 25 2013 */

Formula

G.f.: (-x^4*(x^7+x^6+x^5+2*x^4-x^3+x^2-1))/((x^4+x^2-1)*(x^2-x+1)*(x^2+x-1)). - Conjectured by Simon Plouffe in his 1992 dissertation.
a(n) = Sum_{k=0..floor((n-1)/2)} A102541(n-1, 2*k), n >= 4. - Johannes W. Meijer, Aug 24 2013
a(n) = (1/4) * (2*F(floor((n+1)/2)) + F(n) + A010892(n-1)), with F(n) = A000045(n) the Fibonacci numbers. - Ralf Stephan, from Plouffe's g.f. Aug 25 2013

Extensions

More terms from Johannes W. Meijer, Aug 24 2013

A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

Original entry on oeis.org

1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10
Offset: 1

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Author

Ctibor O. Zizka, Mar 11 2012

Keywords

Comments

OEIS contains a lot of similar sequences, for example A152204, A122196, A173284.
Row sums for this sequence gives A006578.
In general, by given triangle with (A-B,2*A-B,...,A*n-B,...) in every column, shifted down K-times, we have the row sum s(n)= A*(n*n+K*n+nmodK)/(2*K) - B*(n+nmodK)/K. In this sequence K=2,A=3,B=2, in A152204 K=2,A=2,B=1.
No triangle with primes in every column, shifted down by K>=2 in OEIS, no row sums of it in OEIS.
From Johannes W. Meijer, Sep 28 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A143971.
The alternating row sums equal A004524(n+2) + 2*A004524(n+1).
The antidiagonal sums equal A171452(n+1). (End)

Examples

			Triangle:
1
4
7,  1
10, 4
13, 7,  1
16, 10, 4
19, 13, 7,  1
22, 16, 10, 4
25, 19, 13, 7,  1
28, 22, 16, 10, 4
...

Crossrefs

Cf. A008315, A011973, A102541, A122196, A122197, A128099, A152198, A152204, A173284, A207538.
Cf. (Related to triangle sums) A006578, A000217, A002620, A004524, A171452.

Programs

  • Maple
    T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013

Formula

From Johannes W. Meijer, Sep 28 2013: (Start)
T(n, k) = 3*n - 6*k + 4, n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A143971(n-k+1, k), n >= 1 and 1 <= k <= floor((n+1)/2). (End)

A362260 Maximum over 0 <= k <= n/2 of the number of permutations of two symbols occurring k and n-2*k times, respectively, where a permutation and its reversal are counted only once.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 9, 12, 19, 28, 44, 66, 110, 170, 255, 396, 651, 1001, 1519, 2520, 4032, 6216, 9752, 15912, 25236, 38760, 63090, 101850, 160050, 248710, 408760, 653752, 1021735, 1634776, 2656511, 4218786, 6562556, 10737090, 17299646, 27313650, 43249115
Offset: 0

Views

Author

Pontus von Brömssen, Apr 15 2023

Keywords

Comments

Also, a(n) is the maximum number of ways in which a set of integer-sided squares can tile an n X 2 rectangle, up to rotations and reflections.

Examples

			For n = 8, the maximum a(8) = 9 is obtained for k = 2. The corresponding permutations of 2 2's and 4 1's are 221111, 212111, 211211, 211121, 211112, 122111, 121211, 121121, and 112211.

Links

Crossrefs

Row maxima of A102541.
Second column of A362258.
Cf. A001224, A073028, A361224 (rectangular pieces).

Programs

  • Maple
    f:= proc(n) local k, v, m,w;
  m:= 0:
  for k from 0 to n/2 do
    v:= binomial(n-k,k);
    if n:: even and k::even then w:= binomial((n-k)/2,k/2)
    elif (n-k)::odd then w:=binomial((n-k-1)/2, floor(k/2))
    else w:= 0
    fi;
    m:= max(m,(v+w)/2);
  od;
  m
end proc:
map(f, [$0..50]); # Robert Israel, Oct 25 2023

Formula

a(n) >= A073028(n)/2.

A005688 Numbers of Twopins positions.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 20, 30, 45, 69, 104, 157, 236, 356, 540, 821, 1252, 1908, 2909, 4434, 6762, 10319, 15755, 24066, 36766, 56176, 85837, 131172, 200471, 306410, 468371, 715975, 1094516, 1673232, 2557997, 3910683
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

Comments

The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 1 and a(4) =1. The formula for a(n) confirms these values. - Johannes W. Meijer, Aug 24 2013

References

  • R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • Mathematica
    LinearRecurrence[{2,0,-2,1,2,-2,0,0,0,-1},{1,2,3,5,7,10,14,20,30,45},40] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: (x^5*(1-x^2+x^3-2*x^5-x^6-x^7-x^8-x^9))/((1-x^2-x^5)*(1-2*x+x^2-x^5)). - Ralf Stephan, Apr 22 2004
a(n) = sum(A102541(n-k-1, 2*k), k=0..floor((n-1)/3)), n >= 5. - Johannes W. Meijer, Aug 24 2013

Extensions

More terms from Johannes W. Meijer, Aug 24 2013

A257523 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 4, 1, 6, 6, 1, 6, 14, 1, 8, 28, 1, 8, 44, 1, 10, 66, 20, 1, 10, 90, 64, 1, 12, 120, 168, 1, 12, 152, 320, 1, 14, 190, 572, 72, 1, 14, 230, 896, 328, 1, 16, 276, 1360, 984, 1, 16, 324, 1920, 2264, 1, 18, 378, 2660, 4528, 272
Offset: 4

Views

Author

Christopher Hunt Gribble, Apr 27 2015

Keywords

Examples

			The first 9 rows of T(n,k) are:
.\ k    0      1      2     3
n
4       1      2
5       1      2
6       1      4
7       1      4
8       1      6      6
9       1      6     14
10      1      8     28
11      1      8     44
12      1     10     66    20
13      1     10     90    64
14      1     12    120   168
15      1     12    152   320

Links

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592

Programs

  • PARI
    T(n,k)={(4^k*binomial(n-3*k,k) + ((n%2==0||k%2==0)+(k%2==0)+(k==0)) * 4^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=4,15,for(k=0,(n\4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms a(24) and beyond by Andrew Howroyd, May 29 2017
Previous Showing 31-35 of 35 results.

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

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