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 1-9 of 9 results.

A004277 1 together with positive even numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also number of non-attacking bishops on n X n board. - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002
Engel expansion of e^(1/2) (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Numbers n such that a 2n-group (i.e., a group of order 2n) has subgroup C_2. - Lekraj Beedassy, Oct 14 2004
Image of 1/(1-2x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
Position of n in A113322: A113322(a(n-1)) = n for n>0. - Reinhard Zumkeller, Oct 26 2005
Incrementally largest terms in the continued fraction for e. - Nick Hobson, Jan 11 2007
Conjecturally, the differences of two consecutive primes (without repetition). - Juri-Stepan Gerasimov, Nov 09 2009
Equals (1, 2, 2, 2, ...) convolved with (1, 0, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Mar 03 2010
a(n) is the number of 0-dimensional elements (vertices) in an n-cross polytope. - Patrick J. McNab, Jul 06 2015
Numbers k such that in the symmetric representation of sigma(k) there is no pair bars as its ends (Cf. A237593). - Omar E. Pol, Sep 28 2018
Also, the coordination sequence of the L-lattice (see A332419). - Sean A. Irvine, Jul 29 2020

Links

Crossrefs

Cf. A004275, A008486, A030978, A097134.
INVERT transformation yields A098182 without A098182(0). - R. J. Mathar, Sep 11 2008

Programs

Formula

G.f.: (1+x^2)/(1-x)^2. - Paul Barry, Feb 28 2003
Inverse binomial transform of Cullen numbers A002064. a(n)=2n+0^n. - Paul Barry, Jun 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1)*(-1)^k*2^(n-2k). - Paul Barry, Jan 16 2005
Equals binomial transform of [1, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 15 2008
E.g.f.: 1+x*sinh(x) (aerated sequence). - Paul Barry, Oct 11 2009
a(n) = 0^n + 2*n = A000007(n) + A005843(n). - Reinhard Zumkeller, Jan 11 2012

Extensions

Corrected by Charles R Greathouse IV, Mar 18 2010

A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases.

Original entry on oeis.org

8, 25, 16, 56, 69, 32, 105, 194, 191, 64, 176, 435, 676, 529, 128, 273, 846, 1817, 2356, 1465, 256, 400, 1491, 4108, 7587, 8210, 4057, 512, 561, 2444, 8239, 19930, 31677, 28610, 11235, 1024, 760, 3789, 15128, 45465, 96690, 132263, 99700, 31113, 2048
Offset: 1

Views

Author

R. H. Hardin Nov 23 2011

Keywords

Comments

Table starts
....8.....25......56......105.......176........273........400.........561
...16.....69.....194......435.......846.......1491.......2444........3789
...32....191.....676.....1817......4108.......8239......15128.......25953
...64....529....2356.....7587.....19930......45465......93472......177381
..128...1465....8210....31677.....96690.....250913.....577660.....1212729
..256...4057...28610...132263....469116....1384813....3570086.....8291391
..512..11235...99700...552247...2276028....7642875...22063924....56687801
.1024..31113..347434..2305835..11042700...42181611..136360286...387572529
.2048..86161.1210736..9627715..53576350..232803603..842739040..2649819955
.4096.238605.4219166.40199277.259938722.1284861277.5208328180.18116728573

Examples

			Some solutions for n=4 k=3
..1....2....3....0....1....1....2....1....3....3....3....1....2....0....1....1
..0....0....0....2....1....0....3....3....1....3....0....3....2....3....1....0
..0....0....2....2....0....3....0....0....1....2....1....3....2....0....1....1
..3....0....1....3....3....3....3....2....1....2....0....1....2....0....0....1
..3....3....3....0....3....0....1....2....1....1....3....3....3....2....2....3
..1....3....2....0....1....3....3....2....2....1....0....1....2....1....1....0

Links

Crossrefs

Column 1 is A000079(n+2)
Column 2 is A098182(n+3)
Row 1 is A131423(n+1)

Formula

Empirical for columns:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2) +a(n-3)
k=3: a(n) = 4*a(n-1) -2*a(n-2) +a(n-3) -a(n-4)
k=4: a(n) = 5*a(n-1) -4*a(n-2) +3*a(n-3) -3*a(n-4) +a(n-5) -a(n-6)
k=5: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7)
k=6: a(n) = 7*a(n-1) -9*a(n-2) +6*a(n-3) -9*a(n-4) +7*a(n-5) -7*a(n-6) +5*a(n-7) -2*a(n-8) +a(n-9)
k=7: a(n) = 8*a(n-1) -12*a(n-2) +6*a(n-3) -10*a(n-4) +12*a(n-5) -11*a(n-6) +11*a(n-7) -6*a(n-8) +3*a(n-9) -a(n-10)
Empirical for rows:
n=1: a(k) = (2/3)*k^3 + 3*k^2 + (10/3)*k + 1
n=2: a(k) = (5/12)*k^4 + (19/6)*k^3 + (79/12)*k^2 + (29/6)*k + 1
n=3: a(k) = (4/15)*k^5 + (17/6)*k^4 + (28/3)*k^3 + (73/6)*k^2 + (32/5)*k + 1
n=4: a(k) = (61/360)*k^6 + (93/40)*k^5 + (779/72)*k^4 + (521/24)*k^3 + (1801/90)*k^2 + (239/30)*k + 1
n=5: a(k) = (34/315)*k^7 + (163/90)*k^6 + (1981/180)*k^5 + (557/18)*k^4 + (7807/180)*k^3 + (1361/45)*k^2 + (333/35)*k + 1
n=6: a(k) = (277/4032)*k^8 + (1375/1008)*k^7 + (4933/480)*k^6 + (2723/72)*k^5 + (14161/192)*k^4 + (11197/144)*k^3 + (216211/5040)*k^2 + (929/84)*k + 1
n=7: a(k) = (124/2835)*k^9 + (1123/1120)*k^8 + (244/27)*k^7 + (1991/48)*k^6 + (57133/540)*k^5 + (74183/480)*k^4 + (291427/2268)*k^3 + (9739/168)*k^2 + (568/45)*k + 1

A206863 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z z+1 z in any row, column, diagonal or antidiagonal.

Original entry on oeis.org

3, 9, 9, 25, 81, 25, 69, 625, 625, 69, 191, 4761, 11092, 4761, 191, 529, 36481, 192575, 192575, 36481, 529, 1465, 279841, 3425501, 7598355, 3425501, 279841, 1465, 4057, 2146225, 61164640, 314918295, 314918295, 61164640, 2146225, 4057, 11235
Offset: 1

Views

Author

R. H. Hardin Feb 13 2012

Keywords

Comments

Table starts
....3........9..........25.............69...............191
....9.......81.........625...........4761.............36481
...25......625.......11092.........192575...........3425501
...69.....4761......192575........7598355.........314918295
..191....36481.....3425501......314918295.......31208037240
..529...279841....61164640....13093461017.....3092914364659
.1465..2146225..1090023439...541018725304...303316860005399
.4057.16459249.19411973287.22336434688428.29725367062521082

Examples

			Some solutions for n=4 k=3
..2..0..2....2..1..2....1..2..2....0..0..0....0..1..2....1..1..2....2..0..1
..1..1..1....1..2..2....1..1..1....2..0..2....1..2..0....1..1..1....2..0..2
..1..1..0....0..2..1....1..0..1....2..0..1....1..2..0....0..1..2....1..0..2
..1..2..2....1..1..2....2..1..2....0..0..0....2..0..2....2..1..2....1..2..2

Links

Crossrefs

Column 1 is A098182(n+1)

A206957 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z z+1 z in any row, column or nw-to-se diagonal.

Original entry on oeis.org

3, 9, 9, 25, 81, 25, 69, 625, 625, 69, 191, 4761, 11804, 4761, 191, 529, 36481, 218637, 218637, 36481, 529, 1465, 279841, 4107593, 9803278, 4107593, 279841, 1465, 4057, 2146225, 77359677, 449797912, 449797912, 77359677, 2146225, 4057, 11235
Offset: 1

Views

Author

R. H. Hardin Feb 13 2012

Keywords

Comments

Table starts
....3........9..........25.............69...............191
....9.......81.........625...........4761.............36481
...25......625.......11804.........218637...........4107593
...69.....4761......218637........9803278.........449797912
..191....36481.....4107593......449797912.......50967712088
..529...279841....77359677....20694430627.....5791955969105
.1465..2146225..1455935650...950821989599...656572043220119
.4057.16459249.27394495918.43676385937074.74406813170700252

Examples

			Some solutions for n=4 k=3
..0..0..2....2..2..1....1..2..2....1..0..1....0..1..2....2..2..1....2..2..2
..0..0..2....2..0..2....0..0..1....2..0..2....1..2..0....2..0..0....1..2..2
..0..0..0....2..2..0....1..2..0....0..2..2....1..2..0....2..0..0....0..2..2
..1..2..0....0..2..1....1..2..2....2..0..2....2..0..2....2..2..2....0..0..2

Links

Crossrefs

Column 1 is A098182(n+1)
Column 2 is A206857

A207221 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z z+1 z in any row or column.

Original entry on oeis.org

3, 9, 9, 25, 81, 25, 69, 625, 625, 69, 191, 4761, 12534, 4761, 191, 529, 36481, 247163, 247163, 36481, 529, 1465, 279841, 4916282, 12584703, 4916282, 279841, 1465, 4057, 2146225, 97935886, 647820815, 647820815, 97935886, 2146225, 4057, 11235
Offset: 1

Views

Author

R. H. Hardin Feb 16 2012

Keywords

Comments

Table starts
....3........9..........25.............69................191
....9.......81.........625...........4761..............36481
...25......625.......12534.........247163............4916282
...69.....4761......247163.......12584703..........647820815
..191....36481.....4916282......647820815........86510224720
..529...279841....97935886....33403462497.....11574609426324
.1465..2146225..1950327875..1721695415743...1547869995556777
.4057.16459249.38835489653.88730578258613.206971107381598775

Examples

			Some solutions for n=4 k=3
..2..0..1....1..2..2....2..2..0....2..0..2....1..1..2....2..1..0....2..1..2
..2..0..2....0..0..1....0..0..2....1..1..1....1..1..1....2..1..2....1..2..2
..2..0..2....1..2..0....2..0..0....1..1..0....0..1..2....2..1..1....0..2..1
..2..0..1....1..2..2....1..0..1....1..2..2....2..1..2....0..1..1....1..1..2

Links

Crossrefs

Column 1 is A098182(n+1)
Column 2 is A206857

A207989 T(n,k)=Number of nXk 0..2 arrays avoiding the pattern z z+1 z horizontally and z z-1 z vertically.

Original entry on oeis.org

3, 9, 9, 25, 81, 25, 69, 625, 625, 69, 191, 4761, 12468, 4761, 191, 529, 36481, 244325, 244325, 36481, 529, 1465, 279841, 4827482, 12271227, 4827482, 279841, 1465, 4057, 2146225, 95527856, 622569385, 622569385, 95527856, 2146225, 4057, 11235
Offset: 1

Views

Author

R. H. Hardin Feb 22 2012

Keywords

Comments

Table starts
....3........9..........25.............69................191
....9.......81.........625...........4761..............36481
...25......625.......12468.........244325............4827482
...69.....4761......244325.......12271227..........622569385
..191....36481.....4827482......622569385........81269646503
..529...279841....95527856....31638012071.....10629041648848
.1465..2146225..1889730415..1607243845009...1389619561706920
.4057.16459249.37378616643.81641053794111.181656690367717549

Examples

			Some solutions for n=4 k=3
..2..2..0....2..1..0....1..2..2....2..1..0....0..0..0....2..1..2....1..2..2
..0..0..2....2..1..2....0..0..1....0..1..1....2..0..2....1..2..2....2..1..2
..2..0..0....2..1..1....2..2..0....1..0..2....2..0..1....0..2..1....2..0..1
..1..0..1....0..1..1....1..2..2....1..2..2....0..0..0....2..1..0....2..2..1

Links

Crossrefs

Column 1 is A098182(n+1)
Column 2 is A206857

A102620 Number of legal Go positions on a 1 X n board (for which 3^n is a trivial upper bound).

Original entry on oeis.org

1, 5, 15, 41, 113, 313, 867, 2401, 6649, 18413, 50991, 141209, 391049, 1082929, 2998947, 8304961, 22998865, 63690581, 176377839, 488441801, 1352638145, 3745850473, 10373355075, 28726852897, 79553054089, 220305664445, 610090792143, 1689519766073, 4678774170521, 12956893537633, 35881426208451, 99366159258241, 275173945103905, 762037102261925, 2110303520940111
Offset: 1

Views

Author

John Tromp, Jan 31 2005

Keywords

Examples

			a(2)=5 because .. .O .S O. S. are the 5 legal 1 X 2 Go positions, while OO OS SO SS are all illegal, having stones without liberties.

Links

Crossrefs

Cf. A094777.
Cf. A030236, A098182.

Programs

  • Mathematica
    LinearRecurrence[{3,-1,1},{1,5,15},40] (* Harvey P. Dale, Sep 16 2016 *)
  • Maxima
    makelist(sum((2^k)*(binomial(n+k+1,3*k+2)+2*binomial(n+k,3*k+2)+binomial(n+k-1,3*k+2)),k,0,(n-1)/2),n,0,24); /* Emanuele Munarini, Apr 17 2013 */
  • PARI
    Vec(x*(1+x)^2/((1-x)^3-2*x^2)+O(x^66)) \\ Joerg Arndt, Apr 17 2013

Formula

For n >= 4, a(n) = 3*a(n-1) - a(n-2) + a(n-3).
G.f.: x(1+x)^2/((1-x)^3-2x^2). - Josh Simmons (jsimmons10(AT)my.whitworth.edu), May 06 2010
a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * (binomial(n+k+1,3*k+2) + 2*binomial(n+k,3*k+2) + binomial(n+k-1,3*k+2)). - Emanuele Munarini, Apr 17 2013

Extensions

More terms from Joerg Arndt, Apr 17 2013

A098183 a(n) = 3*a(n-1) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 1, 4, 13, 40, 124, 385, 1195, 3709, 11512, 35731, 110902, 344218, 1068385, 3316057, 10292389, 31945552, 99152713, 307750528, 955197136, 2964744121, 9201982891, 28561145809, 88648181548, 275146527535, 854000728414, 2650650366790, 8227097627905, 25535293612129
Offset: 0

Views

Author

Paul Barry, Aug 30 2004

Keywords

Links

Crossrefs

Cf. A098182, A098184.

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n<2, 1, 3*a(n-1)+a(n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 25 2022

Formula

G.f.: (1-x)^2/(1-3*x-x^3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k,3*k) * 3^k.

A098184 a(n) = 3a(n-1)+a(n-2)+a(n-3), a(0)=1, a(1)=1, a(2)=5.

Original entry on oeis.org

1, 1, 5, 17, 57, 193, 653, 2209, 7473, 25281, 85525, 289329, 978793, 3311233, 11201821, 37895489, 128199521, 433695873, 1467182629, 4963443281, 16791208345, 56804250945, 192167404461, 650097672673, 2199264673425
Offset: 0

Views

Author

Paul Barry, Aug 30 2004

Keywords

Comments

Even bisection of the tribonacci sequence A000213. - Oboifeng Dira, Aug 03 2016

Links

Crossrefs

Cf. A000213, A098182, A098183.

Programs

  • Mathematica
    LinearRecurrence[{3,1,1},{1,1,5},30] (* Harvey P. Dale, Nov 29 2011 *)
CoefficientList[Series[(1 - x)^2/(1 - 3 x - x^2 - x^3), {x, 0, 24}], x] (* Michael De Vlieger, Aug 03 2016 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; 1,1,3]^n*[1;1;5])[1,1] \\ Charles R Greathouse IV, Aug 03 2016
  • Sage
    from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(1,1,1,3,1,1)
[next(it) for i in range(32)] # Zerinvary Lajos, Jun 24 2008

Formula

G.f.: (1-x)^2/(1-3*x-x^2-x^3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k, 3*k)*4^k.
a(n)/a(n-1) tends to 3.38297576..., the square of the tribonacci constant A058265. - Gary W. Adamson, Feb 28 2006
Showing 1-9 of 9 results.

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