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 1-6 of 6 results.

A039659 Erroneous version of A039919.

Original entry on oeis.org

0, 1, 5, 21, 50, 355
Offset: 1

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Author

Keywords

A030532 Number of polyhexes of class PF2 with symmetry point group C_s.

Original entry on oeis.org

0, 1, 6, 35, 168, 807, 3738, 17326, 79909, 369330, 1709087, 7929590, 36880231, 171981241, 804008476, 3767969067, 17699758030, 83328230588, 393123455667, 1858351021018, 8801159427825, 41756067216508, 198437454009869, 944521139813575, 4502419756667924
Offset: 4

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Author

N. J. A. Sloane

Keywords

Comments

See reference for precise definition.
Cyvin has incorrect a(13)=369366 and a(14)=1709123 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II.

Links

Crossrefs

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Programs

  • PARI
    L(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*x^2*(1-x)), n); \\ A039658
Lp(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
Mp(n) = N(n) - sum(j=0, n-1, N(j)); \\ A039919
b(n) = N(n+3) - 6*N(n+2) - Mp(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-if (!(n%2),M(n/2))+Lp(n))/2;
a(n) = if (n<=4, 0, b(n-4)); \\ Michel Marcus, Apr 05 2020

Formula

a(n+4) = N(n+3) - 6*N(n+2) - M'(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-e(n)*M(n/2)+L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), M'(n)=A039919(n), L(n)=A039658(n), L'(n)=A039660(n), e(n)=1 if n is even and 0 if n is odd. - Sean A. Irvine, Apr 03 2020

Extensions

a(13) and a(14) corrected, title improved, and more terms from Sean A. Irvine, Apr 03 2020

A026298 Number of polyhexes of class PF2.

Original entry on oeis.org

4, 28, 176, 950, 4908, 24402, 119240, 575348, 2757460, 13157752, 62638788, 297832008, 1415550920, 6728600060, 31998023632, 152271569872, 725231959452, 3457304575812, 16497751608120, 78804354881238, 376806016649964, 1803539487096138, 8641075826669256, 41441524062045660
Offset: 7

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Author

N. J. A. Sloane

Keywords

Comments

See reference for precise definition.
Cyvin et al. has incorrect a(13) = 119204 and a(14) = 575312 due to using incorrect value for A039919(5); cf. A039659. - Sean A. Irvine, Sep 24 2019

Links

Crossrefs

Cf. A026106, A026118, A030519, A030520, A030525, A030529, A030532, A030534, A002212, A039919.

Formula

a(n + 4) = 3 * (N(n+2) - 6*N(n+1) + 8*N(n)) + A039919(floor((n+1)/2)) where N(n) = A002212(n) [from Cyvin]. - Sean A. Irvine, Sep 24 2019

Extensions

Corrected and extended by Sean A. Irvine, Sep 24 2019

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

Original entry on oeis.org

1, 12, 1, 6, 1, 3, 1, 6, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 2, 1, 2, 6, 1, 1, 1, 1, 1, 6, 2, 1, 2, 4, 3, 2, 2, 4, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 3, 2, 2, 4, 2, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2
Offset: 1

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Author

Juri-Stepan Gerasimov, Jun 13 2008

Keywords

Comments

Primes can be classified according to their remainder modulo 30: remainder 1 (A136066), 7 (A132231), 11 (A132232), 13 (A132233), 17 (A039949), 19 (A132234), 23 (A132235), or 29 (A132236). In the sequence A007775 of all numbers (prime or nonprime) in any of these remainder classes, we look for runs of numbers that are successively prime or nonprime and place the lengths of these runs in this sequence.

Examples

			Groups of runs in A007775 are (1), (7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), (49), (53, 59, 61, 67, 71, 73), (77), (79, 83,...), which is 1 nonprime followed by 12 primes followed by 1 nonprime followed by 6 primes etc.

Crossrefs

Cf. A136066, A132231, A132232, A132233, A039919, A132234, A132235, A132236.

Programs

  • Maple
    A007775 := proc(n) option remember ; local a; if n = 1 then 1; else for a from A007775(n-1)+1 do if a mod 2 <>0 and a mod 3 <>0 and a mod 5 <> 0 then RETURN(a) ; fi ; od: fi ; end: A := proc() local al,isp,n; al := 0: isp := false ; n := 1: while n< 300 do a := A007775(n) ; if isprime(a) <> isp then printf("%d,",al) ; al := 1; isp := not isp ; else al := al+1 ; fi ; n := n+1: od: end: A() ; # R. J. Mathar, Jun 16 2008

Extensions

Edited by R. J. Mathar, Jun 16 2008

A140387 Binary encoding of the location of primes in integer sets r+30*n with remainder r=1,7,11,..,29.

Original entry on oeis.org

1, 32, 16, 129, 73, 36, 194, 6, 42, 176, 225, 12, 21, 89, 18, 97, 25, 243, 44, 44, 196, 34, 166, 90, 149, 152, 109, 66, 135, 225, 89, 169, 169, 28, 82, 210, 33, 213, 179, 170, 38, 92, 15, 96, 252, 171, 94, 7, 209, 2, 187, 22, 153, 9, 236, 197, 71, 179, 212, 197, 186
Offset: 1

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Author

Juri-Stepan Gerasimov, Jun 10 2008

Keywords

Comments

Classify all integers 30n+r with r= 1, 7, 11, 13, 17, 19, 23 or 29 as nonprime or prime and assign bit positions 0=LSB, 1, 2, 3, .., 7=MSB to the 8 remainders in the same order. Raise the bit if 30n+r is nonprime, erase it if 30n+r is prime.
The sequence interprets this as a number in base 2 and shows the decimal representation.

Examples

			For n=1, the 8 numbers 31 (r=1), 37 (r=7), 41 (r=11), 43 (r=17), 47 (r=17), 49 (r=19), 53 (r=23) and 59 (r=29) are prime, prime, prime, prime, prime, nonprime, prime, prime, prime, which is rendered into the binary 000100000 = 2^5=32=a(1).

Crossrefs

Cf. A132236, A132235, A132234, A039919, A132233, A132232, A136066, A140378.
Cf. A105052 (analog in base 10, prime = bit 1, remainder 1 = MSB), A140891 (analog in base 14, prime = bit 0, remainder 1 = LSB).

Extensions

Edited by R. J. Mathar, Jun 17 2008

A128744 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 10, 10, 8, 8, 36, 36, 29, 20, 16, 137, 137, 111, 78, 48, 32, 543, 543, 442, 315, 200, 112, 64, 2219, 2219, 1813, 1306, 848, 496, 256, 128, 9285, 9285, 7609, 5527, 3649, 2200, 1200, 576, 256, 39587, 39587, 32521, 23779, 15901, 9802, 5552, 2848
Offset: 1

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Author

Emeric Deutsch, Mar 31 2007

Keywords

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1) (down) and L=(-1,-1) (left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
Row sums yield A002212.

Examples

			T(3,3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
   1;
   1,  2;
   3,  3,  4;
  10, 10,  8,  8;
  36, 36, 29, 20, 16;

Links

  • E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A002212, A039919.

Programs

  • Maple
    g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z*g/(1-t*z-t*z*g): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

Formula

T(n,1) = A002212(n-1).
T(n,2) = A002212(n-1) for n >= 3.
Sum_{k=1..n} k*T(n,k) = A039919(n+1).
G.f.: t*z*g/(1 - t*z - t*z*g), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Showing 1-6 of 6 results.

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

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