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 1-4 of 4 results.

A007751 Even bisection of A007750.

Original entry on oeis.org

0, 7, 120, 1921, 30624, 488071, 7778520, 123968257, 1975713600, 31487449351, 501823476024, 7997688167041, 127461187196640, 2031381306979207, 32374639724470680, 515962854284551681, 8223031028828356224
Offset: 0

Views

Author

John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)

Keywords

Crossrefs

Programs

  • GAP
    a:=[0,7,120];; for n in [4..30] do a[n]:=17*a[n-1]-17*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 10 2020
  • Magma
    I:=[0,7,120]; [n le 3 select I[n] else 17*Self(n-1) -17*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 10 2020
    
  • Maple
    seq(simplify((4*ChebyshevU(n,8) -11*ChebyshevU(n-1,8) -4)/7)), n = 0..30); # G. C. Greubel, Feb 10 2020
  • Mathematica
    Table[(4*ChebyshevU[n,8] -11*ChebyshevU[n-1,8] -4)/7, {n,0,30}] (* G. C. Greubel, Feb 10 2020 *)
    LinearRecurrence[{17,-17,1},{0,7,120},20] (* Harvey P. Dale, Dec 01 2022 *)
  • PARI
    a(n)=local(w); w=8+3*quadgen(28); imag(w^n)+4*(real(w^n)-1)/7
    
  • PARI
    vector(31, n, my(m=n-1); (4*polchebyshev(m,2,8) -11*polchebyshev(m-1,2,8) -4)/7 ) \\ G. C. Greubel, Feb 10 2020
    
  • Sage
    [(4*chebyshev_U(n,8) -11*chebyshev_U(n-1,8) -4)/7 for n in (0..30)] # G. C. Greubel, Feb 10 2020
    

Formula

G.f.: x*(7 + x)/((1-x)*(1-16*x+x^2)).
a(n) = 16*a(n-1) - a(n-2) + 8.
a(n) = (4*ChebyshevU(n,8) - 11*ChebyshevU(n-1,8) -4)/7. - G. C. Greubel, Feb 10 2020
E.g.f.: (cosh(x) + sinh(x))*(-4 + (cosh(7*x) + sinh(7*x))*(4*cosh(3*sqrt(7)*x) + sqrt(7)*sinh(3*sqrt(7)*x)))/7. - Stefano Spezia, Feb 20 2020

Extensions

Edited by Michael Somos, Jul 27 2002

A007752 Odd bisection of A007750.

Original entry on oeis.org

1, 24, 391, 6240, 99457, 1585080, 25261831, 402604224, 6416405761, 102259887960, 1629741801607, 25973608937760, 413948001202561, 6597194410303224, 105141162563649031, 1675661406608081280
Offset: 1

Views

Author

John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)

Keywords

References

  • Mentioned in a problem on p. 334 of Two-Year College Math. Jnl., Vol. 25, 1994.

Crossrefs

Programs

  • GAP
    a:=[1,24,391];; for n in [4..30] do a[n]:=17*a[n-1]-17*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Mar 04 2020
  • Magma
    I:=[1,24,391]; [n le 3 select I[n] else 17*Self(n-1) -17*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 04 2020
    
  • Maple
    seq( simplify( (4*ChebyshevU(n,8) - 53*ChebyshevU(n-1,8) -4)/7), n=1..20); # G. C. Greubel, Mar 04 2020
  • Mathematica
    Table[(4*ChebyshevU[n, 8] -53*ChebyshevU[n-1, 8] -4)/7, {n,20}] (* G. C. Greubel, Mar 04 2020 *)
  • PARI
    a(n)=local(w); w=8+3*quadgen(28); imag(1/w^n)+4*(real(1/w^n)-1)/7
    
  • PARI
    vector(30, n, (4*polchebyshev(n,2,8) -53*polchebyshev(n-1,2,8) -4)/7 ) \\ G. C. Greubel, Mar 04 2020
    
  • Sage
    [(4*chebyshev_U(n,8) -53*chebyshev_U(n-1,8) -4)/7 for n in (1..30)] # G. C. Greubel, Mar 04 2020
    

Formula

G.f.: x*(1+7*x)/((1-x)*(1-16*x+x^2)).
a(n) = 16*a(n-1) - a(n-2) + 8.
a(n) = (4*ChebyshevU(n, 8) -53*ChebyshevU(n-1, 8) -4)/7. - G. C. Greubel, Mar 04 2020
E.g.f.: (exp(8*x)*(4*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 4*exp(x))/7. - Stefano Spezia, Mar 14 2020
a(n) = A097830(n-1)+7*A097830(n-2). - R. J. Mathar, Jul 04 2024

Extensions

Edited by Michael Somos, Jul 27 2002

A073352 Positive integers making n^2*(n-1)*(2*n-1)^2*(7*n-1)/36 a square.

Original entry on oeis.org

1, 4, 55, 868, 13825, 220324, 3511351, 55961284, 891869185, 14213945668, 226531261495, 3610286238244, 57538048550401, 916998490568164, 14614437800540215, 232914006318075268, 3712009663288664065
Offset: 0

Views

Author

Michael Somos, Jul 27 2002

Keywords

Examples

			G.f. = 1 + 4*x + 55*x^2 + 868*x^3 + 13825*x^4 + 220324*x^5 + 3511351*x^6 + ...
		

Crossrefs

Programs

  • GAP
    a:=[1,4,55];; for n in [4..30] do a[n]:=17*a[n-1]-17*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 09 2020
  • Magma
    I:=[1,4,55]; [n le 3 select I[n] else 17*Self(n-1) - 17*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 09 2020
    
  • Maple
    seq( simplify((4 +3*ChebyshevU(n,8) -24*ChebyshevU(n-1,8))/7), n=0..30); # G. C. Greubel, Feb 09 2020
  • Mathematica
    LinearRecurrence[{17,-17,1},{1,4,55},30] (* Harvey P. Dale, Dec 09 2018 *)
  • PARI
    {a(n) = if( n<0, a(-n), if( n<1, 1, 16*a(n-1) - a(n-2) - 8))}
    
  • PARI
    {a(n) = (4 + 3 * real((8 + 3 * quadgen(28))^n)) / 7}
    
  • Sage
    [(4 +3*chebyshev_U(n,8) -24*chebyshev_U(n-1,8))/7 for n in (0..30)] # G. C. Greubel, Feb 09 2020
    

Formula

G.f.: (1 - 13*x + 4*x^2)/((1-x)*(1-16*x+x^2)).
a(n) = (4 + 3*ChebyshevU(n, 8) - 24*ChebyshevU(n-1, 8))/7. - G. C. Greubel, Feb 09 2020

A073351 n^2(n+1)(2n+1)^2(7n+1)/36.

Original entry on oeis.org

0, 4, 125, 1078, 5220, 18150, 50869, 122500, 263568, 519840, 956725, 1664234, 2762500, 4407858, 6799485, 10186600, 14876224, 21241500, 29730573, 40876030, 55304900, 73749214, 97057125, 126204588, 162307600, 206635000, 260621829
Offset: 0

Views

Author

Michael Somos, Jul 27 2002

Keywords

References

  • Mentioned in a problem on p. 334 of Two-Year College Math. Jnl., Vol. 25, 1994.

Crossrefs

Cf. A007750.

Formula

a(n)=sum(i=1, n, i^2)*sum(i=n+1, 2*n, i^2).
G.f.: x(4+97x+287x^2+159x^3+13x^4)/(1-x)^7.
Showing 1-4 of 4 results.