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.

A082999 a(n) = A046195(n) mod 5.

Original entry on oeis.org

1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1, 1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4, 1, 1
Offset: 1

Views

Author

Jon Perry, May 30 2003

Keywords

Comments

Sequence only consists of 0, 1, 4 mod 5.

Examples

			a(2)=1 because A046195(2)=6=1 mod 5.

Links

Crossrefs

Cf. A046195.

Programs

  • Magma
    I:=[1,1,4,1,4,0,4,0,4,0,4,1,4]; [n le 13 select I[n] else + Self(n-1) - Self(n-3) + Self(n-4) - Self(n-6) + Self(n-7) - Self(n-9) + Self(n-10) - Self(n-12) + Self(n-13): n in [1..100]]; // Vincenzo Librandi, Aug 07 2015
  • Maple
    A046195 := proc(n) option remember; if n <= 7 then op(n,[1, 6, 49, 961, 8214, 70225, 1385329 ]) ; else procname(n-1)+1442*procname(n-3) -1442*procname(n-4)-procname(n-6) +procname(n-7) ; end if; end proc:
A082999 := proc(n) A046195(n) mod 5 ; end proc: seq(A082999(n),n=1..120) ;
# R. J. Mathar, Jul 27 2010
  • Mathematica
    LinearRecurrence[{1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1}, {1, 1, 4, 1, 4, 0, 4, 0, 4, 0, 4, 1, 4}, 100] (* Vincenzo Librandi, Aug 07 2015 *)

Formula

a(n)= +a(n-1) -a(n-3) +a(n-4) -a(n-6) +a(n-7) -a(n-9) +a(n-10) -a(n-12) +a(n-13). - R. J. Mathar, Jul 27 2010

Extensions

More terms from R. J. Mathar, Jul 27 2010

A036354 Heptagonal square numbers.

Original entry on oeis.org

1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, 729252434211108535809, 53306479301521270428241, 20744638830126197732344369, 1516379800105728357531817761, 110843467413344235941816109721
Offset: 1

Views

Author

Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)

Keywords

Comments

From Ant King, Nov 11 2011: (Start)
This sequence is also the union of the three sequences defined by:
a(3n-2) = ((10 - sqrt(10)) * (3 + sqrt(10))^(4*n-3) - (10 + sqrt(10)) * (-3 + sqrt(10))^(4*n-3))^2 / 1600.
a(3n-1) = 9/160 * ((3 + sqrt(10))^(4*n-2) - (-3 + sqrt(10))^(4*n-2))^2.
a(3n) = ((20 - 7*sqrt(10)) * (3 + sqrt(10))^(4*n) + (20 + 7*sqrt(10)) * (-3 + sqrt(10))^(4*n))^2 / 1600.
Equivalent short forms for these subsequences are:
a(3n-2) = floor((10 - sqrt(10))^2 * (3 + sqrt(10))^(8*n - 6) / 1600).
a(3n-1) = floor( 9/160 * (3 + sqrt(10))^(8*n - 4)).
a(3n) = floor((20 - 7*sqrt(10))^ 2 * (3 + sqrt(10))^(8*n) / 1600).
(End)
Also heptagonal numbers (A000566) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 19 2015

Links

Crossrefs

Cf. A046195, A046196, A253920.

Programs

  • Maple
    A036354 := proc(n)
if n <= 7 then
    op(n,[1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609]);
else
    procname(n-1) +2079362 *(procname(n-3)-procname(n-4)) -procname(n-6) +procname(n-7) ;
end if;
end proc:
seq(A036354(n),n=1..12) ;
  • Mathematica
    LinearRecurrence[{ 1, 0, 2079362, -2079362, 0, -1, 1 }, {1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609 }, 13] (* Ant King, Nov 11 2011 *)
  • PARI
    Vec(-x*(x^6+80*x^5+5848*x^4+222070*x^3+5848*x^2+80*x+1)/((x-1)*(x^6-2079362*x^3+1)) + O(x^100)) \\ Colin Barker, Jan 19 2015

Formula

O.g.f.: -x*(1 + 80*x + 5848*x^2 + 222070*x^3 + 5848*x^4 + 80*x^5 + x^6) / ( (x-1)*(x^6 - 2079362*x^3 + 1) ).
From Richard Choulet, May 08 2009: (Start)
With the first values, for n>=0, a(n+9) = 2079363*(a(n+6) - a(n+3)) + a(n).
On every bisection modulo 2: a(n+1) = 1039681*a(n) + 116964 + 164388*sqrt(40*a(n)^2 + 9*a(n)).
On every bisection modulo 2: a(n+2) = 2079362*a(n+1) - a(n) + 233928. (End)
From Ant King, Nov 11 2011: (Start)
a(n) = a(n-1) + 2079362*a(n-3) - 2079362*a(n-4) - a(n-6) + a(n-7).
a(n) = 2079362*a(n-3) - a(n-6) + 233928.
(End)
From Jonathan Pappas, Jan 16 2022: (Start)
Define the three sequences
b(n) = 1442*b(n-1) - b(n-2) for n >= 2, with b(0) = -77, b(1) = 1;
c(n) = 1442*c(n-1) - c(n-2) for n >= 2, with c(0) = -9, c(1) = 9; and
d(n) = 1442*d(n-1) - d(n-2) for n >= 2, with d(0) = -1, d(1) = 77.
Then, for n >= 1,
a(3n - 2) = b(n)^2,
a(3n - 1) = c(n)^2, and
a(3n) = d(n)^2.
(End)

Extensions

More terms from Eric W. Weisstein
One more term from Richard Choulet, May 08 2009

A046196 Indices of square numbers which are also heptagonal.

Original entry on oeis.org

1, 9, 77, 1519, 12987, 111035, 2190397, 18727245, 160112393, 3158550955, 27004674303, 230881959671, 4554628286713, 38940721617681, 332931625733189, 6567770830889191, 56152493568021699, 480087173425298867, 9470720983513926709, 80971856784365672277
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A036354, A046195.

Programs

  • Maple
    for n from 1 to 10000 do m:=sqrt((5*n*n-3*n)/2):
if (trunc(m)=m) then print(n,m): end if: end do: # Paul Weisenhorn, May 01 2009
  • Mathematica
    LinearRecurrence[{ 0, 0, 1442, 0, 0, -1 } , {1, 9, 77, 1519, 12987, 111035 }, 17] (* Ant King, Nov 11 2011 *)
  • PARI
    Vec(x*(x+1)*(x^4+8*x^3+69*x^2+8*x+1)/(x^6-1442*x^3+1) + O(x^50)) \\ Colin Barker, Jun 23 2015

Formula

From Paul Weisenhorn, May 01 2009: (Start)
a(n+6) = 1442*a(n+3)-a(n) with
a(-2)=-77; a(-1)=-9; a(0)=-1; a(1)=1; a(2)=9; a(3)=77;
A = (721+sqrt(10)*228)^k; B = (721-sqrt(10)*228)^k;
a(3*k+1) = (7*(A-B)/sqrt(10)+2*(A+B))/4;
a(3*k+2) = (57*(A-B)/sqrt(10)+18*(A+B))/4;
a(3*k) = (7*(A-B)/sqrt(10)-2*(A+B))/4;
(End)
G.f.: x * (1 + x) * (1 + 8*x + 69*x^2 + 8*x^3 + x^4) / (1-1442*x^3 + x^6). - Ant King, Nov 11 2011

A253920 Indices of centered octagonal numbers (A016754) which are also heptagonal numbers (A000566).

Original entry on oeis.org

1, 5, 39, 760, 6494, 55518, 1095199, 9363623, 80056197, 1579275478, 13502337152, 115440979836, 2277314143357, 19470360808841, 166465812866595, 3283885415444596, 28076246784010850, 240043586712649434, 4735360491756963355, 40485928392182836139
Offset: 1

Views

Author

Colin Barker, Jan 19 2015

Keywords

Comments

Also positive integers y in the solutions to 5*x^2 - 8*y^2 - 3*x + 8*y - 2 = 0, the corresponding values of x being A046195.

Examples

			5 is in the sequence because the 5th centered octagonal number is 81, which is also the 6th heptagonal number.

Links

Crossrefs

Cf. A000566, A016754, A046195, A036354.

Programs

  • Magma
    I:=[1,5,39,760,6494,55518,1095199]; [n le 7 select I[n] else Self(n-1)+1442*Self(n-3)-1442*Self(n-4)-Self(n-6)+Self(n-7): n in [1..25]]; // Vincenzo Librandi, Jan 20 2015
  • Mathematica
    CoefficientList[Series[(4 x^5 + 34 x^4 + 721 x^3 - 34 x^2 -4 x - 1)/((x-1) (x^6 - 1442 x^3 + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 20 2015 *)
LinearRecurrence[{1,0,1442,-1442,0,-1,1},{1,5,39,760,6494,55518,1095199},20] (* Harvey P. Dale, Jul 04 2017 *)
  • PARI
    Vec(x*(4*x^5+34*x^4+721*x^3-34*x^2-4*x-1)/((x-1)*(x^6-1442*x^3+1)) + O(x^100))

Formula

a(n) = a(n-1)+1442*a(n-3)-1442*a(n-4)-a(n-6)+a(n-7).
G.f.: x*(4*x^5+34*x^4+721*x^3-34*x^2-4*x-1) / ((x-1)*(x^6-1442*x^3+1)).

Extensions

Corrected by Vincenzo Librandi, Jan 20 2015

A251927 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to a heptagonal number H(m) for some m.

Original entry on oeis.org

0, 8, 76, 1518, 12986, 111034, 2190396, 18727244, 160112392, 3158550954, 27004674302, 230881959670, 4554628286712, 38940721617680, 332931625733188, 6567770830889190, 56152493568021698, 480087173425298866, 9470720983513926708, 80971856784365672276
Offset: 1

Views

Author

Colin Barker, Dec 11 2014

Keywords

Comments

Also nonnegative integers x in the solutions to 2*x^2-5*y^2+4*x+3*y+2+2 = 0, the corresponding values of y being A046195.

Examples

			8 is in the sequence because T(8)+T(9) = 36+45 = 81 = H(6).

Links

Crossrefs

Cf. A000217, A000566, A046195.

Programs

  • Mathematica
    LinearRecurrence[{1,0,1442,-1442,0,-1,1},{0,8,76,1518,12986,111034,2190396},20] (* Harvey P. Dale, Dec 08 2016 *)
  • PARI
    concat(0, Vec(2*x^2*(x^5+4*x^4+34*x^3-721*x^2-34*x-4)/((x-1)*(x^6-1442*x^3+1)) + O(x^100)))

Formula

G.f.: 2*x^2*(x^5+4*x^4+34*x^3-721*x^2-34*x-4) / ((x-1)*(x^6-1442*x^3+1)).

A083000 Values of x for which 9y^2 = x^2 + 2xy - 2x has integer solutions with positive y.

Original entry on oeis.org

3, 18, 338, 2883, 24642, 486098, 4155987, 35532450, 700951682, 5992929075, 51237766962
Offset: 1

Views

Author

Jon Perry, May 30 2003

Keywords

Comments

From the x and y values, one can derive some z's such that 10z^2-9 is a square (A052454): z = x+y-1. Other A052454 values can be derived from A046195.

Crossrefs

Cf. A046195, A052454.

Extensions

Edited by Don Reble, Nov 07 2005
Showing 1-6 of 6 results.

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