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.

Previous Showing 21-28 of 28 results.

A220989 a(n) = 12^(2n+1) - 6 * 12^n + 1: the left Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

7, 1657, 247969, 35821441, 5159655937, 743006877697, 106993187463169, 15407021359595521, 2218611104160546817, 319479999339664244737, 46005119908998197280769, 6624737266944778960896001, 953962166440636632998608897
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220990.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[12^(2n+1) - 6 * 12^n + 1, {n, 0, 20}]

Formula

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * a(n) * A220990(n).
G.f.: -(1008*x^2+558*x+7) / ((x-1)*(12*x-1)*(144*x-1)). [Colin Barker, Jan 03 2013]

A378963 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the short leg of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

Original entry on oeis.org

3, 4, 5, 7, 24, 25, 15, 112, 113, 31, 480, 481, 63, 1984, 1985, 127, 8064, 8065, 255, 32512, 32513, 511, 130560, 130561, 1023, 523264, 523265, 2047, 2095104, 2095105, 4095, 8384512, 8384513, 8191, 33546240, 33546241, 16383, 134201344, 134201345
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

Keywords

Comments

The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).

Examples

			Triples begin:
  3, 4, 5;
  7, 24, 25;
  15, 112, 113;
  31, 480, 481;

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz, and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Crossrefs

Cf. A000225 (short leg), A092440 (hypotenuse), A378395, A365577.

Programs

  • Mathematica
    a=Table[2^(n+1)-1,{n,1,13}];Apply[Join,Map[{#,(#^2-1)/2,(#^2+1)/2}&,a]]

A112830 Table of number of domino tilings of generalized Aztec pillows of type (1, ..., 1, 3, 1, ..., 1)_n.

Original entry on oeis.org

1, 1, 5, 1, 10, 25, 1, 17, 65, 113, 1, 26, 146, 346, 481, 1, 37, 292, 932, 1637, 1985, 1, 50, 533, 2248, 5013, 7218, 8065, 1, 65, 905, 4937, 13897, 24201, 30529, 32513, 1, 82, 1450, 10018, 35218, 74530, 108970, 126034, 130561, 1, 101, 2216, 19016, 82436
Offset: 0

Views

Author

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

Keywords

Comments

The number of tilings of a generalized Aztec pillow of type (k 1's followed by a 3 followed by n-k-1 1's) is entry (n,k+1).

Examples

			The number of tilings of a generalized Aztec pillow of type (1,1,3,1)_n is entry (4,3) = 346.

Links

Crossrefs

A092440 (main diagonal), A092441 (first subdiagonal), A002522 (column k = 1), A066455 (column k = 2). Cf. A264960.

Programs

  • Maple
    matrix(11,11,[seq([seq(((2^n-sum(binomial(n,j),j=0..k))^2+(binomial(n-1,k))^2)/2,n=k+1..k+11)],k=0..10)]);

Formula

T(2*n,n) = A264960(n). - Peter Bala, Nov 29 2015

A171663 Expansion of (1 + 4*x - 6*x^2 - 16*x^3 + 20*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

Original entry on oeis.org

1, 5, 5, 13, 25, 41, 113, 145, 481, 545, 1985, 2113, 8065, 8321, 32513, 33025, 130561, 131585, 523265, 525313, 2095105, 2099201, 8384513, 8392705, 33546241, 33562625, 134201345, 134234113, 536838145, 536903681, 2147418113, 2147549185
Offset: 0

Views

Author

Jonathan Vos Post, Dec 14 2009

Keywords

Links

Crossrefs

Cf. A000040, A000215, A019434.
Cf. A092440, A085601 (bisections). - R. J. Mathar, Jan 25 2010

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2)) )); // G. C. Greubel, Jun 01 2019
  • Mathematica
    Flatten[Table[2^(2*n+1) + 1 + 2^(n+1) {-1, 1}, {n, 0, 40}]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
  • PARI
    my(x='x+O('x^40)); Vec((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))) \\ G. C. Greubel, Jun 01 2019
  • Sage
    ((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 01 2019

Formula

G.f.: (1 + 4*x - 6*x^2 - 16*x^3 + 20*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)). - Colin Barker, Apr 27 2013

Extensions

More terms from R. J. Mathar and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
New name from Joerg Arndt, Jun 03 2019

A220985 The left Aurifeuillian factor of 10^(20n+10) + 1.

Original entry on oeis.org

3541, 904806804901, 99004980069800499001, 9990004998000699800049990001, 999900004999800006999800004999900001, 99999000004999980000069999800000499999000001, 9999990000004999998000000699999800000049999990000001
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220986.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1.
Aurifeuillian factorization: 10^(20n+10) + 1 = (10^(4n+2) + 1) * a(n) * A220986(n).

A220986 The right Aurifeuillian factor of 10^(20n + 10) + 1.

Original entry on oeis.org

27961, 1105207205101, 101005020070200501001, 10010005002000700200050010001, 1000100005000200007000200005000100001, 100001000005000020000070000200000500001000001, 10000010000005000002000000700000200000050000010000001
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220985.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979, A220980, A220981, A220982
Cf. A220983, A220984, A220985, A220987, A220988, A220989, A220990

Programs

  • Mathematica
    a[n_] := 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1

Formula

a(n) = 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1
Aurifeuillian factorization: 10^(20n + 10) + 1 = (10^(4n + 2) + 1) * A220985(n) * a(n)

A325914 Primes of the form (2^k-1)*2^(k+1)+1.

Original entry on oeis.org

5, 113, 140737471578113, 9444732965601851473921, 604462909806215075725313, 10384593717069655112945804582584321, 2854495385411919762116496381035264358442074113, 187072209578355573530071639244871112681892570202113
Offset: 1

Views

Author

Gary Wright, Sep 08 2019

Keywords

Comments

The next two terms are 883423532389192164791648750371459256584513952652893606156996040365965313
and 3533694129556768659166595001485837028996511802181406170435598282024550401. - N. J. A. Sloane, Sep 13 2019

Crossrefs

These are the primes in A092440. See A006598 for the values of k.

Extensions

Edited by N. J. A. Sloane, Sep 13 2019

A344917 a(n) = numerator(4^(n + 1)*zeta(-n, 1/4)).

Original entry on oeis.org

1, 1, -1, -7, 5, 31, -61, -127, 1385, 511, -50521, -1414477, 2702765, 8191, -199360981, -118518239, 19391512145, 5749691557, -2404879675441, -91546277357, 370371188237525, 162912981133, -69348874393137901, -1982765468311237, 15514534163557086905, 22076500342261
Offset: 0

Views

Author

Peter Luschny, Jul 09 2021

Keywords

Examples

			Rational sequence starts: 1, 1/6, -1, -7/60, 5, 31/126, -61, -127/120, 1385, ...

Crossrefs

Cf. A344918 (denominators), A092440, A163982.

Programs

  • Maple
    seq(numer(4^(n+1)*Zeta(0, -n, 1/4)), n=0..25);
  • SageMath
    def a(n): return 4^(n+1)*hurwitz_zeta(-n, 1/4) if n > 0 else 1
print([a(n).numerator() for n in (0..25)])

Formula

a(n)/A344918(n) - 2*A092440(n)*zeta(-n) = -A163982(n) for n >= 0.
Previous Showing 21-28 of 28 results.

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

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