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.

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A173174 a(n) = cosh(2*n*arcsinh(sqrt(n))).

Original entry on oeis.org

1, 3, 49, 1351, 51841, 2550251, 153090001, 10850138895, 886731088897, 82094249361619, 8491781781142001, 970614726270742103, 121485428812828080001, 16525390478051500325307, 2427469037137019032095121, 382956978214541873571486751, 64576903826545426454350012417, 11591229031806966336496244914595
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311 - A146313, A173115, A173116 A173121, A173127 - A173131, A173133, A173134, A173148, A173151, A173170, A173171.
Cf. A322746.
Main diagonal of A322790.

Programs

  • Magma
    [&+[Binomial(2*n, 2*k)*(n+1)^(n-k)*n^k: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Dec 29 2018
  • Maple
    A173174 := proc(n) cosh(2*n*arcsinh(sqrt(n))) ; expand(%) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Table[Round[N[Cosh[(2 n) ArcSinh[Sqrt[n]]], 100]], {n, 0, 30}] (* Artur Jasinski *)
Join[{1}, a[n_]:=Sum[Binomial[2 n, 2 k] (n + 1)^(n - k) n^k, {k, 0, n}]; Array[a, 25]] (* Vincenzo Librandi, Dec 29 2018 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k)} \\ Seiichi Manyama, Dec 26 2018
  • PARI
    {a(n) = polchebyshev(n, 1, 2*n+1)} \\ Seiichi Manyama, Dec 29 2018

Formula

a(n) = Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k. - Seiichi Manyama, Dec 26 2018
a(n) = T_{n}(2*n+1) where T_{n}(x) is a Chebyshev polynomial of the first kind. - Seiichi Manyama, Dec 29 2018

Extensions

More terms from Seiichi Manyama, Dec 26 2018

A173171 a(n) = - sin^2((2n-1)*arccos(sqrt n)) = sin^2((2n-1)*arcsin(sqrt n)) - 1.

Original entry on oeis.org

-1, 0, 49, 23762, 25421763, 48225038404, 142786923879605, 608447515452613206, 3527836867501829594887, 26710782540478226038759688, 255922222218837615280903143609, 3026917140685147530327256796600410
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170.

Programs

  • Mathematica
    Table[Round[-N[ Sin[(2 n - 1) ArcCos[Sqrt[n]]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010; Typo fixed by Vincenzo Librandi, Jun 29 2014 *)

A173175 a(n) = sinh^2( 2n*arcsinh(sqrt n)).

Original entry on oeis.org

0, 8, 2400, 1825200, 2687489280, 6503780163000, 23436548406180000, 117725514040791821024, 786292024016459316676608, 6739465778247681589030301160, 72110357818535214970387726284000, 942092946853627620313318842336862608, 14758709413836719039368938494112056160000
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A322699.

Programs

  • Maple
    A173175 := proc(n) sinh(2*n*arcsinh(sqrt(n))) ; %^2 ; expand(%); simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Table[Round[N[Sinh[(2 n) ArcSinh[Sqrt[n]]]^2, 100]], {n, 0, 20}]
  • PARI
    {a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ Seiichi Manyama, Jan 02 2019
  • PARI
    {a(n) = 1/2*(-1+sum(k=0, 2*n, binomial(4*n, 2*k)*(n+1)^(2*n-k)*n^k))} \\ Seiichi Manyama, Jan 02 2019

Formula

From Seiichi Manyama, Jan 02 2019: (Start)
a(n) = A322699(n,2*n).
a(n) = (T_{2*n}(2*n+1) - 1)/2 where T_{n}(x) is a Chebyshev polynomial of the first kind.
a(n) = 1/2 * (-1 + Sum_{k=0..2*n} binomial(4*n,2*k)*(n+1)^(2*n-k)*n^k). (End)
a(n) ~ exp(1) * 2^(4*n - 2) * n^(2*n). - Vaclav Kotesovec, Jan 02 2019

Extensions

a(11)-a(12) from Seiichi Manyama, Jan 02 2019

A173194 a(n) = -sin^2 (2*n*arccos n) = - sin^2 (2*n*arcsin n).

Original entry on oeis.org

0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520
Offset: 0

Views

Author

Artur Jasinski, Feb 12 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A173174, A173175, A173176.

Programs

  • Maple
    A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *)
Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *)
Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    {a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019
  • PARI
    {a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019

Formula

4*a(n) = ( (n+sqrt(n^2-1))^(2*n) - (n-sqrt(n^2-1))^(2*n) )^2. - Artur Jasinski, Feb 17 2010
From Seiichi Manyama, Jan 05 2019: (Start)
a(n) = (n^2-1) * n^2 * (Sum_{k=0..n-1} binomial(2*n,2*k+1)*(n^2-1)^(n-1-k)*n^(2*k))^2.
For n > 0, a(n) = (n^2-1) * U_{2*n-1}(n)^2 where U_{n}(x) is a Chebyshev polynomial of the second kind. (End)
a(n) ~ 2^(4*n - 2) * n^(4*n). - Vaclav Kotesovec, Jan 05 2019

Extensions

a(9) from Seiichi Manyama, Jan 05 2019

A173150 a(n) = sinh^2 (2n*arccosh(sqrt n)).

Original entry on oeis.org

0, 0, 288, 235224, 354079488, 865363202000, 3134808545188320, 15796198853361763368, 105717380511014096025600, 907380314352243226001152800, 9718304978537581699085289156000
Offset: 0

Views

Author

Artur Jasinski, Feb 11 2010

Keywords

Comments

Also a(n) = -sin(2n*arccos(sqrt(n)))^2 = -sin(2n*arcsin(sqrt(n)))^2.

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116, A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148.

Programs

  • Maple
    A173150 := proc(n) sinh(2*n*arccosh(sqrt(n))) ; %^2 ; expand(%) ; simplify(%) ;end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Table[Round[-Sin[2 n ArcCos[Sqrt[n]]]^2], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010 *)

Extensions

Minor edits by Vaclav Kotesovec, Apr 05 2016

A190840 a(n+1) = 4*a(n)*(a(n)+1) for a(0) = 1.

Original entry on oeis.org

1, 8, 288, 332928, 443365544448, 786292024016459316676608, 2473020588127600939387543243786675530709484249088
Offset: 0

Views

Author

Alexander Zhukov, Aug 08 2011

Keywords

Comments

For n>0, subsequence of A132592: both a(n)/2 and a(n)+1 are squares.
All terms (n > 0) are divisible by 8, yielding all terms of A185097, which is indexed from n=1, thus having the first term A185097(1) = 1.
The next term has 98 digits. - Harvey P. Dale, Jan 01 2014
For n>0, subsequence of A060355: both a(n) and a(n)+1 are powerful numbers. - Bernard Schott, Apr 24 2023

Crossrefs

Cf. A000217, A001601, A060355, A132592, A185097.

Programs

  • Mathematica
    NestList[4#(#+1)&,1,7] (* Harvey P. Dale, Jan 01 2014 *)

Formula

a(n+1) = 4*a(n)*(a(n)+1) for a(0) = 1.
a(n) = sinh(2^(n-2)*arccosh(17))^2. - Alexander R. Povolotsky, Aug 14 2011
a(n) = 8*A185097(n) for n > 0. - Alexander R. Povolotsky, Aug 14 2011
a(n) = (1 + sqrt(2))^(2^(n+1))/4 + (1 - sqrt(2))^(2^(n+1))/4 - 1/2. Therefore 2*a(n) + 1 = A001601(n+1). - Bruno Berselli, Feb 01 2017

A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.

Original entry on oeis.org

0, 2, 3, 8, 15, 18, 24, 32, 35, 48, 50, 63, 72, 80, 98, 99, 120, 128, 143, 162, 168, 195, 200, 224, 242, 255, 288, 323, 338, 360, 392, 399, 440, 450, 483, 512, 528, 575, 578, 624, 648, 675, 722, 728, 783, 800, 840, 882, 899, 960, 968, 1023, 1058, 1088, 1152, 1155, 1224
Offset: 0

Views

Author

Karl-Heinz Hofmann, Sep 12 2022

Keywords

Comments

Characteristics of the terms:
- Any x combined with any y is a solution.
This special case is valid only for k = 0 (exception: x = y = 0).
- Any x is possible and if x is odd: y = x. If x is even: y = x + 3.
This special case is valid only for k = 2 (see A356880).
- Only even x combined with y = x + 1 gives a solution.
Those terms are the terms of A132411.
- Only odd x combined with y = x gives a solution.
Those terms are the terms of A001105.
- Any x is possible and if x is odd: y = x. If x is even: y = x + 1.
Those terms are the terms of A132592.

Examples

			Squares that can be produced with k = 8: 8^0 + 8^1 = 9; 8^1 + 8^1 = 16; 8^2 + 8^3 = 576; 8^3 + 8^3 = 1024; 8^4 + 8^5 = 36864; 8^5 + 8^5 = 65536; 8^6 + 8^7 = 2359296, ....

Crossrefs

Cf. A132411 is a subsequence (except A132411(1)), A001105 is a subsequence.
Cf. A132592 is a subsequence.
Cf. A356880 (k = 2), A270473 (k = 3).

Programs

  • Mathematica
    Select[Range[0, 1225], IntegerQ[Sqrt[# + 1]] || IntegerQ[Sqrt[#/2]] &] (* Amiram Eldar, Sep 18 2022 *)
  • Python
    from gmpy2 import is_square
print([n for n in range(0,1225) if is_square(n+1) or (n % 2 == 0 and is_square(n//2))])
Previous Showing 21-27 of 27 results.

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