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
Cf.
A132592,
A146311,
A146312,
A146313,
A173115,
A173116,
A173121,
A173127,
A173128,
A173129,
A173130,
A173131,
A173133,
A173134,
A173148,
A173151,
A173170,
A173171,
A322699.
-
A173175 := proc(n) sinh(2*n*arcsinh(sqrt(n))) ; %^2 ; expand(%); simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
-
Table[Round[N[Sinh[(2 n) ArcSinh[Sqrt[n]]]^2, 100]], {n, 0, 20}]
-
{a(n) = (polchebyshev(2*n, 1, 2*n+1)-1)/2} \\ Seiichi Manyama, Jan 02 2019
-
{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
A382209
Numbers k such that 10+k and 10*k are perfect squares.
Original entry on oeis.org
90, 136890, 197402490, 284654260890, 410471246808090, 591899253243012090, 853518312705176632890, 1230772815021611461622490, 1774773545742851022483004890, 2559222222188376152809031436090, 3690396669622092669499600847844090, 5321549438372835441042271613559748890
Offset: 1
90 is a term because 10+90=100 is a square and 10*90=900 is a square.
(3,1) is a solution to x^2 - 10*y^2 = -1, from which a(n) = 100*y^2-10 = 10*x^2 = 90.
Cf. 5*
A075796^2 (5+k and 5*k are squares).
Cf. 5*
A081071 (20+k and 20*k are squares).
Cf.
A245226 (m such that k+m and k*m are squares).
-
CoefficientList[Series[ 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)),{x,0,11}],x] (* or *) LinearRecurrence[{1443,-1443,1},{90,136890,197402490},12] (* James C. McMahon, May 08 2025 *)
-
from itertools import islice
def A382209_gen(): # generator of terms
x, y = 30, 10
while True:
yield x**2//10
x, y = x*19+y*60, x*6+y*19
A382209_list = list(islice(A382209_gen(),30)) # Chai Wah Wu, Apr 24 2025
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
Cf.
A132592,
A146311,
A146312,
A146313,
A173115,
A173116 A173121,
A173127,
A173128,
A173129,
A173130,
A173131,
A173133,
A173134,
A173148,
A173151,
A173170,
A173171,
A173174,
A173175,
A173176.
-
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
-
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 *)
-
{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
-
{a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ 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
Cf.
A132592,
A146311,
A146312,
A146313,
A173115,
A173116,
A173121,
A173127,
A173128,
A173129,
A173130,
A173131,
A173133,
A173134,
A173148.
-
A173150 := proc(n) sinh(2*n*arccosh(sqrt(n))) ; %^2 ; expand(%) ; simplify(%) ;end proc: # R. J. Mathar, Feb 26 2011
-
Table[Round[-Sin[2 n ArcCos[Sqrt[n]]]^2], {n, 0, 20}] (* Artur Jasinski, Feb 11 2010 *)
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