A099761 -id:A099761 - cp's OEIS frontend

A007204 Crystal ball sequence for D_4 lattice.

1, 25, 169, 625, 1681, 3721, 7225, 12769, 21025, 32761, 48841, 70225, 97969, 133225, 177241, 231361, 297025, 375769, 469225, 579121, 707281, 855625, 1026169, 1221025, 1442401, 1692601, 1974025, 2289169, 2640625, 3031081, 3463321, 3940225, 4464769, 5040025

Offset: 0

N. J. A. Sloane and J. H. Conway, Apr 28 1994

Equals binomial transform of [1, 24, 120, 192, 96, 0, 0, 0, ...]. - Gary W. Adamson, Aug 13 2009
Hypotenuse of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + A069074(n-1)^2 = a(n)^2. - Martin Renner, Nov 12 2011
Numbers n such that n*x^4 + x^2 + 1 is reducible. - Arkadiusz Wesolowski, Nov 04 2013

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for crystal ball sequences
Index entries for sequences related to D_4 lattice
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016754, A057769, A060300, A069074.

Cf. A000290, A000583, A001844, A005563, A099761.

[(2*n^2+2*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Nov 18 2016

A007204:=n->(2*n^2+2*n+1)^2; seq(A007204(n), n=0..30);

Table[(2n^2+2n+1)^2,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,25,169,625,1681},40] (* Harvey P. Dale, Mar 03 2013 *)

a(n)=(2*n^2+2*n+1)^2 \\ Charles R Greathouse IV, Feb 08 2017

G.f.: (1 + 54*x^2 + 20*x + 20*x^3 + x^4)/(1-x)^5.
a(0)=1, a(1)=25, a(2)=169, a(3)=625, a(4)=1681, a(n)=5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Mar 03 2013
Sum_{n>=0} 1/a(n) = Pi*(sinh(Pi) - Pi)/(2*(cosh(Pi) + 1)) = 1.0487582722070177... - Ilya Gutkovskiy, Nov 18 2016
a(n) = A016754(n) + A060300(n). - Bruce J. Nicholson, Apr 14 2017
a(n) = A001844(n)^2 = (2*n^2+2*n+1)^2. - Bruce J. Nicholson, May 15 2017
a(n) = A000583(n+1) + A099761(n) + 2*A005563(n-1)*A000290(n). - Charlie Marion, Jan 14 2021
E.g.f.: exp(x)*(1 + 24*x + 60*x^2 + 32*x^3 + 4*x^4). - Stefano Spezia, Jun 06 2021

More terms from Harvey P. Dale, Mar 03 2013

A244730 a(n) = 2*n^4.

Original entry on oeis.org

0, 2, 32, 162, 512, 1250, 2592, 4802, 8192, 13122, 20000, 29282, 41472, 57122, 76832, 101250, 131072, 167042, 209952, 260642, 320000, 388962, 468512, 559682, 663552, 781250, 913952, 1062882, 1229312, 1414562, 1620000, 1847042, 2097152, 2371842, 2672672

Offset: 0

Vincenzo Librandi, Jul 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A141046, A219056.

Magma
```
[2*n^4: n in [0..40]];
```

I:=[0,2,32,162, 512]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[2 n^4, {n, 0, 40}] (* or *) CoefficientList[Series[2(x + 11 x^2 + 11 x^3 + x^4)/(1 - x)^5, {x, 0, 40}], x]
LinearRecurrence[{5,-10,10,-5,1},{0,2,32,162,512},40] (* Harvey P. Dale, Jun 17 2022 *)

G.f.: 2*(x + 11*x^2 + 11*x^3 + x^4)/(1 - x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>4.
a(n) = (A082044(n) + A099761(n+1)-2)/2. - Bruce J. Nicholson, Jun 12 2017

A156701 a(n) = 4n^4 + 17n^2 + 4.

Original entry on oeis.org

4, 25, 136, 481, 1300, 2929, 5800, 10441, 17476, 27625, 41704, 60625, 85396, 117121, 157000, 206329, 266500, 339001, 425416, 527425, 646804, 785425, 945256, 1128361, 1336900, 1573129, 1839400, 2138161, 2471956, 2843425, 3255304, 3710425

Offset: 0

Reinhard Zumkeller, Feb 13 2009

a(n) = A087475(n)*A053755(n).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016850, A053755, A087475, A099761.

[4*n^4+17*n^2+4: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

Table[4n^4+17n^2+4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{4,25,136,481,1300},50] (* Harvey P. Dale, Nov 08 2017 *)

a(n)=4*n^4+17*n^2+4 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = (2*(n^2 - 1))^2 + (5*n)^2.
G.f.: (-4-25*x^4-11*x^3-51*x^2-5*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
E.g.f.: exp(x)*(4 + 21*x + 45*x^2 + 24*x^3 + 4*x^4). - Stefano Spezia, Jul 08 2023

A171638 Denominator of 1/(n-2)^2 - 1/(n+2)^2.

Original entry on oeis.org

0, 25, 9, 441, 64, 2025, 225, 5929, 576, 13689, 1225, 27225, 2304, 48841, 3969, 81225, 6400, 127449, 9801, 190969, 14400, 275625, 20449, 385641, 28224, 525625, 38025, 700569, 50176, 915849, 65025, 1177225, 82944, 1490841

Offset: 2

Paul Curtz, Dec 13 2009

Fifth column of an array of denominators related to the energies of the hydrogen spectrum, mentioned in A171522. At n=2, the defining formula has a pole and is replaced by 0 to conform with A171621 and A099761.

G. C. Greubel, Table of n, a(n) for n = 2..5000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A099761, A171621.

[0] cat [Denominator((1/(n-2)^2 -1/(n+2)^2)): n in [3..350]]; // Bruno Berselli, Apr 05 2011

A061037 := proc(n) 1/4-1/n^2 ; numer(%) ; end proc:
A171621 := proc(n) if n mod 4 = 2 then 4*A061037(n) ; else A061037(n) ; end if; end proc:
A171638 := proc(n) A171621(n)^2 ; end proc:
seq(A171638(n),n=2..90) ; # R. J. Mathar, Apr 02 2011

Table[If[n == 2, 0, Denominator[1/(n-2)^2 - 1/(n+2)^2]], {n, 2, 50}] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,25,9,441,64,2025,225,5929,576,13689},50] (* Harvey P. Dale, Sep 07 2021 *)

for(n=2,100, print1(if(n==2,0, denominator(1/(n-2)^2 - 1/(n+2)^2)), ", ")) \\ G. C. Greubel, Sep 20 2018

a(n) = (A171621(n))^2.
a(2*n+2) = A099761(n).
G.f.: -((x(25+9*x+316*x^2+19*x^3+70*x^4-5*x^5-36*x^6+x^7+9*x^8))/((-1+x)^5 (1+x)^5)). - Harvey P. Dale, Sep 07 2021
Sum_{n>=3} 1/a(n) = 19*Pi^2/192 - 115/144. - Amiram Eldar, Aug 14 2022

A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

Original entry on oeis.org

0, 0, 0, 12, 48, 120, 240, 420, 672, 1008, 1440, 1980, 2640, 3432, 4368, 5460, 6720, 8160, 9792, 11628, 13680, 15960, 18480, 21252, 24288, 27600, 31200, 35100, 39312, 43848, 48720, 53940, 59520, 65472, 71808, 78540, 85680, 93240, 101232, 109668, 118560

Offset: 0

Indranil Ghosh, Dec 25 2016

Consider all Pythagorean triples (X,Y,Z=Y+2) ordered by increasing Z; A005843, A005563, A002522 and A007531 give the X, Y, Z and area A values of related triangles; for n >= 2 altitude h(n) = a(n+1)/A002522(n) or h(n)/2 is irreducible fraction in Q\Z. - Ralf Steiner, Mar 29 2020

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A015237 (where the entries can be repeated), A005843, A005563, A002522, A016742, A099761, A007531.

Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* G. C. Greubel, Dec 25 2016 *)

for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ G. C. Greubel, Dec 25 2016

a(n)=12*binomial(n,3) \\ Charles R Greathouse IV, Dec 25 2016

def t(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            if a!=b:
                for c in range(0,n+1):
                    if a!=c and b!=c:
                        for d in range(0,n+1):
                            if d!=a and d!=b and d!=c:
                                if (a*d-b*c)==(a*d+b*c):
                                    s+=1
    return s
for i in range(0,201):
    print(str(i)+" "+str(t(i)))

a = lambda n: 2*n*(n-1)*(n-2) # David Radcliffe, Jun 14 2025

a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.
a(n) = 2*n*(n-1)*(n-2). - David Radcliffe, Jun 14 2025
a(n) == 0 (mod 12).
From G. C. Greubel, Dec 25 2016: (Start)
G.f.: (12*x^3)/(1 - x)^4.
E.g.f.: 2*x^3*exp(x).
a(n) = 2*n*(n-1)*(n-2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = 12 * A000292(n-2) for n>1. - Alois P. Heinz, Jan 30 2017
a(n+1) = sqrt(A016742(n)*A099761(n-1)) for n>=2. - Ralf Steiner, Mar 29 2020
From Amiram Eldar, Jun 30 2025: (Start)
Sum_{n>=3} 1/a(n) = 1/8.
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2) - 5/8. (End)

cp's OEIS Frontend

A007204 Crystal ball sequence for D_4 lattice.

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A244730 a(n) = 2*n^4.

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Magma

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A156701 a(n) = 4*n^4 + 17*n^2 + 4.

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Magma

Mathematica

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A171638 Denominator of 1/(n-2)^2 - 1/(n+2)^2.

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Magma

Maple

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A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

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Mathematica

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PARI

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Python

Formula

A156701 a(n) = 4n^4 + 17n^2 + 4.