A078001 -id:A078001 - cp's OEIS frontend

A104862 First differences of A014292.

0, 1, 1, 1, 1, 0, -2, -5, -9, -13, -15, -12, 0, 25, 65, 117, 169, 196, 158, 3, -321, -841, -1519, -2200, -2560, -2079, -79, 4121, 10881, 19720, 28638, 33435, 27351, 1547, -52895, -140772, -256000, -372775, -436655, -359763, -26871

Offset: 0

Gerald McGarvey, Apr 24 2005

sign

Real part of the sequence of complex numbers defined by c(n) = c(n-1) + i*c(n-2) for n > 1, c(0) = 1, c(1) = 1.

a(n) = real part of the sequence b of quaternions defined by b(0)=1, b(1)=1, b(n) = b(n-1) + b(n-2)*(0,s,s,s) where s = 1/sqrt(3).

Michael De Vlieger, Table of n, a(n) for n = 0..6285

Cf. A078001, A014292.

Differences@ LinearRecurrence[{2, -1, 0, -1}, {0, 0, 1, 2}, 42] (* Michael De Vlieger, Mar 19 2021 *)

a = [0]*1000
a[1]=1
for n in range(1,55):
    print(a[n-1], end=", ")
    s=sum(a[k] for k in range(n-2))
    a[n+1] = a[n]-s
# from Alex Ratushnyak, May 03 2012

G.f.: Re(1/(1-x-ix^2)) = (1-x)/(1-2x+x^2+x^4). - Paul Barry, Apr 25 2005
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*cos(Pi*k/2). - Paul Barry, Apr 25 2005
a(0)=0, a(1)=1, a(n+1) = a(n) - Sum_{k=0..n-3} a(k). - Alex Ratushnyak, May 03 2012

A348322 a(n) = Sum_{k=0..n} (-1)^k * binomial(n^2 - k,n*k).

Original entry on oeis.org

1, 1, -2, -48, 1626, 931040, -479909170, -5499596761127, 43158334880135692, 9081843551946977373216, -1095541637114516172591381711, -4049135740387789992460066844854898, 7569951149407063102291625516677078697579

Offset: 0

Seiichi Manyama, Oct 12 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..57

Cf. A078001, A348315, A348321.

a[n_] := Sum[(-1)^k * Binomial[n^2 - k, n*k], {k, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, Oct 12 2021 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n^2-k, n*k));

a(n) = polcoef((1-x)^(n-1)/((1-x)^n+x^(n+1)+x*O(x^n^2)), n^2);

a(n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n + x^(n+1)).

A077856 Expansion of (1-x)^(-1)/(1-2*x+x^2+x^3).

Original entry on oeis.org

1, 3, 6, 9, 10, 6, -6, -27, -53, -72, -63, 0, 136, 336, 537, 603, 334, -471, -1878, -3618, -4886, -4275, -45, 9072, 22465, 35904, 40272, 22176, -31823, -126093, -242538, -327159, -285686, -1674, 609498, 1506357, 2404891, 2693928, 1476609, -2145600, -8461736, -16254480, -21901623

Offset: 0

N. J. A. Sloane, Nov 17 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,0,1).

Cf. A078001.

I:=[1,3,6,9]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, May 27 2016

LinearRecurrence[{3, -3, 0, 1}, {1, 3, 6, 9}, 50] (* Vincenzo Librandi, May 27 2016 *)

Vec((1-x)^(-1)/(1-2*x+x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = Sum_{k=1..floor(n/3+1)} (-1)^k*binomial(n-k+3, 2*k). - Vladeta Jovovic, Feb 10 2003

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-4). - Chai Wah Wu, May 25 2016

cp's OEIS Frontend

A104862 First differences of A014292.

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A348322 a(n) = Sum_{k=0..n} (-1)^k * binomial(n^2 - k,n*k).

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Author

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PARI

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A077856 Expansion of (1-x)^(-1)/(1-2*x+x^2+x^3).

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula