A190958
a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.
Original entry on oeis.org
0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264
Offset: 0
Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
-
I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011
-
LinearRecurrence[{2,-10}, {0,1}, 50]
-
a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016
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[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022
A072547
Main diagonal of the array in which first column and row are filled alternatively with 1's or 0's and then T(i,j) = T(i-1,j) + T(i,j-1).
Original entry on oeis.org
1, 0, 2, 6, 22, 80, 296, 1106, 4166, 15792, 60172, 230252, 884236, 3406104, 13154948, 50922986, 197519942, 767502944, 2987013068, 11641557716, 45429853652, 177490745984, 694175171648, 2717578296116, 10648297329692, 41757352712480
Offset: 1
The array begins:
1 0 1 0 1..
0 0 1 1 2..
1 1 2 3 5..
0 1 3 6 11..
so sequence begins : 1, 0, 2, 6, ...
- L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
- David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, and Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.
- Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015. [It is only a conjecture that this is the same sequence. It would be nice to have a proof.]
- Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
- Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
- Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
-
a072547 n = a108561 (2 * (n - 1)) (n - 1)
-- Reinhard Zumkeller, Jan 03 2014
-
R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( x*(1 + Sqrt(1-4*x))/(Sqrt(1-4*x)*(3-Sqrt(1-4*x))) )); // G. C. Greubel, Feb 17 2019
-
taylor( (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^(-1),z=0,42); for n from -1 to 40 do a(n):=sum('(-1)^(p)*binomial(2n-p+1,1+n-p)',p=0..n+1): od:seq(a(n),n=-1..40):od; # Richard Choulet, Jan 25 2010
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CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x]) /(2*x))^(-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
a[n_] := Binomial[2 n - 2, n] Hypergeometric2F1[1, 2 - n, n + 1, 1/2] / 2 + (-2)^(1 - n); Table[a[n], {n, 1, 26}] (* Peter Luschny, Dec 03 2021 *)
-
a(n) = (-1)^n*sum(k=0, n, binomial(-n, k));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 02 2015
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a=(x*(1+sqrt(1-4*x))/(sqrt(1-4*x)*(3-sqrt(1-4*x)))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 17 2019
A088137
Generalized Gaussian Fibonacci integers.
Original entry on oeis.org
0, 1, 2, 1, -4, -11, -10, 13, 56, 73, -22, -263, -460, -131, 1118, 2629, 1904, -4079, -13870, -15503, 10604, 67717, 103622, 4093, -302680, -617639, -327238, 1198441, 3378596, 3161869, -3812050, -17109707, -22783264, 5762593, 79874978, 142462177, 45299420, -336787691
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
- Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
- Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 18.
- Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
- Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 6.
- Wikipedia, Lucas sequence
- Index entries for linear recurrences with constant coefficients, signature (2,-3).
- Index entries for Lucas sequences
-
[n le 2 select n-1 else 2*Self(n-1)-3*Self(n-2): n in [1..50]]; // G. C. Greubel, Oct 22 2018
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A[0]:= 0: A[1]:= 1:
for n from 2 to 100 do A[n]:= 2*A[n-1] - 3*A[n-2] od:
seq(A[n],n=0..100); # Robert Israel, Aug 05 2014
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LinearRecurrence[{2,-3},{0,1},40] (* Harvey P. Dale, Nov 03 2014 *)
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x='x+O('x^50); concat([0], Vec(x/(1-2*x+3*x^2))) \\ G. C. Greubel, Oct 22 2018
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[lucas_number1(n,2,3) for n in range(0, 38)] # Zerinvary Lajos, Apr 23 2009
A087455
Expansion of (1 - x)/(1 - 2*x + 3*x^2) in powers of x.
Original entry on oeis.org
1, 1, -1, -5, -7, 1, 23, 43, 17, -95, -241, -197, 329, 1249, 1511, -725, -5983, -9791, -1633, 26107, 57113, 35905, -99529, -306773, -314959, 290401, 1525679, 2180155, -216727, -6973919, -13297657, -5673557, 28545857, 74112385, 62587199, -97162757, -382087111, -472685951
Offset: 0
G.f. = 1 + x - x^2 - 5*x^3 - 7*x^4 + x^5 + 23*x6 + 43*x^7 + 17*x^8 - 95*x^9 + ...
- Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
- S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).
- Robert Israel, Table of n, a(n) for n = 0..3500
- Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
- A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
- F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
- M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
- Wikipedia, Lucas sequence
- Index entries for linear recurrences with constant coefficients, signature (2,-3).
- Index entries for sequences related to Benford's law
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[n le 2 select 1 else 2*Self(n-1) -3*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 03 2024
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Digits:=100; a:=n->round(abs(evalf((3^(n/2))*cos(n*arctan(sqrt(2))))));
# alternative:
a:= gfun:-rectoproc({a(n) = 2*a(n-1) - 3*a(n-2),a(0)=1,a(1)=1},a(n),remember):
map(a, [$0..100]); # Robert Israel, Jun 23 2015
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CoefficientList[Series[(1-x)/(1-2*x+3*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, Apr 01 2014 *)
a[ n_] := ChebyshevT[ n, 1/Sqrt[3]] Sqrt[3]^n // Simplify; (* Michael Somos, May 15 2015 *)
LinearRecurrence[{2,-3},{1,1},50] (* Harvey P. Dale, Jul 30 2019 *)
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{a(n) = real( (1 + quadgen(-8))^n )}; /* Michael Somos, Jul 26 2006 */
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{a(n) = real( subst( poltchebi(n), 'x, quadgen(12) / 3) * quadgen(12)^n)}; /* Michael Somos, Jul 26 2006 */
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a(n)=simplify(polchebyshev(n,,quadgen(12)/3)*quadgen(12)^n) \\ Charles R Greathouse IV, Jun 26 2013
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[sqrt(3)^n*chebyshev_T(n, 1/sqrt(3)) for n in range(41)] # G. C. Greubel, Jan 03 2024
A207538
Triangle of coefficients of polynomials v(n,x) jointly generated with A207537; see Formula section.
Original entry on oeis.org
1, 2, 4, 1, 8, 4, 16, 12, 1, 32, 32, 6, 64, 80, 24, 1, 128, 192, 80, 8, 256, 448, 240, 40, 1, 512, 1024, 672, 160, 10, 1024, 2304, 1792, 560, 60, 1, 2048, 5120, 4608, 1792, 280, 12, 4096, 11264, 11520, 5376, 1120, 84, 1, 8192, 24576, 28160, 15360
Offset: 1
First seven rows:
1
2
4...1
8...4
16..12..1
32..32..6
64..80..24..1
(2, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, ...) begins:
1
2, 0
4, 1, 0
8, 4, 0, 0
16, 12, 1, 0, 0
32, 32, 6, 0, 0, 0
64, 80, 24, 1, 0, 0, 0
128, 192, 80, 8, 0, 0, 0, 0
- Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358.
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207538, |A133156| *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)
t[n_, k_] := t[n, k] = 2^(n - 2 k) * (n - k)!/((n - 2 k)! k!) ; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]} ] // Flatten (* Zagros Lalo, Jul 31 2018 *)
A138230
Expansion of (1-x)/(1 - 2*x + 4*x^2).
Original entry on oeis.org
1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592
Offset: 0
- Michael De Vlieger, Table of n, a(n) for n = 0..3322
- Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
- Index entries for linear recurrences with constant coefficients, signature (2,-4).
-
[2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023
-
CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* Harvey P. Dale, Nov 11 2014 *)
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[2^n*chebyshev_T(n,1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023
A100192
a(n) = Sum_{k=0..n} binomial(2*n,n+k)*2^k.
Original entry on oeis.org
1, 4, 18, 82, 374, 1704, 7752, 35214, 159750, 723880, 3276908, 14821668, 66991436, 302605528, 1366182276, 6165204102, 27811282374, 125415953208, 565408947756, 2548400193852, 11483706241044, 51739037228688, 233070330199296, 1049777052815052, 4727770393417884
Offset: 0
-
CoefficientList[Series[Sqrt[1-4*x]*(Sqrt[1-4*x]-3*x+1)/((1-4*x)*(2-9*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
A168175
Expansion of 1/(1 - 4*x + 7*x^2).
Original entry on oeis.org
1, 4, 9, 8, -31, -180, -503, -752, 513, 7316, 25673, 51480, 26209, -255524, -1205559, -3033568, -3695359, 6453540, 51681673, 161551912, 284435937, 6880364, -1963530103, -7902282960, -17864421119, -16141703756, 60484132809
Offset: 0
G.f. = 1 + 4*x + 9*x^2 + 8*x^3 - 31*x^4 - 180*x^5 - 503*x^6 - 752*x^7 + ... - _Michael Somos_, Feb 23 2020
- Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.
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I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012
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CoefficientList[Series[1/(1-4x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {4,-7},{1,4},30] (* Harvey P. Dale, Nov 28 2014 *)
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{a(n) = my(s=1, t=1); if( n<0, n=-2-n; s=-1; t=1/7); s * t^(n+1) * polcoeff(1 / (1 - 4*x + 7*x^2) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2020 */
A213421
Real part of Q^n, Q being the quaternion 2+i+j+k.
Original entry on oeis.org
1, 2, 1, -10, -47, -118, -143, 254, 2017, 6290, 11041, 134, -76751, -307942, -694511, -622450, 2371777, 13844258, 38774593, 58188566, -38667887, -561991510, -1977290831, -3975222754, -2059855199, 19587138482
Offset: 0
- Stanislav Sykora, Table of n, a(n) for n = 0..1000
- Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras Vol. 29, No. 3 (2019), Article 54.
- Wikipedia, Lucas sequence
-
#A213421
seq(simplify(1/2*((2+I*sqrt(3))^n+(2-I*sqrt(3))^n)), n = 0 .. 25); # Peter Bala, Mar 29 2015
-
QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);}
Q=QuaternionToN(2,1,1,1,1000);
for(n=1,#Q[,1],write("A213421.txt",n-1," ",Q[n,1]));
A100067
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*2^(n-2*k).
Original entry on oeis.org
1, 2, 6, 14, 38, 92, 240, 590, 1510, 3740, 9476, 23564, 59372, 147968, 371636, 927374, 2324870, 5805740, 14538660, 36322340, 90898228, 227153192, 568235696, 1420236524, 3551943388, 8878506392, 22201466280, 55498465400, 138766221800, 346895496200, 867316299260, 2168213189390
Offset: 0
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m:=2; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022
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CoefficientList[Series[x/(Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]+x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)
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my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)*(sqrt(1-4*x^2)+x-1))) \\ Joerg Arndt, May 12 2013
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m=2; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022
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