A001834 -id:A001834 - cp's OEIS frontend

A335312 A(n, k) = k! [x^k] exp(2x)(ysinh(xy) + cosh(x*y)) and y = sqrt(n). Square array read by ascending antidiagonals, for n >= 0 and k >= 0.

1, 1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 14, 27, 16, 1, 6, 19, 48, 81, 32, 1, 7, 24, 71, 164, 243, 64, 1, 8, 29, 96, 265, 560, 729, 128, 1, 9, 34, 123, 384, 989, 1912, 2187, 256, 1, 10, 39, 152, 521, 1536, 3691, 6528, 6561, 512, 1, 11, 44, 183, 676, 2207, 6144, 13775, 22288, 19683, 1024

Offset: 0

Peter Luschny, Jun 24 2020

			[0] 1, 2, 4,    8,  16,   32,    64,   128,    256,     512, ...  [A000079]
[1] 1, 3, 9,   27,  81,  243,   729,  2187,   6561,   19683, ...  [A000244]
[2] 1, 4, 14,  48, 164,  560,  1912,  6528,  22288,   76096, ...  [A007070]
[3] 1, 5, 19,  71, 265,  989,  3691, 13775,  51409,  191861, ...  [A001834]
[4] 1, 6, 24,  96, 384, 1536,  6144, 24576,  98304,  393216, ...  [A164908]
[5] 1, 7, 29, 123, 521, 2207,  9349, 39603, 167761,  710647, ...  [A048876]
[6] 1, 8, 34, 152, 676, 3008, 13384, 59552, 264976, 1179008, ...  [A335749]

Cf. A000079 (n=0), A000244 (n=1), A007070 (n=2), A001834 (n=3), A164908 (n=4), A048876 (n=5), A335749 (n=6).

Cf. A335537, A118800.

Arow := proc(n, len) local H; H := (x, y) -> exp(2*x)*(y*sinh(x*y) + cosh(x*y)):
series(H(x, sqrt(n)), x, len+1): seq(k!*coeff(%, x, k), k=0..len-1) end:
A := (n, k) -> Arow(n, k+2)[k+1]: seq(lprint(Arow(n, 9)), n=0..6);
# Alternative:
A := proc(n, k) option remember; if k = 0 then return 1 fi;
if k = 1 then return n+2 fi; 4*A(n, k-1) + (n-4)*A(n, k-2) end;

The Taylor series of exp(2*x)*(y*sinh(x*y) + cosh(x*y)) starts: 1 + x*(y^2 + 2) + x^2*((5*y^2)/2 + 2) + (1/6)*x^3*(y^4 + 18*y^2 + 8) + x^4*((3*y^4)/8 + (7*y^2)/3 + 2/3) + O(x^5). The coefficient polynomials expand in even powers (cf. A118800).
A(n, k) = k! [x^k] (c*exp(x*(1 + c)) + d*exp(x*(1 + d)))/2 where c = 1 + sqrt(n) and d = 1 - sqrt(n).
A(n, k) = 4*A(n, k-1) + (n-4)*A(n, k-2) if k >= 2. A(n, 0) = 1, A(n, 1) = n + 2.

A055269 a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 30, 116, 437, 1635, 6106, 22792, 85065, 317471, 1184822, 4421820, 16502461, 61588027, 229849650, 857810576, 3201392657, 11947760055, 44589647566, 166410830212, 621053673285, 2317803862931, 8650161778442, 32282843250840, 120481211224921, 449642001648847

Offset: 0

Barry E. Williams, May 10 2000

Also partial sums of A054491.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001834, A054491.

I:=[1,7,30]; [n le 3 select I[n] else 5*Self(n-1) - 5*Self(n-2) + Self(n-3): n in [1..40]]; // G. C. Greubel, Mar 16 2020

A055269:= n-> simplify((5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2); seq( A055269(n), n=0..40); # G. C. Greubel, Mar 16 2020

LinearRecurrence[{5,-5,1},{1,7,30},40] (* or *) CoefficientList[ Series[ (1+2*x)/(1-5*x+5*x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *)
Table[(5*ChebyshevU[n, 2] -3*ChebyshevU[n-1, 2] - 3)/2, {n,0,40}] (* G. C. Greubel, Mar 16 2020 *)

[(5*chebyshev_U(n, 2) - 3*chebyshev_U(n-1, 2) - 3)/2 for n in (0..40)] # G. C. Greubel, Mar 16 2020

G.f.: (1+2*x)/((1-x)*(1-4*x+x^2)).
a(n) = ( ( (17 - 5*(2-sqrt(3)))*(2+sqrt(3))^n + (5*(2+sqrt(3))-17)*(2-sqrt(3))^n )/(4*sqrt(3)) ) - 3/2.
a(n) = (5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2. - G. C. Greubel, Mar 16 2020

Corrected by T. D. Noe, Nov 07 2006

A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

Original entry on oeis.org

4, 17, 65, 244, 912, 3405, 12709, 47432, 177020, 660649, 2465577, 9201660, 34341064, 128162597, 478309325, 1785074704, 6661989492, 24862883265, 92789543569, 346295291012, 1292391620480, 4823271190909, 18000693143157

Offset: 0

Creighton Dement, Dec 30 2004

nonn

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

Cf. A061278, A092184, A001834, A001353, A102206.

a[0] = 4; a[1] = 17; a[2] = 65; a[n_] := a[n] = 5a[n - 1] - 5a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 22}] (* Or *)
CoefficientList[ Series[(3x - 4)/((x - 1)(x^2 - 4x + 1)), {x, 0, 22}], x] (* Robert G. Wilson v, Jan 12 2005 *)
LinearRecurrence[{5,-5,1},{4,17,65},30] (* or *) With[{c=Sqrt[3]},Table[ Simplify[ ((3-7c)(2-c)^x+(2+c)^x (3+7c)-6)/12],{x,30}]] (* Harvey P. Dale, Mar 15 2013 *)

G.f.: (3x-4)/((x-1)(x^2-4x+1))
(1/2) [A001353(n+1) + 5*A001353(n) - 1 ]. - Ralf Stephan, May 17 2007
a(n)=1/12*((3-7*Sqrt[3])*(2-Sqrt[3])^n+(3+7*Sqrt[3])*(2+Sqrt[3])^n-6). - Harvey P. Dale, Mar 15 2013

More terms from Robert G. Wilson v, Jan 12 2005

A107388 Expansion of x(3x-1)(2x-1) / ( (1-x)(1+x)(x^2-4*x+1) ).

Original entry on oeis.org

0, 1, -1, 2, 4, 21, 75, 286, 1064, 3977, 14839, 55386, 206700, 771421, 2878979, 10744502, 40099024, 149651601, 558507375, 2084377906, 7779004244, 29031639077, 108347552059, 404358569166, 1509086724600, 5631988329241, 21018866592359, 78443478040202

Offset: 0

Roger L. Bagula, May 24 2005, corrected Sep 04 2008

One of the components of the n-th power of the 4X4 matrix with rows (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,4,0,-4) multiplied by the vector (0,1,1,2).

Index entries for linear recurrences with constant coefficients, signature (4,0,-4,1).

CoefficientList[Series[x(3x-1)(2x-1)/((1-x)(1+x)(x^2-4x+1)),{x,0,40}],x] (* or *) LinearRecurrence[ {4,0,-4,1},{0,1,-1,2},40] (* Harvey P. Dale, Jun 17 2023 *)

a(n) = (-1)^(n+1)+(3*A001353(n+1)-11*A001353(n)-1)/2. - R. J. Mathar, Jul 10 2012

a(n) -a(n-2) = A001834(n-3). - R. J. Mathar, Dec 17 2017

Sign of a(2) flipped. - R. J. Mathar, Jul 10 2012

A254308 a(n) = a(n-1) + (if a(n-1) is odd a(n-2) else a(n-3)) with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 11, 19, 30, 41, 71, 112, 153, 265, 418, 571, 989, 1560, 2131, 3691, 5822, 7953, 13775, 21728, 29681, 51409, 81090, 110771, 191861, 302632, 413403, 716035, 1129438, 1542841, 2672279, 4215120, 5757961, 9973081, 15731042, 21489003, 37220045

Offset: 0

Russell Walsmith, Feb 23 2015

nonn

Every third iteration is a tribonacci-type recursion: a(n) = a(n-1) + a(n-3) otherwise it is Fibonacci-type a(n) = a(n-1) + a(n-2).

			For n = 7, a(n-1) is even so 8 + 3 = 11.
G.f. = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 19*x^8 + 30*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Russell Walsmith, Fibonacci-Tribonacci Fusion
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Cf. A001834, A001835, A052530.

a254308 n = a254308_list !! n
a254308_list = 0 : 1 : 1 : zipWith3 (\u v w -> u + if odd u then v else w)
               (drop 2 a254308_list) (tail a254308_list) a254308_list
-- Reinhard Zumkeller, Feb 24 2015

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+2*x^2-x^3+x^4)/(1-4*x^3+x^6))); // G. C. Greubel, Aug 03 2018

CoefficientList[Series[x*(1+x+2*x^2-x^3+x^4)/(1-4*x^3+x^6), {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2018 *)
nxt[{a_,b_,c_}]:={b,c,If[OddQ[c],c+b,c+a]}; NestList[nxt,{0,1,1},50][[All,1]] (* or *) LinearRecurrence[{0,0,4,0,0,-1},{0,1,1,2,3,5},50] (* Harvey P. Dale, May 12 2022 *)

{a(n) = polcoeff( x * if( n<0, n=-n; -(1 - x + 2*x^2 + x^3 + x^4), (1 + x + 2*x^2 - x^3 + x^4)) / (1 - 4*x^3 + x^6) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2015 */

Two identities: a(3n)/2 + a(3n-3)/2 = a(3n-1); a(3n)/2 - a(3n-3)/2 = a(3n-2).
G.f.: x * (1 + x + 2*x^2 - x^3 + x^4) / (1 - 4*x^3 + x^6). - Michael Somos, Feb 23 2015
0 = a(n) - 4*a(n+3) + a(n+6) for all n in Z. - Michael Somos, Feb 23 2015
a(3*n) = A052530(n). a(3*n-2) = A001835(n). a(3*n+2) = A001834(n). - Michael Somos, Feb 23 2015

A341671 Solutions y of the Diophantine equation 3*(x^2+x+1) = y^2.

Original entry on oeis.org

3, 39, 543, 7563, 105339, 1467183, 20435223, 284625939, 3964327923, 55215964983, 769059181839, 10711612580763, 149193516948843, 2077997624703039, 28942773228893703, 403120827579808803, 5614748812888429539, 78203362552858204743, 1089232326927126436863, 15171049214426911911339

Offset: 1

Bernard Schott, Feb 17 2021

nonn

Corresponding x are in A028231.
This equation belongs to the family of equations studied by Kustaa A. Inkeri, y^m = a * (x^q-1)/(x-1) with here: m=2, a=3, q=3. This equation is exhibed in A307745 by Giovanni Resta to prove that this sequence has infinitely many terms.
This Diophantine equation 3*(x^2+x+1) = y^2 has infinitely many solutions because the Pell-Fermat equation u^2 - 3*v^2 = -2 also has infinitely many solutions. The corresponding (u,v) are in (A001834, A001835) and for each pair (u,v), the corresponding solutions of 3*(x^2+x+1) = y^2 are x = (3*u*v-1)/2 and y = 3*(u^2+1)/2.
Note that if y = 3*z, this equation becomes 3*z^2 = x^2+x+1 with solutions (x, z) = (A028231, A001570).

			The first few values for (x,y) are (1,3), (22,39), (313,543), (4366,7563), (60817,105339), ...

Kustaa A. Inkeri, On the Diophantine equation a(x^n-1)/(x-1) = y^m, Acta Arithmetica, Vol. 21, No. 1 (1972), pp. 299-311.

Cf. A001834, A001835, A001570, A028231, A307745.

Subsequence of A158235, for a(n)>3.

f[x_] := Sqrt[3*(x^2 + x + 1)]; f /@ LinearRecurrence[{15, -15, 1}, {1, 22, 313}, 20] (* Amiram Eldar, Feb 17 2021 *)

a(n) = 3*A001570(n). - Hugo Pfoertner, Feb 17 2021

a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).

More terms from Amiram Eldar, Feb 17 2021

A342568 1/a(n) is the current through the resistor at the central rung of an electrical ladder network made of 6*n+1 one-ohm resistors, fed by 1 volt at diametrically opposite ends of the ladder.

Original entry on oeis.org

1, 7, 45, 239, 1157, 5307, 23497, 101467, 430089, 1796975, 7422437, 30373191, 123327373, 497484067, 1995542913, 7965875891, 31663779857, 125391332055, 494914083229, 1947613051807, 7643917792917, 29928228744587, 116921300015417, 455866012012299, 1774126607984665

Offset: 0

Hugo Pfoertner, Jun 10 2021

			   a(0)=1         a(1)=7                      a(2)=45
   o 1V       o-----o-----o 1V       o-----o-----o-----o-----o 1V
   |          |     |     |          |     |     |     |     |
   |          |     |     |          |     |     |     |     |
  1 A         |   1/7 A   |          |     |   1/45 A  |     |
   |          |     |     |          |     |     |     |     |
   |          |     |     |          |     |     |     |     |
   o 0V    0V o-----o-----o       0V o-----o-----o-----o-----o

Bisection (odd indices) of A093652.
A001834(n)/a(n) is the total resistance of the network between diametrically opposite ends of the ladder.
Cf. A082630.

a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4); a(0)=1, a(1)=7, a(2)=45, a(3)=239.

G.f.: (3*x^3 - 7*x^2 + x - 1)/(x^2 - 4*x + 1)^2. - Joerg Arndt, Jun 11 2021

A087945 Expansion of (1-2x-x^2)/((1-2x)(1-4x+x^2)).

Original entry on oeis.org

1, 4, 14, 50, 182, 670, 2482, 9226, 34358, 128078, 477698, 1782202, 6650086, 24816094, 92610194, 345616490, 1289839382, 4813708270, 17964928162, 67045873306, 250218302918, 933826814078, 3485087904818, 13006522708042

Offset: 0

Paul Barry, Sep 16 2003

First differences of A087944. Binomial transform of A052948(n+1). a(n)=(2/3)A001834+2^n/3.

Index entries for linear recurrences with constant coefficients, signature (6,-9,2).

CoefficientList[Series[(1-2x-x^2)/((1-2x)(1-4x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-9,2},{1,4,14},30] (* Harvey P. Dale, Aug 21 2021 *)

a(0)=1, a(2)=4, a(2)=14, a(n)=6a(n-1)-9a(n-2)+2a(n-3), n>2; a(n)=(2^n+(1-sqrt(3))(2-sqrt(3))^n+(1+sqrt(3))(2+sqrt(3))^n)/3.

A106851 Expansion of (-3x^3 - 7x^2 + 2x)/((1-4x-x^2)(1-4x+x^2)).

Original entry on oeis.org

0, 2, 9, 37, 152, 626, 2585, 10701, 44400, 184610, 769065, 3209461, 13415048, 56153618, 235357241, 987609501, 4148575200, 17443003202, 73402179657, 309116995525, 1302649664888, 5492768393906, 23173154692697, 97810060234605

Offset: 0

Roger L. Bagula, May 30 2005

Index entries for linear recurrences with constant coefficients, signature (8, -16, 0, 1).

CoefficientList[Series[(-3 x^3-7x^2+2x)/((1-4x-x^2)(1-4x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-16,0,1},{0,2,9,37},31] (* Harvey P. Dale, Aug 05 2011 *)

G.f.: (-3*x^3 - 7*x^2 + 2*x)/((1-4*x-x^2)*(1-4*x+x^2)).
a(n) = (1/2) * [A001834(n-1) + Fibonacci(3n+1) ]. - Ralf Stephan, Nov 18 2010
a(0)=0, a(1)=2, a(2)=9, a(3)=37, a(n)=8*a(n-1)-16*a(n-2)+a(n-4) [Harvey P. Dale, Aug 05 2011]

Edited by N. J. A. Sloane, Apr 09 2007

New name from Joerg Arndt, Dec 26 2022

A255445 Number of ON cells after n generations of the odd-rule cellular automaton defined by OddRule 037 when started with a single ON cell.

Original entry on oeis.org

1, 5, 5, 19, 5, 25, 19, 71, 5, 25, 25, 95, 19, 95, 71, 265, 5, 25, 25, 95, 25, 125, 95, 355, 19, 95, 95, 361, 71, 355, 265, 989, 5, 25, 25, 95, 25, 125, 95, 355, 25, 125, 125, 475, 95, 475, 355, 1325, 19, 95, 95, 361, 95, 475, 361, 1349, 71, 355, 355, 1349

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 23 2015

nonn

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Run length transform of A001834.

(* b = a1834 *) b[0] = 1; b[1] = 5; b[n_] := b[n] = 4 b[n-1] - b[n-2];
Table[Times @@ (b[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 59}] (* Jean-François Alcover, Sep 15 2018 *)

cp's OEIS Frontend

A335312 A(n, k) = k! [x^k] exp(2*x)*(y*sinh(x*y) + cosh(x*y)) and y = sqrt(n). Square array read by ascending antidiagonals, for n >= 0 and k >= 0.

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A055269 a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.

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A102207 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.

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A107388 Expansion of x*(3*x-1)*(2*x-1) / ( (1-x)*(1+x)*(x^2-4*x+1) ).

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A254308 a(n) = a(n-1) + (if a(n-1) is odd a(n-2) else a(n-3)) with a(0) = 0, a(1) = 1.

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A341671 Solutions y of the Diophantine equation 3*(x^2+x+1) = y^2.

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A342568 1/a(n) is the current through the resistor at the central rung of an electrical ladder network made of 6*n+1 one-ohm resistors, fed by 1 volt at diametrically opposite ends of the ladder.

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A087945 Expansion of (1-2x-x^2)/((1-2x)(1-4x+x^2)).

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A335312 A(n, k) = k! [x^k] exp(2x)(ysinh(xy) + cosh(x*y)) and y = sqrt(n). Square array read by ascending antidiagonals, for n >= 0 and k >= 0.

A107388 Expansion of x(3x-1)(2x-1) / ( (1-x)(1+x)(x^2-4*x+1) ).

A106851 Expansion of (-3x^3 - 7x^2 + 2x)/((1-4x-x^2)(1-4x+x^2)).