A060006 -id:A060006 - cp's OEIS frontend

A276276 a(n) = a(n-2)+a(n-3) with a(1)=2 a(2)=1 a(3)=0.

2, 1, 0, 3, 1, 3, 4, 4, 7, 8, 11, 15, 19, 26, 34, 45, 60, 79, 105, 139, 184, 244, 323, 428, 567, 751, 995, 1318, 1746, 2313, 3064, 4059, 5377, 7123, 9436, 12500, 16559, 21936, 29059, 38495, 50995, 67554, 89490

Offset: 1

Nicolas Bègue, Aug 26 2016

Sequence is obtained from modulo 3 periodic sequence of Padovan numbers 1112201210010, it satisfies the same recurrence a(n) = a(n-2) + a(n-3).

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

Cf. A000931, A060006.

RecurrenceTable[{a[n] == a[n - 2] + a[n - 3], a[1] == 2, a[2] == 1, a[3] == 0}, a, {n, 1, 43}] (* or *)
CoefficientList[Series[(2 x^2 - x - 2)/(x^3 + x^2 - 1), {x, 0, 42}], x] (* Michael De Vlieger, Aug 28 2016 *)

a(n)=([0,1,0; 0,0,1; 1,1,0]^(n-1)*[2;1;0])[1,1] \\ Charles R Greathouse IV, Aug 28 2016

G.f.: (2x^2-x-2)/(x^3+x^2-1).

a(n) = A000931(n) - A000931(n-9), for n>2.

More terms from Charles R Greathouse IV, Aug 28 2016

A278766 Engel expansion of plastic constant (real root of x^3 - x - 1).

Original entry on oeis.org

1, 4, 4, 6, 6, 27, 74, 86, 372, 853, 947, 1475, 3686, 9084, 19174, 30737, 1530833, 2401466, 2521253, 3649563, 3802245, 9320024, 1092256819, 2114664794, 2878948610, 8842525055, 13769551820, 26996892389, 215947176106, 269439735691, 13694290818678, 18312336654245, 19649485782723, 63266709043539

Offset: 1

Ilya Gutkovskiy, Nov 28 2016

			(1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3) = 1.324717957244... = 1/1 + 1/(1*4) + 1/(1*4*4) + 1/(1*4*4*6) + 1/(1*4*4*6*6) + 1/(1*4*4*6*6*27) + ...

Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Plastic Constant
Index entries for sequences related to Engel expansions

Cf. A006784 (for definition of Engel expansion).

Cf. A060006, A072117.

EngelExp[A_, n_]:=Join[Array[1&, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]], Expand[ #[[1]]#[[2]]-1]}&, {Ceiling[1/(A-Floor[A])], A-Floor[A]}, n-1]]; EngelExp[N[(1/2 + Sqrt[23/108])^(1/3) + (1/2 - Sqrt[23/108])^(1/3), 7! ], 40]

A301509 Decimal expansion of (1 + (1 + (1 + (1 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

Original entry on oeis.org

1, 5, 1, 7, 6, 0, 0, 1, 6, 7, 8, 7, 7, 7, 1, 8, 8, 9, 1, 3, 7, 0, 6, 5, 8, 0, 4, 3, 0, 6, 3, 4, 1, 6, 2, 7, 1, 8, 8, 5, 1, 5, 7, 0, 0, 6, 7, 9, 1, 9, 7, 6, 0, 3, 3, 8, 6, 7, 7, 0, 4, 3, 0, 1, 4, 7, 2, 1, 3, 5, 2, 2, 5, 3, 2, 3, 5, 2, 2, 8, 2, 1, 1, 2, 1, 7, 7, 8, 5, 6, 1, 4, 5, 1, 3, 9, 3, 3, 8, 8

Offset: 1

Ilya Gutkovskiy, Mar 22 2018

			1.517600167877718891370658043063416271885...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A001622, A060006, A060007, A160155, A296547.

A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

Original entry on oeis.org

1, 3, 7, 14, 25, 40, 56, 63, 37, -71, -350, -945, -2064, -3952, -6783, -10381, -13625, -13330, -2359, 33208, 117672, 288959, 598325, 1099385, 1812546, 2640543, 3197152, 2497824, -1541375, -12816925, -37865849, -86422322, -170718343, -301444536, -476474600, -655816385, -713055419, -351058887, 1028750562, 4501424879, 11797832400, 25361896880, 47988600961

Offset: 0

Raul Prisacariu, Dec 18 2023

sign

Whittaker's Root Series Formula is applied to the polynomial equation -1+2x+3x^2+x^3. The following infinite series involving the Plastic Ratio (rho) is obtained: rho - 1 = 1/2 - 3/(2*7) + 7/(7*21) - 14/(21*65) + 25/(65*200) - 40/(200*616) + 56/(616*1897) - ...

The terms of the sequence appear in the numerators of the infinite sequence (with alternating signs). The denominators of the sequence are formed by multiplying consecutive terms from the sequence A218836.

			a(0) = 1,
a(1) = 3*a(0) = 3*1 = 3,
a(2) = 3*a(1) - 2*a(0) = 3*3 - 2*1 = 7,
a(3) = 3*a(2) - 2*a(1) - a(0) = 3*7 - 2*3 - 1 = 14.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A218836 (denominator), A060006.

a:=proc(n) local c1,c2,c3;
 option remember;
c1:=3; c2:=2; c3:=1;
if n=0 then 1
elif n=1 then 3
elif n=2 then 7
else c1*a(n-1)-c2*a(n-2)-c3*a(n-3); fi;
end; # N. J. A. Sloane, Dec 31 2023
[seq(a(n),n=0..30)];

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

a(n) = determinant of the n X n Toeplitz Matrix((3,2,-1,0,0,...,0),(3,1,0,0,0,...,0)).

A372244 Decimal expansion of the maverick constant lambda^*.

Original entry on oeis.org

2, 0, 1, 9, 8, 0, 0, 8, 8, 7, 0, 9, 0, 4, 6, 7, 3, 3, 7, 4, 8, 7, 1, 7, 1, 1, 8, 2, 6, 5, 2, 4, 7, 3, 0, 8, 6, 1, 4, 4, 4, 2, 0, 3, 4, 3, 6, 3, 0, 6, 8, 4, 4, 7, 4, 2, 7, 2, 0, 0, 8, 2, 4, 3, 2, 6, 4, 5, 2, 6, 2, 2, 3, 4, 0, 8, 9, 5, 1, 2, 3, 1, 9, 1, 6, 8, 1

Offset: 1

Eric W. Weisstein, Apr 24 2024

			2.019800887090467337...

Hricha Acharya and Zilin Jiang, Beyond the Classification Theorem of Cameron, Goethals, Seidel, and Shult, arXiv:2404.13136 [math.CO], 2024.
Eric Weisstein's World of Mathematics, Maverick Graph.

Cf. A060006 (decimal digits of the plastic constant).

RealDigits[Root[-1 + 4 #^2 - 5 #^4 + #^6& , 2, 0], 10, 87][[1]]

lambda^* = rho^(1/2) + rho^(-1/2), where rho is the plastic constant with decimal digits A060006.

cp's OEIS Frontend

A276276 a(n) = a(n-2)+a(n-3) with a(1)=2 a(2)=1 a(3)=0.

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A278766 Engel expansion of plastic constant (real root of x^3 - x - 1).

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A301509 Decimal expansion of (1 + (1 + (1 + (1 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

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A368205 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

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A372244 Decimal expansion of the maverick constant lambda^*.

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A368205 a(n) = 3a(n-1) - 2a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.