A084158 -id:A084158 - cp's OEIS frontend

A084159 Pell oblongs.

1, 3, 21, 119, 697, 4059, 23661, 137903, 803761, 4684659, 27304197, 159140519, 927538921, 5406093003, 31509019101, 183648021599, 1070379110497, 6238626641379, 36361380737781, 211929657785303, 1235216565974041, 7199369738058939, 41961001862379597, 244566641436218639

Offset: 0

Paul Barry, May 18 2003

Essentially the same as A046727.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 60 at p. 123.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. E. Trier, "Almost Isosceles" Right-Angled Triangles, Eureka, No. 4, May 1940, pp. 9 - 11.
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A046727 (same sequence except for first term).

Cf. A001333, A001654, A078057, A084158, A084175, A090390, A182432, A211955.

[Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)+2*(-1)^n)/4): n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

b[n_]:= Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]];
Join[{1}, Table[b[n+1], {n,50}]*Table[b[n], {n,50}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
LinearRecurrence[{5,5,-1},{1,3,21},30] (* Harvey P. Dale, Aug 04 2019 *)

[(lucas_number2(2*n+1, 2, -1) + 2*(-1)^n)/4 for n in range(31)] # G. C. Greubel, Oct 11 2022

a(n) = ((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) + 2*(-1)^n)/4.
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). - Paul Curtz, May 17 2008
G.f.: (1-x)^2/((1+x)*(1-6*x+x^2)). - R. J. Mathar, Sep 17 2008
a(n) = A078057(n)*A001333(n). - R. J. Mathar, Jul 08 2009
a(n) = A001333(n)*A001333(n+1).
From Peter Bala, May 01 2012: (Start)
a(n) = (-1)^n*R(n,-4), where R(n,x) is the n-th row polynomial of A211955.
a(n) = (-1)^n*1/u*T(n,u)*T(n+1,u) with u = sqrt(-1) and T(n,x) the Chebyshev polynomial of the first kind.
a(n) = (-1)^n + 4*Sum_{k = 1..n} (-1)^(n-k)*8^(k-1)*binomial(n+k,2*k).
Recurrence equations: a(n) = 6*a(n-1) - a(n-2) + 4*(-1)^n, with a(0) = 1 and a(1) = 3; a(n)*a(n-2) = a(n-1)*(a(n-1)+4*(-1)^n).
Sum_{k >= 0} (-1)^k/a(k) = 1/sqrt(2).
1 - 2*(Sum_{k = 0..n} (-1)^k/a(k))^2 = (-1)^(n+1)/A090390(n+1). (End)
a(n) = (A001333(2*n+1) + (-1)^n)/2. - G. C. Greubel, Oct 11 2022
E.g.f.: exp(-x)*(1 + exp(4*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/2. - Stefano Spezia, Aug 03 2024

A163960 Decimal expansion of 2*(sqrt(2) - 1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 02 2010

Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)). (See A195284.) - Clark Kimberling, Sep 14 2011

			0.82842712474619009760337744841939615713934375075389614635335...

J. M. Steele, Probability Theory and Combinatorial Optimization, SIAM, 1997, p. 3.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for algebraic numbers, degree 2.

Cf. A084158, A156035, A195284.

Essentially the same digit sequence as A010466, A086178, A090488 and A157258.

RealDigits[2(Sqrt[2]-1),10,120][[1]] (* Harvey P. Dale, May 27 2016 *)

2*(sqrt(2)-1) \\ G. C. Greubel, Aug 13 2017

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 4^k). - Amiram Eldar, May 06 2022

Equals Sum_{k>=1} (-1)^(k+1)/A084158(k). - Amiram Eldar, Dec 02 2024

A096979 Sum of the areas of the first n+1 Pell triangles.

Original entry on oeis.org

0, 1, 6, 36, 210, 1225, 7140, 41616, 242556, 1413721, 8239770, 48024900, 279909630, 1631432881, 9508687656, 55420693056, 323015470680, 1882672131025, 10973017315470, 63955431761796, 372759573255306, 2172602007770041

Offset: 0

Paul Barry, Jul 17 2004

Convolution of A059841(n) and A001109(n+1).

Partial sums of A084158.

S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
Index entries for linear recurrences with constant coefficients, signature (6,0,-6,1).

Cf. A096977, A064831, A096978.

Accumulate[LinearRecurrence[{5,5,-1},{0,1,5},30]] (* Harvey P. Dale, Sep 07 2011 *)
LinearRecurrence[{6, 0, -6, 1},{0, 1, 6, 36},22] (* Ray Chandler, Aug 03 2015 *)

G.f.: x/((1-x)*(1+x)*(1-6*x+x^2)).
a(n) = 6*a(n-1)-6*a(n-3)+a(n-4).
a(n) = (3-2*sqrt(2))^n*(3/32-sqrt(2)/16)+(3+2*sqrt(2))^n*(sqrt(2)/16+3/32)-(-1)^n/16-1/8.
a(n) = Sum_{k=0..n} (sqrt(2)*(sqrt(2)+1)^(2*k)/8-sqrt(2)*(sqrt(2)-1)^(2*k)/8)*(1+(-1)^(n-k))/2.
a(n) = Sum_{k=0..n} A000129(k)*A000129(k+1)/2. [corrected by Jason Yuen, Jan 14 2025]
a(n) = (A001333(n+1)^2 - 1)/8 = ((A000129(n) + A000129(n+1))^2 - 1)/8. - Richard R. Forberg, Aug 25 2013
a(n) = A002620(A000129(n+1)) = A000217(A048739(n-1)), n > 0. - Ivan N. Ianakiev, Jun 21 2014

A383720 a(0)=3, a(1)=5, a(2)=35; a(n) = 5a(n-1) + 5a(n-2) - a(n-3) for n > 2.

Original entry on oeis.org

3, 5, 35, 197, 1155, 6725, 39203, 228485, 1331715, 7761797, 45239075, 263672645, 1536796803, 8957108165, 52205852195, 304278004997, 1773462177795, 10336495061765, 60245508192803, 351136554095045, 2046573816377475, 11928306344169797, 69523264248641315

Offset: 0

Seiichi Manyama, May 07 2025

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

Cf. A000129, A002203, A047946, A084158.

LinearRecurrence[{5, 5, -1}, {3, 5, 35}, 25] (* Paolo Xausa, Jul 03 2025 *)

my(N=30, x='x+O('x^N)); Vec((3-10*x-5*x^2)/((1+x)*(1-6*x+x^2)))

G.f.: (3 - 10*x - 5*x^2)/((1 + x) * (1 - 6*x + x^2)).

a(n) = Pell(3*n)/Pell(n) for n > 0.

A213688 a(n) = Sum_{i=0..n} A000129(i)^3.

Original entry on oeis.org

0, 1, 9, 134, 1862, 26251, 369251, 5196060, 73113372, 1028784997, 14476099149, 203694183170, 2866194639170, 40330419190351, 567492063162119, 7985219303802744, 112360562315573112, 1581033091723823881, 22246823846444284881, 313036566941955454910

Offset: 0

N. J. A. Sloane, Jun 18 2012

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,18,-42,11,1).

Cf. A000129, A048739 (partial sums of A000129), A084158 (sum of squares of A000129).

Cf. A110272 (cubes of the Pell numbers).

A110272:=func; [&+[A110272(i): i in [0..n]]: n in [0..19]]; // Bruno Berselli, Jun 21 2012

LinearRecurrence[{13, 18, -42, 11, 1}, {0, 1, 9, 134, 1862}, 20] (* Bruno Berselli, Jun 21 2012 *)

G.f.: x*(1-4*x-x^2)/((1-x)*(1+2*x-x^2)*(1-14*x-x^2)). [Bruno Berselli, Jun 18 2012]

a(n) = ((3+sqrt(2))*(1+sqrt(2))^(3n+1)+(3-sqrt(2))*(1-sqrt(2))^(3n+1)-21*(-1)^n*((1+sqrt(2))^n+(1-sqrt(2))^n)+32)/224. [Bruno Berselli, Jun 18 2012]

A218990 Power ceiling-floor sequence of 3+sqrt(8).

Original entry on oeis.org

6, 34, 199, 1159, 6756, 39376, 229501, 1337629, 7796274, 45440014, 264843811, 1543622851, 8996893296, 52437736924, 305629528249, 1781339432569, 10382407067166, 60513102970426, 352696210755391, 2055664161561919, 11981288758616124, 69832068390134824

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(8), and the limit p3(r) = 5.854315472394508538153482993682502287049948...

			a(0) = ceiling(r) = 6, where r = 3+sqrt(8);
a(1) = floor(6*r) = 34; a(2) = ceiling(35*r) = 199.

R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.~

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A214992, A001653, A084158, A001109.

x = 3 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A001653 *)
t2 = Table[p2[n], {n, 0, z}]  (* A084158 *)
t3 = Table[p3[n], {n, 0, z}]  (* A218990 *)
t4 = Table[p4[n], {n, 0, z}]  (* A001109 *)
LinearRecurrence[{5,5,-1},{6,34,199},30] (* Harvey P. Dale, Mar 21 2024 *)

Vec((6 + 4*x - x^2) / ((1 + x)*(1 - 6*x + x^2)) + O(x^50)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=3+sqrt(8) and a(0) = ceiling(x).
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
G.f.: (6 + 4*x - x^2)/(1 - 5*x - 5*x^2 + x^3).
a(n) = (1/16)*(2*(-1)^n + (47-33*sqrt(2))*(3-2*sqrt(2))^n + (3+2*sqrt(2))^n*(47+33*sqrt(2))). - Colin Barker, Nov 13 2017

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).

Original entry on oeis.org

1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673

Offset: 0

Creighton Dement, Apr 04 2005

A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).
Floretion Algebra Multiplication Program, FAMP Code:
1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,-1)

Cf. A000129, A001109, A001333, A046729, A077444.

Cf. A077445, A079291, A082639, A084158, A090390.

[Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // G. C. Greubel, Aug 21 2022

CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x,0,30}], x] (* Robert G. Wilson v, Apr 06 2005 *)
LinearRecurrence[{5,5,-1}, {1,13,69}, 30] (* Harvey P. Dale, Jun 03 2017 *)

[lucas_number1(2*n+2,2,-1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022

a(n) = 2 * A001109(n+1) - (-1)^n.
G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013
From G. C. Greubel, Aug 21 2022: (Start)
a(n) = A000129(2*n+2) - (-1)^n.
E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)

cp's OEIS Frontend

A084159 Pell oblongs.

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A163960 Decimal expansion of 2*(sqrt(2) - 1).

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A096979 Sum of the areas of the first n+1 Pell triangles.

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A383720 a(0)=3, a(1)=5, a(2)=35; a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3) for n > 2.

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A213688 a(n) = Sum_{i=0..n} A000129(i)^3.

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Magma

Mathematica

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A218990 Power ceiling-floor sequence of 3+sqrt(8).

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Mathematica

PARI

Formula

A105058 Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).

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Magma

Mathematica

A383720 a(0)=3, a(1)=5, a(2)=35; a(n) = 5a(n-1) + 5a(n-2) - a(n-3) for n > 2.

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).