A098317 -id:A098317 - cp's OEIS frontend

A229780 Decimal expansion of (3+sqrt(5))/10.

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

Offset: 0

Joost Gielen, Sep 29 2013

sqrt((3+sqrt(5))/10) = sqrt(phi^2/5) = (5+sqrt(5))/10 = (3+sqrt(5))/10 + 2/10 = 0.723606797... .
Essentially the same as A134972, A134945, A098317 and A002163. - R. J. Mathar, Sep 30 2013
Equals one tenth of the limit of (G(n+2)+G(n+1)+G(n-1)+G(n-2))/G(n), where G(n) is any nonzero sequence satisfying the recurrence G(n+1) = G(n) + G(n-1) including A000032 and A000045, as n --> infinity. - Richard R. Forberg, Nov 17 2014
3+sqrt(5) is the perimeter of a golden rectangle with a unit width. - Amiram Eldar, May 18 2021
Constant x such that x = sqrt(x) - 1/5. - Andrea Pinos, Jan 15 2024

			0.5236067977499...

Cf. A094874, A187798.

RealDigits[GoldenRatio^2/5,10,120][[1]] (* Harvey P. Dale, Dec 02 2014 *)

(3+sqrt(5))/10 = (phi/sqrt(5))^2 = phi^2/5 where phi is the golden ratio.

A261540 a(n) = n^7 + 7n^5 + 14n^3 + 7*n.

Original entry on oeis.org

0, 29, 478, 4287, 24476, 101785, 337434, 946043, 2333752, 5206581, 10714070, 20633239, 37597908, 65378417, 109216786, 176222355, 275832944, 420346573, 625528782, 911300591, 1302512140, 1829807049, 2530582538, 3450050347, 4642403496, 6172093925, 8115226054

Offset: 0

Raphael Ranna, Aug 24 2015

Also numbers of the form (n-th metallic mean)^7 - 1/(n-th metallic mean)^7, see link to Wikipedia.

Raphael Ranna, Table of n, a(n) for n = 0..100
Wikipedia, Metallic mean
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458, A176522, A261391.

[n^7 + 7*n^5 + 14*n^3 + 7*n: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Table[n^7 + 7 n^5 + 14 n^3 + 7 n, {n, 0, 30}] (* Bruno Berselli, Aug 24 2015 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 29, 478, 4287, 24476, 101785, 337434, 946043}, 30] (* Vincenzo Librandi, Aug 24 2015 *)

a(n)=n^7+7*n^5+14*n^3+7*n \\ Charles R Greathouse IV, Aug 24 2015

[n^7+7*n^5+14*n^3+7*n for n in (0..30)] # Bruno Berselli, Aug 24 2015

a(n) = -a(-n) = ( (n+sqrt(n^2+4))/2 )^7 - 1/( (n+sqrt(n^2+4))/2 )^7.

G.f.: x*(29 + 246*x + 1275*x^2 + 1940*x^3 + 1275*x^4 + 246*x^5 + 29*x^6)/(1 - x)^8. - Bruno Berselli, Aug 24 2015

Offset changed from 1 to 0 and initial 0 added by Bruno Berselli, Aug 25 2015

A115605 Expansion of -x^2(2 + x - 2x^2 - x^3 + 2x^4) / ( (x-1)(1+x)(1 + x + x^2)(x^2 - x + 1)(x^2 + 4x - 1)*(x^2 - x - 1) ).

Original entry on oeis.org

0, 0, 2, 7, 31, 128, 549, 2315, 9826, 41594, 176242, 746496, 3162334, 13395658, 56745250, 240376201, 1018250793, 4313378176, 18271765435, 77400436781, 327873517634, 1388894499108, 5883451527348, 24922700587008

Offset: 0

Roger L. Bagula, Mar 13 2006

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1,0,1,-3,-6,3,1).

Cf. A000045, A079962.

A000035 := proc(n)
        n mod 2 ;
end proc:
A061347 := proc(n)
        op((n mod 3)+1,[-2,1,1]) ;
end proc:
A001076 := proc(n)
        option remember;
        if n <=1 then
                n;
        else
                4*procname(n-1)+procname(n-2) ;
        end if;
end proc:
A039834 := proc(n)
        (-1)^(n+1)*combinat[fibonacci](n) ;
end proc:
A087204 := proc(n)
        op((n mod 6)+1,[2,1,-1,-2,-1,1]) ;
end proc:
A115605 := proc(n)
        -A000035(n+1)/6 +A061347(n+2)/12 + A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12 ;
end proc: # R. J. Mathar, Dec 16 2011

LinearRecurrence[{3,6,-3,-1,0,1,-3,-6,3,1},{0,0,2,7,31,128,549,2315,9826,41594},30] (* Harvey P. Dale, Dec 16 2011 *)

concat([0,0],Vec((2+x-2*x^2-x^3+2*x^4)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)*(x^2+4*x-1)*(x^2-x-1))+O(x^99))) \\ Charles R Greathouse IV, Sep 27 2012

Lim_{n->infinity} a(n+1)/a(n) = phi^3 = A098317.

a(n) = -A000035(n+1)/6 +A061347(n+2)/12 +A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12. - R. J. Mathar, Dec 16 2011

A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

Original entry on oeis.org

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

Offset: 0

Martin Janecke, Jan 26 2012

			0.0416662....

Martin Janecke, Edle Reihe

Cf. A001620, A205326, continued fractions A001622, A014176, A098316, A098317, A098318.

digits = 10; dn = 1000000; Clear[f]; f[n_] := NSum[2/(k + Sqrt[k^2+4]) - 1/k, {k, 1, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits+10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] + EulerGamma // RealDigits[#, 10, digits+2]& // First; f[dn]; f[n = 2*dn]; While[f[n] != f[n-dn], n = n+dn]; Prepend[ f[n][[1 ;; digits]], 0] (* Jean-François Alcover, Feb 25 2013 *)

lim_{n->infinity} (1/[1;1,...] + 1/[2;2,...] + 1/[3;3,...] + ... + 1/[n;n,...] - log(n)).

lim_{n->infinity} (sum_{k=1...n} (2/(k + sqrt(k^2 + 4))) - log(n)).

More terms from Jean-François Alcover, Feb 25 2013

More terms from Jon E. Schoenfield, Jan 05 2014

A205326 Decimal expansion of the sum of [0;n,n,n,...]^2 for n=1..infinity.

Original entry on oeis.org

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

Offset: 0

Martin Janecke, Jan 26 2012

This is the total area of all squares with sides parallel to the axes of the Cartesian coordinate system, the lower left vertex at (n,0) and the upper right vertex on f(x)=1/x for n=1..infinity.

			0.9155879199018197251998168538031900897353...

Martin Janecke, Edle Reihe

Cf. A013661, A205325, continued fractions A001622, A014176, A098316, A098317, A098318.

zeta(2)+sumpos(n=1,4/(n+sqrt(n^2+4))^2-1/n^2) \\ Charles R Greathouse IV, Jan 26 2012

Sum_{n>=1} 1/[n;n,n,...]^2.

Sum_{n>=1} 4/(n + sqrt(n^2 + 4))^2.

a(-5)-a(-86) from Charles R Greathouse IV, Jan 26 2012

A261574 a(n) = n(n^2 + 3)(n^6 + 6n^4 + 9n^2 + 3).

Original entry on oeis.org

0, 76, 2786, 46764, 439204, 2744420, 12813606, 48229636, 153992264, 432083484, 1092730090, 2537720636, 5489037036, 11179326964, 21624372014, 40001698260, 71163830416, 122319408236, 203920464114, 330799604044, 523606640180, 810600392196, 1229857906486

Offset: 0

Raphael Ranna, Aug 24 2015

Also numbers of the form (n-th metallic mean)^9 - 1/(n-th metallic mean)^9, see link to Wikipedia.

Colin Barker, Table of n, a(n) for n = 0..1000 (first 100 terms from Raphael Ranna)
Wikipedia, Metallic mean
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458, A176522, A261391, A261540.

[n*(n^2+3)*(n^6+6*n^4+9*n^2+3): n in [0..25]]; // Bruno Berselli, Aug 25 2015

Table[n (n^2 + 3) (n^6 + 6 n^4 + 9 n^2 + 3), {n, 0, 25}] (* Bruno Berselli, Aug 25 2015 *)

concat(0, Vec(2*x*(38*x^8 +1013*x^7 +11162*x^6 +43907*x^5 +69200*x^4 +43907*x^3 +11162*x^2 +1013*x +38) / (x -1)^10 + O(x^50))) \\ Colin Barker, Aug 25 2015

a(n) = -a(-n) = ( (n+sqrt(n^2+4))/2 )^9-1/( (n+sqrt(n^2+4))/2 )^9.

G.f.: 2*x*(38*x^8 +1013*x^7 +11162*x^6 +43907*x^5 +69200*x^4 +43907*x^3 +11162*x^2 +1013*x +38) / (x -1)^10. - Colin Barker, Aug 25 2015

Formula in Name by Colin Barker, Aug 25 2015

Offset changed from 1 to 0 and initial 0 added by Bruno Berselli, Aug 25 2015

A300763 a(n) = ceiling(n/g^3), where g = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18

Offset: 0

Jeffrey Shallit, Jul 02 2018

nonn

Altug Alkan, Table of n, a(n) for n = 0..1000

Cf. A001622, A019446, which is ceiling(n/g), A189663, which is ceiling(n/g^2) (but shifted by one).

Cf. A098317.

Array[Ceiling[#/GoldenRatio^3] &, 90, 0] (* Robert G. Wilson v, Jul 02 2018 *)

a(n) = my(t=2+sqrt(5)); ceil(n/t); \\ Altug Alkan, Jul 02 2018

a(n) = ceiling(n/t) where t = 2*g + 1 = 2 + sqrt(5). - Altug Alkan, Jul 02 2018

A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 4, 16, 1, 64, 8, 256, 48, 1, 1024, 256, 12, 4096, 1280, 96, 1, 16384, 6144, 640, 16, 65536, 28672, 3840, 160, 1, 262144, 131072, 21504, 1280, 20, 1048576, 589824, 114688, 8960, 240, 1, 4194304, 2621440, 589824, 57344, 2240, 24, 16777216, 11534336, 2949120, 344064, 17920, 336, 1

Offset: 0

Shara Lalo, Jun 11 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.

			Triangle begins:
         1;
         4;
        16,        1;
        64,        8;
       256,       48,        1;
      1024,      256,       12;
      4096,     1280,       96,       1;
     16384,     6144,      640,      16;
     65536,    28672,     3840,     160,      1;
    262144,   131072,    21504,    1280,     20;
   1048576,   589824,   114688,    8960,    240,    1;
   4194304,  2621440,   589824,   57344,   2240,   24;
  16777216, 11534336,  2949120,  344064,  17920,  336,  1;
  67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.

Shara Lalo, Left justified triangle
Shara Lalo, Skew diagonals in triangle A013611

Row sums give A001076.
Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
Cf. A013611.
Cf. A098317.

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

G.f.: 1 / (1 - 4*t*x - t^2).

A354855 a(n) = floor(n(2+sqrt(5))^n), equivalently, floor(nphi^(3n)), where phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

0, 4, 35, 228, 1287, 6820, 34667, 171332, 829455, 3952836, 18604979, 86693156, 400623383, 1838490212, 8387044091, 38065809540, 171999313951, 774138335108, 3472202765123, 15525625108324, 69229056160039, 307921937307684, 1366491508589195, 6051666872017348

Offset: 0

Jiale Wang, Jun 09 2022

Stefano Spezia, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (8,-13,-16,13,8,1).

Cf. A000032, A000045, A001622, A004976, A098317, A128439.

a[n_] := Floor[n * GoldenRatio^(3*n)]; Array[a, 25, 0] (* Amiram Eldar, Jun 09 2022 *)

a(n) = floor((2+sqrt(5))^n*n).
a(n) = floor(n*phi^(3n)) where phi=(1+sqrt(5))/2 is the golden ratio.
a(n) = floor(n*F(3n-1)+n*phi*F(3n)), where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = n*L(3n) when n is odd and a(n) = n*L(3n)-1 when n is even (n>=2), where L(n) = A000032(n) is the n-th Lucas number.
G.f.: x*(4 + 3*x - 18*x^3 - 4*x^4 - x^5)/((1 - x)*(1 + x)*(1 - 4*x - x^2)^2). - Stefano Spezia, Jun 12 2022

A382008 Decimal expansion of the isoperimetric quotient of a rhombic triacontahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 20 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.88720000254802085800544409398426003785738986572116...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A344171 (surface area), A344172 (volume).

Cf. A000796, A098317, A381684.

First[RealDigits[Pi*(2 + Sqrt[5])/15, 10, 100]]

Equals 36*Pi*A344172^2/(A344171^3).

Equals Pi*(2 + sqrt(5))/15 = A000796*A098317/15.

cp's OEIS Frontend

A229780 Decimal expansion of (3+sqrt(5))/10.

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A261540 a(n) = n^7 + 7*n^5 + 14*n^3 + 7*n.

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A115605 Expansion of -x^2*(2 + x - 2*x^2 - x^3 + 2*x^4) / ( (x-1)*(1+x)*(1 + x + x^2)*(x^2 - x + 1)*(x^2 + 4*x - 1)*(x^2 - x - 1) ).

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A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

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A205326 Decimal expansion of the sum of [0;n,n,n,...]^2 for n=1..infinity.

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A261574 a(n) = n*(n^2 + 3)*(n^6 + 6*n^4 + 9*n^2 + 3).

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A300763 a(n) = ceiling(n/g^3), where g = (1+sqrt(5))/2 is the golden ratio.

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A261540 a(n) = n^7 + 7n^5 + 14n^3 + 7*n.

A115605 Expansion of -x^2(2 + x - 2x^2 - x^3 + 2x^4) / ( (x-1)(1+x)(1 + x + x^2)(x^2 - x + 1)(x^2 + 4x - 1)*(x^2 - x - 1) ).

A261574 a(n) = n(n^2 + 3)(n^6 + 6n^4 + 9n^2 + 3).

A354855 a(n) = floor(n(2+sqrt(5))^n), equivalently, floor(nphi^(3n)), where phi = (1+sqrt(5))/2 is the golden ratio.