A098316 -id:A098316 - cp's OEIS frontend

A006273 Numerators of a continued fraction for (3+sqrt(13))/2.

Original entry on oeis.org

3, 10, 1297, 2186871697, 10458512317535240383929505297

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

For denominators see A006274.

Cf. A098316, A002814, A006268, A006271, A006275, A006276.

a := proc (n) option remember; if n = 1 then 10 else a(n-1)^3 + 3*a(n-1)^2 - 3 end if; end proc:
seq(a(n), n = 1..5); # Peter Bala, Jan 19 2022

From Peter Bala, Jan 19 2022: (Start)
a(n) = (11/2 + 3/2*sqrt(13))^3^(n-1) + (11/2 - 3/2*sqrt(13))^3^(n-1) - 1.
a(1) = 10 and a(n) = a(n-1)^3 + 3*a(n-1)^2 - 3 for n >= 2.
a(1) = 10 and a(n) = 13*(Product_{k = 1..n-1} a(k))^2 - 3 for n >= 2.
a(n) = A006268(n-1)^2 + 1 for n >= 1.
13 - 9*Product_{n = 1..N} (1 + 2/a(n))^2 = 52/(a(N+1) + 3). Therefore
sqrt(13) = 3*(1 + 2/10) * (1 + 2/1297) * (1 + 2/2186871697) * ... The convergence is cubic: the first six factors of the product give sqrt(13) correct to more than 750 decimal places.
3/sqrt(13) = (1 - 2/(10+2)) * (1 - 2/(1297+2)) * (1 - 2/(2186871697+2)) * .... (End)

A186425 Antidiagonal sums of A179748.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 30, 45, 68, 104, 161, 251, 393, 618, 976, 1547, 2459, 3917, 6251, 9993, 15999, 25647, 41157, 66108, 106272, 170961, 275202, 443250, 714265, 1151486, 1857057, 2995991, 4834907, 7804653, 12601553, 20351114, 32872743, 53107823, 85811996, 138674777, 224130364, 362286475

Offset: 1

Mats Granvik, Feb 21 2011

nonn

a(n+1)/a(n) tends to the golden ratio. [Note added by Joerg Arndt, Mar 16 2013: this is only a conjecture so far!]
Grows slower than the Fibonacci sequence. More complicated than the Fibonacci sequence.
The divisibility related table A051731 can be described by the recurrence:
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - 1*(Sum_{i=1..k-1} T(n-i,k)).
The silver means can be found as limiting ratios of the antidiagonal sums of the tables described by the following similar recurrences:
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 0*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the golden ratio, A001622.
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 1*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the silver ratio, A014176.
T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 2*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the bronze ratio, A098316
The limiting ratio becomes apparent after the first 275 terms or so of the antidiagonal sums.
The empirical observation that the ratio a(n+1)/a(n) tends to the golden ratio 1.6180339887498... has been verified up to a(1500)/a(1499) which gives the first 65 digits of A001622. - Mats Granvik, Sep 16 2017

Mats Granvik, Does this ratio converge to the Golden ratio?

Cf. A001622, A179748, cumulative sums of A186426.

Clear[a,t]; nn = 58; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; a = Table[Total[Table[t[n - k + 1, k], {k, 1, nn}]], {n, 1, nn}]; a (* Mats Granvik, Apr 27 2013 *)

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

A006274 First differences of A006268.

Original entry on oeis.org

33, 46728, 102266868085272, 1069559300034650646049671038948382825526728

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

Cf. A006268, A006273, A098316.

Definition corrected by N. J. A. Sloane, May 23 2023 using a formula suggested by R. J. Mathar, Apr 26 2007.

A117917 a(n) = 3*a(n-1) + a(n-2) + n.

Original entry on oeis.org

1, 4, 15, 52, 175, 582, 1927, 6370, 21045, 69514, 229597, 758316, 2504557, 8272000, 27320571, 90233728, 298021771, 984299058, 3250918963, 10737055966, 35462086881, 117123316630, 386832036793, 1277619427032, 4219690317913

Offset: 0

Gary W. Adamson, Apr 02 2006

a(n)/a(n-1) tends to 3.30277563... = exp(ArcSinh(3/2)) (A098316).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6,1,1).

Cf. A006190.

I:=[1,4,15,52]; [n le 4 select I[n] else 5*Self(n-1) -6*Self(n-2) +Self(n-3) +Self(n-4): n in [1..31]]; // G. C. Greubel, Oct 22 2021

LinearRecurrence[{5,-6,1,1},{1,4,15,52},30] (* Harvey P. Dale, Mar 21 2018 *)

[(2*lucas_number1(n+2, 3, -1) + 8*lucas_number1(n+1, 3, -1) - 3*n -5)/9 for n in (0..30)] # G. C. Greubel, Oct 22 2021

O.g.f.: (1 - x + x^2)/((1-x)^2*(1 - 3*x - x^2)). - R. J. Mathar, Mar 17 2008

a(n) = ( -3*n - 5 + 2*A006190(n) + 14*A006190(n+1) )/9. - R. J. Mathar, Oct 21 2012

More terms from R. J. Mathar, Mar 17 2008

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

A351898 Decimal expansion of metallic ratio for N = 14.

Original entry on oeis.org

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

Offset: 2

A.H.M. Smeets, Feb 24 2022

Decimal expansion of continued fraction [14; 14, 14, 14, ...].
Also largest solution of x^2 - 14 x - 1 = 0.
Essentially the same digit sequence as A010503, A157214, A174968 and A268683.
The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

			14.0710678118654752440084436210484903928483593...

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.
Wikipedia, Metallic mean.

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

PARI
```
(1+sqrt(2))^3
```

Equals 2 + 5*A014176.
Equals A014176^3.
Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

cp's OEIS Frontend

A006273 Numerators of a continued fraction for (3+sqrt(13))/2.

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Maple

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A186425 Antidiagonal sums of A179748.

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

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Magma

Mathematica

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Sage

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A006274 First differences of A006268.

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A117917 a(n) = 3*a(n-1) + a(n-2) + n.

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Magma

Mathematica

Sage

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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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Mathematica

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

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PARI

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

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

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