A176439 -id:A176439 - cp's OEIS frontend

A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

2, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, 48229636, 344362251, 2458765393, 17555720002, 125348805407, 894997357851, 6390330310364, 45627309530399, 325781497023157, 2326097788692498, 16608466017870643, 118585359913786999, 846705985414379636

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 18 2003

a(n+1)/a(n) converges to (7+sqrt(53))/2 = 7.14005... = A176439.
Lim a(n)/a(n+1) as n approaches infinity = 0.1400549... = 2/(7+sqrt(53)) = (sqrt(53)-7)/2 = 1/A176439 = A176439 - 7.
From Johannes W. Meijer, Jun 12 2010: (Start)
In general sequences with recurrence a(n) = k*a(n-1)+a(n-2) with a(0)=2 and a(1)=k [and a(-1)=0] have generating function (2-k*x)/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n+1) = b(5n), a(3n+2)=b(5*n+3), a(3n+3)=2*b(5n+4) where b(n) is the sequence of numerators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n)/2, for n>=1, is the sequence of numerators of continued fraction convergents to sqrt(k^2/4+1).]
For the sequence given above k=7 which implies that it is associated with A041090.
For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0)=1 [and a(-1)=0] see A054413; a sequence that is associated with A041091.
For more information follow the Khovanova link and see A087130, A140455 and A178765.
(End)

			a(4) = 7*a(3) + a(2) = 7*364 + 51 = 2599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (7,1).

Cf. A058316, A176439.

Cf. A000032 (k=1), A006497 (k=3), A087130 (k=5), A086902 (k=7), A087798 (k=9), A001946 (k=11), A088316 (k=13), A090301 (k=15), A090306 (k=17). - Johannes W. Meijer, Jun 12 2010

I:=[2,7]; [n le 2 select I[n] else 7*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == 7 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{7,1},{2,7},30] (* Harvey P. Dale, May 25 2023 *)

a(n)=([0,1; 1,7]^n*[2;7])[1,1] \\ Charles R Greathouse IV, Apr 06 2016

a(n) = ((7+sqrt(53))/2)^n + ((7-sqrt(53))/2)^n.
E.g.f. : 2exp(7x/2)cosh(sqrt(53)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)53^k7^(n-2k)}. a(n)=2T(n, 7i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
G.f.: (2-7x)/(1-7x-x^2). - Philippe Deléham, Nov 16 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 7*A097837(n), a(2n) = A099368(n).
a(3n+1) = A041090(5n), a(3n+2) = A041090(5*n+3), a(3n+3) = 2*A041090(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53). (End)

A010727 Constant sequence: the all 7's sequence.

Original entry on oeis.org

7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

N. J. A. Sloane

a(n) = A153466(n) mod 9. - Paul Curtz, Dec 27 2008
Continued fraction expansion of A176439.  - Bruno Berselli, Mar 15 2011
Final digit of 16^(2^n) + 1. That is, the last digit of every Fermat number F(n) is 7, where n >= 2. - Arkadiusz Wesolowski, Jul 28 2011
Decimal expansion of 7/9. - Arkadiusz Wesolowski, Sep 12 2011

Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1015
Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000012 (the all 1's sequence), A153466, A176439.

ContinuedFraction[(7 + Sqrt@ 53)/2, 105] (* Or *)
CoefficientList[ Series[7/(1 - x), {x, 0, 104}], x] (* Robert G. Wilson v *)
PadRight[{},90,7] (* or *) Table[7,{90}] (* Harvey P. Dale, Jun 05 2013 *)

a(n)=7 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 7/(1-x).  - Bruno Berselli, Mar 15 2011
a(n) = 7. - Arkadiusz Wesolowski, Sep 12 2011
E.g.f.: 7*e^x. - Vincenzo Librandi, Jan 28 2012

A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

Original entry on oeis.org

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Paul Curtz, Aug 15 2015

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.
b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.
Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
   0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
-1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
-1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
-1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
-1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
-1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
-1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
The numerators are almost A165795(n).
Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).
A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
a(2*k) = 1 + k^2.
a(2*k+1) = 5 + 4*k*(k+1).
a(2*k+1) = 4*a(2*k) + 4*k + 1.
a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019
a(-n) = a(n).
a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
a(n) = A168077(n) + period 2: repeat 1, 4.
a(n) = A171621(n) + period 2: repeat 2, 8.
From Colin Barker, Aug 15 2015: (Start)
a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
a(n) = n^2/4 + 1 for n even;
a(n) = n^2 + 4 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015
Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

New name by Peter Luschny, Mar 18 2022

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jul 01 2014

Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014

The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 2017

			11.09016994374947424102293417182819058860154589902881431067724311352630...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

RealDigits[GoldenRatio^5, 10, 103] // First

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A261391 a(n) = n^5 + 5n^3 + 5n.

Original entry on oeis.org

0, 11, 82, 393, 1364, 3775, 8886, 18557, 35368, 62739, 105050, 167761, 257532, 382343, 551614, 776325, 1069136, 1444507, 1918818, 2510489, 3240100, 4130511, 5206982, 6497293, 8031864, 9843875, 11969386, 14447457, 17320268, 20633239, 24435150, 28778261, 33718432, 39315243, 45632114

Offset: 0

Raphael Ranna, Aug 17 2015

Also numbers of the form (n-th metallic mean)^5 - 1/(n-th metallic mean)^5, see link to Wikipedia.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Metallic mean.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

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

Array[#^5 + 5 #^3 + 5 # &, 34] (* Michael De Vlieger, Aug 18 2015 *)
Table[n^5 + 5*n^3 + 5*n, {n,0, 50}] (* G. C. Greubel, Aug 21 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,11,82,393,1364,3775},40] (* Harvey P. Dale, May 07 2018 *)

concat(0, Vec(x*(11*x^4+16*x^3+66*x^2+16*x+11)/(x-1)^6 + O(x^100))) \\ Colin Barker, Aug 18 2015

a(n) = ( (n+sqrt(n^2+4))/2 )^5 - 1/( (n+sqrt(n^2+4))/2 )^5.
a(n) = -a(-n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Colin Barker, Aug 18 2015
G.f.: x*(11*x^4+16*x^3+66*x^2+16*x+11) / (x-1)^6. - Colin Barker, Aug 18 2015
E.g.f.: (x^5 + 15*x^4 + 70*x^3 + 120*x^2 + 71*x + 11)*e^x. - G. C. Greubel, Aug 21 2015

Offset changed from 1 to 0, initial 0 added and b-file adapted from Bruno Berselli, Aug 25 2015

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

A317014 Triangle read by rows: T(0,0) = 1; T(n,k) = 7 * 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, 7, 49, 1, 343, 14, 2401, 147, 1, 16807, 1372, 21, 117649, 12005, 294, 1, 823543, 100842, 3430, 28, 5764801, 823543, 36015, 490, 1, 40353607, 6588344, 352947, 6860, 35, 282475249, 51883209, 3294172, 84035, 735, 1, 1977326743, 403536070, 29647548, 941192, 12005, 42

Offset: 0

Zagros Lalo, Jul 19 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A027466 ((7+x)^n).
The coefficients in the expansion of 1/(1-7x-x^2) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 7.14005494464025913554... ((7+sqrt(53))/2), a metallic mean (see A176439), when n approaches infinity.

			Triangle begins:
1;
7;
49, 1;
343, 14;
2401, 147, 1;
16807, 1372, 21;
117649, 12005, 294, 1;
823543, 100842, 3430, 28;
5764801, 823543, 36015, 490, 1;
40353607, 6588344, 352947, 6860, 35;
282475249, 51883209, 3294172, 84035, 735, 1;
1977326743, 403536070, 29647548, 941192, 12005, 42;
13841287201, 3107227739, 259416045, 9882516, 168070, 1029, 1;
96889010407, 23727920916, 2219448385, 98825160, 2117682, 19208, 49;
678223072849, 179936733613, 18643366434, 951192165, 24706290, 302526, 1372, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 96.

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

Row sums give A054413.
Cf. A013614, A027466, A176439.
Cf. A000420 (column 0), A027473 (column 1), A027474 (column 2), A140107 (column 3), A139641 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 7*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

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

A086902 a(n) = 7*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 7.

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A010727 Constant sequence: the all 7's sequence.

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A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

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A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

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A261391 a(n) = n^5 + 5*n^3 + 5*n.

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

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A261391 a(n) = n^5 + 5n^3 + 5n.

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).