A302332 -id:A302332 - cp's OEIS frontend

A188646 Array of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2-1)+x)^2 for integers x>=1.

1, 1, 1, 1, 13, 1, 1, 181, 33, 1, 1, 2521, 1121, 61, 1, 1, 35113, 38081, 3781, 97, 1, 1, 489061, 1293633, 234361, 9505, 141, 1, 1, 6811741, 43945441, 14526601, 931393, 20021, 193, 1, 1, 94875313, 1492851361, 900414901, 91267009, 2842841, 37441, 253, 1

Offset: 0

Charles L. Hohn, Apr 06 2011

Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; constant k=f(x, y); and initial term a(0)=1; then for all integers x>=1 and y=[+-]1, k may be irrational, but sequence a(n)=a(n-1)*k-((k-1)/(k^n)) always produces integer sequences; y=-1 results shown here; y=1 results are A188647.

Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is (1/n) * T_{2*k+1}(n), with the Chebyshev polynomials of the first kind (type T). - Seiichi Manyama, Jan 01 2019

			Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   1,      1,         1,            1, ...
   2 | 1,  13,    181,      2521,        35113, ...
   3 | 1,  33,   1121,     38081,      1293633, ...
   4 | 1,  61,   3781,    234361,     14526601, ...
   5 | 1,  97,   9505,    931393,     91267009, ...
   6 | 1, 141,  20021,   2842841,    403663401, ...
   7 | 1, 193,  37441,   7263361,   1409054593, ...
   8 | 1, 253,  64261,  16322041,   4145734153, ...
   9 | 1, 321, 103361,  33281921,  10716675201, ...
  10 | 1, 397, 158005,  62885593,  25028308009, ...
  11 | 1, 481, 231841, 111746881,  53861764801, ...
  12 | 1, 573, 328901, 188788601, 108364328073, ...
  13 | 1, 673, 453601, 305726401, 206059140673, ...
  14 | 1, 781, 610741, 477598681, 373481557801, ...
  15 | 1, 897, 805505, 723342593, 649560843009, ...
  ...

Row 1-8 give A000012, A001570, A077420, A302329, A302330, A302331, A302332, A253880.
Column 1 is A082109(n-1).
Cf. A188644, A188647 (f(x, y) as above with y=1).
Diagonal gives A322904.

A[n_, k_] := 1/n ChebyshevT[2k+1, n];
Table[A[n-k, k], {n, 1, 9}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jan 02 2019, after Seiichi Manyama *)

A(n,k) = 2 * A188644(n,k) - A(n,k-1).

A(n,k) = Sum_{j=0..k} binomial(2*k+1,2*j+1)*(n^2-1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019

Edited and extended by Seiichi Manyama, Jan 01 2019

A302329 a(0)=1, a(1)=61; for n>1, a(n) = 62*a(n-1) - a(n-2).

Original entry on oeis.org

1, 61, 3781, 234361, 14526601, 900414901, 55811197261, 3459393815281, 214426605350161, 13290990137894701, 823826961944121301, 51063980650397625961, 3165142973362708688281, 196187800367837541047461, 12160478479832564836254301, 753753477949251182306719201

Offset: 0

Bruno Berselli, Apr 05 2018

Centered hexagonal numbers (A003215) with index in A145607. Example: 35 is a member of A145607, therefore A003215(35) = 3781 is a term of this sequence.
Also, centered 10-gonal numbers (A062786) with index in A182432. Example: 28 is a member of A182432 and A062786(28) = 3781.
a(n) is a solution to the Pell equation (4*a(n))^2 - 15*b(n)^2 = 1. The corresponding b(n) are A258684(n). - Klaus Purath, Jul 19 2025

Colin Barker, Table of n, a(n) for n = 0..500
Christian Aebi and Grant Cairns, Less than Equable Triangles on the Eisenstein lattice, arXiv:2312.10866 [math.CO], 2023.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (62,-1).

Cf. A003215, A145607; A062786, A182432.
Fourth row of the array A188646.
First bisection of A041449, A042859.
Similar sequences of the type cosh((2*n+1)*arccosh(k))/k: A000012 (k=1), A001570 (k=2), A077420 (k=3), this sequence (k=4), A302330 (k=5), A302331 (k=6), A302332 (k=7), A253880 (k=8).

Mathematica
```
LinearRecurrence[{62, -1}, {1, 61}, 20]
```

x='x+O('x^99); Vec((1-x)/(1-62*x+x^2)) \\ Altug Alkan, Apr 06 2018

G.f.: (1 - x)/(1 - 62*x + x^2).
a(n) = a(-1-n).
a(n) = cosh((2*n + 1)*arccosh(4))/4.
a(n) = ((4 + sqrt(15))^(2*n + 1) + 1/(4 + sqrt(15))^(2*n + 1))/8.
a(n) = (1/4)*T(2*n+1, 4), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 08 2022
E.g.f.: exp(31*x)*(4*cosh(8*sqrt(15)*x) + sqrt(15)*sinh(8*sqrt(15)*x))/4. - Stefano Spezia, Jul 25 2025

A067902 a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.

Original entry on oeis.org

2, 14, 194, 2702, 37634, 524174, 7300802, 101687054, 1416317954, 19726764302, 274758382274, 3826890587534, 53301709843202, 742397047217294, 10340256951198914, 144021200269567502, 2005956546822746114, 27939370455248878094, 389145229826661547202, 5420093847118012782734

Offset: 0

Lekraj Beedassy, May 13 2003

Solves for x in x^2 - 3*y^2 = 4. [Complete nonnegative solutions are in A003500 and A052530. - Wolfdieter Lang, Sep 05 2021]
For n>0, a(n)+2 is the number of dimer tilings of a 4 X 2n Klein bottle (cf. A103999).
This is the Lucas sequence V(14,1). In addition to the comment above: If x = a(n) then y(n) = (a(n+1) - a(n-1))/24, n >= 1. - Klaus Purath, Aug 17 2021

			G.f. = 2 + 14*x + 194*x^2 + 2702*x^3 + 37634*x^4 + 524174*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..850
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (14,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A067900, A302332.
Row 2 * 2 of array A188644.
Cf. A001075, A001353, A003500, A052530.

m:=7;; a:=[2,14];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019

[Floor((2+Sqrt(3))^(2*n)+(1+Sqrt(3))^(-n)): n in [0..19]]; // Vincenzo Librandi, Mar 31 2011

a := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(14) fi: 14*a(n-1)-a(n-2): end: for n from 0 to 30 do printf(`%d,`,a(n)) od:
seq( simplify(2*ChebyshevT(n, 7)), n=0..20); # G. C. Greubel, Dec 23 2019

a[0]=2; a[1]=14; a[n_]:= 14a[n-1] -a[n-2]; Table[a[n], {n,0,20}] (* Robert G. Wilson v, Jan 30 2004 *)
2*ChebyshevT[Range[21] -1, 7] (* G. C. Greubel, Dec 23 2019 *)

vector( 21, n, 2*polchebyshev(n-1, 1, 7) ) \\ G. C. Greubel, Dec 23 2019

[lucas_number2(n,14,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

[2*chebyshev_T(n,7) for n in (0..20)] # G. C. Greubel, Dec 23 2019

G.f.: 2*(1-7*x)/(1-14*x+x^2). - N. J. A. Sloane, Nov 22 2006
a(n) = p^n + q^n, where p = 7 + 4*sqrt(3) and q = 7 - 4*sqrt(3). - Tanya Khovanova, Feb 06 2007
a(n) = 2*A011943(n+1). - R. J. Mathar, Sep 27 2014
From Peter Bala, Oct 16 2019: (Start)
Let F(x) = Product_{n >= 0} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 7 - 4*sqrt(3). This sequence gives the partial denominators in the simple continued fraction expansion of 1 + F(alpha) = 2.07140228197873197080... = 2 + 1/(14 + 1/(194 + 1/(2702 + ...))). Cf. A005248.
12*Sum_{n >= 1} 1/(a(n) - 16/a(n)) = 1.
16*Sum_{n >= 1} (-1)^(n+1)/(a(n) + 12/a(n)) = 1.
Series acceleration formula for sum of reciprocals:
Sum_{n >= 1} 1/a(n) = 1/12 - 16*Sum_{n >= 1} 1/(a(n)*(a(n)^2 - 16)).
Sum_{n >= 1} 1/a(n) = ( (theta_3(7-4*sqrt(3)))^2 - 1 )/4, where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122). Cf. A153415 and A003499.
(End)
From Klaus Purath, Aug 17 2021: (Start)
a(n) = (a(n-1)*a(n-2) + 2688)/a(n-3), n >= 3.
a(n) = (a(n-1)^2 + 192)/a(n-2), n >= 2.
a(2*n) = A302332(n-1) + A302332(n), n >= 1.
a(2*n+1) = 14*A302332(n). (End)
a(n) = A003500(2*n) = S(2*n,4) - S(2*n-2, 4) = 2*T(2*n,2), for n >= 0, with Chebyshev S and T. S(n, 4) = A001353(n+1) and T(n, 2) = A001075(n). - Wolfdieter Lang, Sep 06 2021

A084232 RMS values associated with A084231.

Original entry on oeis.org

1, 195, 37829, 7338631, 1423656585, 276182038859, 53577891882061, 10393834843080975, 2016350381665827089, 391161580208327374291, 75883330210033844785365, 14720974899166357560986519, 2855793247108063332986599321, 554009168964065120241839281755

Offset: 0

Ignacio Larrosa Cañestro, May 20 2003

From Klaus Purath, Aug 20 2025: (Start)
Solutions to the Pell equation (7*b(n))^2 - 3*(4*a(n))^2 = 1. The corresponding b(n) are given by A302332.
For any two consecutive terms (x,y), x^2 - 194*x*y + y^2 - 196 = 0. By analogy to this, for three consecutive terms (x, y, z), y^2 - x*z - 196 = 0. (End)

			a(1)=195 because 195 = sqrt((Sum_{k=1..337}k^2)/337) and 337 = A084231(1).

Indranil Ghosh, Table of n, a(n) for n = 0..436
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (194,-1).

Cf. A084231, A217855, A302332.

LinearRecurrence[{194,-1},{1,195},20] (* Harvey P. Dale, Nov 10 2021 *)

a(n) = ((7+4*sqrt(3))^(2*n+1)-(7-4*sqrt(3))^(2*n+1))/(8*sqrt(3)). [simplified by Bruno Berselli, Oct 19 2012]
a(n) = floor(((7*sqrt(3) + 12)/24)*(56*sqrt(3) + 97)^n).
a(n+2) = 194*a(n+1) - a(n).
G.f.: (1-x)/(1-194*x+x^2). - Philippe Deléham, Nov 18 2008
a(n)^2 = (Sum_{i=1..A084231(n+1)}i^2)/A084231(n+1). - Bruno Berselli, Oct 17 2012

A322904 a(n) = Sum_{k=0..n} binomial(2n+1,2k+1)(n^2-1)^(n-k)n^(2*k).

Original entry on oeis.org

1, 1, 181, 38081, 14526601, 8943235489, 8138661470941, 10287228590683393, 17254778510170993681, 37095265466946847758401, 99474891266913130060486021, 325534304813775692747248543681, 1276941308627620432293188401109401, 5914558735952850788377566338591400673

Offset: 0

Seiichi Manyama, Dec 30 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..193
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Diagonal of A188646.

Cf. A253880, A302329, A302330, A302331, A302332.

[&+[Binomial(2*n+1,2*k+1)*(n^2-1)^(n-k)*n^(2*k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Jan 03 2019

a[0] = 1; a[n_] := 1/n ChebyshevT[2n+1, n];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 02 2019 *)

{a(n) = sum(k=0, n, binomial(2*n+1, 2*k+1)*(n^2-1)^(n-k)*n^(2*k))}

a(n) = if (n==0, 1, polchebyshev(2*n+1, 1, n)/n); \\ Michel Marcus, Jan 02 2019

For n > 0, a(n) = (1/n) * T_{2*n+1}(n) where T_{n}(x) is a Chebyshev polynomial of the first kind.
For n > 0, a(n) = (1/n) * cosh((2*n+1)*arccosh(n)).
a(n) ~ 4^n * n^(2*n). - Vaclav Kotesovec, Jan 03 2019

cp's OEIS Frontend

A188646 Array of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2-1)+x)^2 for integers x>=1.

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A302329 a(0)=1, a(1)=61; for n>1, a(n) = 62*a(n-1) - a(n-2).

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A067902 a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.

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A084232 RMS values associated with A084231.

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A322904 a(n) = Sum_{k=0..n} binomial(2*n+1,2*k+1)*(n^2-1)^(n-k)*n^(2*k).

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A322904 a(n) = Sum_{k=0..n} binomial(2n+1,2k+1)(n^2-1)^(n-k)n^(2*k).