A041449 -id:A041449 - cp's OEIS frontend

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

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

A041448 Numerators of continued fraction convergents to sqrt(240).

Original entry on oeis.org

15, 31, 945, 1921, 58575, 119071, 3630705, 7380481, 225045135, 457470751, 13949167665, 28355806081, 864623350095, 1757602506271, 53592698538225, 108942999582721, 3321882686019855, 6752708371622431, 205903133834692785, 418558976041008001, 12762672415064932815

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,62,0,-1).

Cf. A041449, A040224.

Numerator[Convergents[Sqrt[240], 30]] (* or *) CoefficientList[Series[(15 + 31 x + 945 x^2 + 1921 x^3 + 945 x^4 - 31 x^5 + 15 x^6 - x^7)/(1 - 3842 x^4 + x^8), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 02 2013 *)
a0[n_] := (-15-4*Sqrt[15]+(-15+4*Sqrt[15])*(31+8*Sqrt[15])^(2*n))/(2*(31+8*Sqrt[15])^n) // Simplify
a1[n_] := (1+(31+8*Sqrt[15])^(2*n))/(2*(31+8*Sqrt[15])^n) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]}&,Range[10]]] (* Gerry Martens, Jul 10 2015 *)

G.f.: -(x+1)*(x^2-16*x-15) / ((x^2-8*x+1)*(x^2+8*x+1)). - Vincenzo Librandi, Nov 02 2013, simplified by Colin Barker, Dec 28 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((-15-4*sqrt(15))/(31+8*sqrt(15))^n+(-15+4*sqrt(15))*(31+8*sqrt(15))^n)/2.
a1(n) = (1/(31+8*sqrt(15))^n+(31+8*sqrt(15))^n)/2. (End)

More terms from Colin Barker, Dec 28 2013

cp's OEIS Frontend

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

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