A145607 -id:A145607 - 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

A145608 Numbers a(n)=k such that (1/3)(5(2k+1)^2-2) is A057080(n)^2.

Original entry on oeis.org

0, 3, 27, 216, 1704, 13419, 105651, 831792, 6548688, 51557715, 405913035, 3195746568, 25160059512, 198084729531, 1559517776739, 12278057484384, 96664942098336, 761041479302307, 5991666892320123, 47172293659258680, 371386682381749320, 2923921165394735883, 23019982640776137747

Offset: 0

Richard Choulet, Oct 14 2008

Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A057080, A131571, A145607.

RecurrenceTable[{a[0]==0,a[1]==3,a[n]==8a[n-1]-a[n-2]+3},a,{n,30}] (* or *) LinearRecurrence[{9,-9,1},{0,3,27},30] (* Harvey P. Dale, May 06 2013 *)

a(n+2) = 8*a(n+1) - a(n) + 3.
a(n) = (A070997(n)-1)/2 = 3*A076765(n-1). - R. J. Mathar, Oct 16 2008
G.f.: -3*x / ( (x-1)*(x^2-8*x+1) ). - R. J. Mathar, Nov 27 2011
a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=27. - Harvey P. Dale, May 06 2013

Made definition and sequence consistent. Changed offset to 0. - R. J. Mathar, Oct 16 2008

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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A145608 Numbers a(n)=k such that (1/3)(5(2k+1)^2-2) is A057080(n)^2.

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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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Programs

Mathematica

PARI

Formula

A145608 Numbers a(n)=k such that (1/3)*(5*(2k+1)^2-2) is A057080(n)^2.

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Extensions

A145608 Numbers a(n)=k such that (1/3)(5(2k+1)^2-2) is A057080(n)^2.