A087404 -id:A087404 - cp's OEIS frontend

A087405 First differences of A087404: a(n) = A087404(n) - A087404(n-1), a(0) = A087404(0).

2, 2, 22, 98, 502, 2498, 12502, 62498, 312502, 1562498, 7812502, 39062498, 195312502, 976562498, 4882812502, 24414062498, 122070312502, 610351562498, 3051757812502, 15258789062498, 76293945312502, 381469726562498, 1907348632812502, 9536743164062498

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 01 2003

Index entries for linear recurrences with constant coefficients, signature (4,5).

CoefficientList[Series[((1 - x)(2 - 4x))/(1 - 4x - 5x^2), {x, 0, 25}], x]
LinearRecurrence[{4,5},{2,2,22},30] (* Harvey P. Dale, Oct 23 2024 *)

G.f.: ((1-x)(2-4x))/(1-4x-5x^2).
a(n) = 4*5^(n-1)+2(-1)^n for n > 0.
E.g.f.: 2*(5*cosh(x) + 2*cosh(5*x) - 5*sinh(x) + 2*sinh(5*x) - 2)/5. - Stefano Spezia, Aug 29 2025

A081340 (5^n+(-1)^n)/2.

Original entry on oeis.org

1, 2, 13, 62, 313, 1562, 7813, 39062, 195313, 976562, 4882813, 24414062, 122070313, 610351562, 3051757813, 15258789062, 76293945313, 381469726562, 1907348632813, 9536743164062, 47683715820313, 238418579101562

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A081341, A087404.

CoefficientList[Series[(1 - 2 x) / ((1 + x) (1 - 5 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n)=(5^n+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

[lucas_number2(n,4,-5)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.
G.f.: (1-2*x)/((1+x)*(1-5*x)).
E.g.f.: exp(2*x) * cosh(3*x).
a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = sum( k=0..n, A201730(n,k)*8^k ). - Philippe Deléham, Dec 06 2011

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Bruno Berselli, Jan 09 2013

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Bruno Berselli, Table of n, a(n) for n = 0..200
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).
Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).
Cf. A052539, A024036.

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

G.f.: (2-3*x)/((1+x)*(1-4*x)).
a(n) = 4^n+(-1)^n.
a(n) = A086341(A047524(n)) for n>0, a(0)=2.
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019

A274074 a(n) = 6^n+(-1)^n.

Original entry on oeis.org

2, 5, 37, 215, 1297, 7775, 46657, 279935, 1679617, 10077695, 60466177, 362797055, 2176782337, 13060694015, 78364164097, 470184984575, 2821109907457, 16926659444735, 101559956668417, 609359740010495, 3656158440062977, 21936950640377855, 131621703842267137

Offset: 0

Colin Barker, Jun 09 2016

Michael De Vlieger, Table of n, a(n) for n = 0..1285
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (5,6).

Sequences of the type k^n+(-1)^n: A014551 (k=2), A102345 (k=3), A201455 (k=4), A087404 (k=5), this sequence (k=6).

Array[6^# + (-1)^# &, 23, 0] (* or *)
LinearRecurrence[{5, 6}, {2, 5}, 23] (* or *)
CoefficientList[ Series[(5x -2)/(6x^2 + 5x -1), {x, 0, 23}], x] (* Robert G. Wilson v, Jan 01 2017 *)

PARI
```
Vec((2-5*x)/((1+x)*(1-6*x)) + O(x^30))
```

O.g.f.: (2-5*x) / ((1+x)*(1-6*x)).
E.g.f.: exp(-x) + exp(6*x).
a(n) = 5*a(n-1)+6*a(n-2) for n>1.

cp's OEIS Frontend

A087405 First differences of A087404: a(n) = A087404(n) - A087404(n-1), a(0) = A087404(0).

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A081340 (5^n+(-1)^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A274074 a(n) = 6^n+(-1)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.