A218723 -id:A218723 - cp's OEIS frontend

A218748 a(n) = (45^n - 1)/44.

0, 1, 46, 2071, 93196, 4193821, 188721946, 8492487571, 382161940696, 17197287331321, 773877929909446, 34824506845925071, 1567102808066628196, 70519626362998268821, 3173383186334922096946, 142802243385071494362571, 6426100952328217246315696

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 45 (A009989).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (46,-45).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009989.

[n le 2 select n-1 else 46*Self(n-1) - 45*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{46, -45}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218748(n):=(45^n-1)/44$ makelist(A218748(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218748(n)=45^n\44
```

G.f.: x/((1-x)*(1-45*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 46*a(n-1) - 45*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 45*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(45^n/44).  - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(23*x)*sinh(22*x)/22. - Elmo R. Oliveira, Aug 27 2024

A218749 a(n) = (46^n - 1)/45.

Original entry on oeis.org

0, 1, 47, 2163, 99499, 4576955, 210539931, 9684836827, 445502494043, 20493114725979, 942683277395035, 43363430760171611, 1994717814967894107, 91757019488523128923, 4220822896472063930459, 194157853237714940801115, 8931261248934887276851291, 410838017451004814735159387

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 46 (A009990).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (47,-46).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009990.

[n le 2 select n-1 else 47*Self(n-1) - 46*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{47, -46}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
(46^Range[0,20]-1)/45 (* Harvey P. Dale, Aug 17 2017 *)

A218749(n):=(46^n-1)/45$ makelist(A218749(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218749(n)=46^n\45
```

From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-46*x)).
a(n) = 47*a(n-1) - 46*a(n-2) with a(0)=0, a(1)=1.
a(n) = 46*a(n-1) + 1 with a(0)=0.
a(n) = floor(46^n/45). (End)
E.g.f.: exp(x)*(exp(45*x) - 1)/45. - Elmo R. Oliveira, Aug 29 2024

A218751 a(n) = (48^n - 1)/47.

Original entry on oeis.org

0, 1, 49, 2353, 112945, 5421361, 260225329, 12490815793, 599559158065, 28778839587121, 1381384300181809, 66306446408726833, 3182709427618887985, 152770052525706623281, 7332962521233917917489, 351982201019228060039473, 16895145648922946881894705, 810966991148301450330945841

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 48 (A009992).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (49,-48).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009992.

[n le 2 select n-1 else 49*Self(n-1)-48*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{49, -48}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218751(n):=floor((48^n-1)/47)$ makelist(A218751(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218751(n)=48^n\47
```

a(n) = floor(48^n/47).
From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-48*x)).
a(n) = 49*a(n-1) - 48*a(n-2) with a(0)=0, a(1)=1.
a(n) = 48*a(n-1) + 1 with a(0)=0. (End)
E.g.f.: exp(x)*(exp(47*x) - 1)/47. - Elmo R. Oliveira, Aug 29 2024

A282137 Expansion of (24x^2-10x-1)/(16x^3-16x^2+x-1).

Original entry on oeis.org

1, 11, -29, -189, 451, 3011, -7229, -48189, 115651, 771011, -1850429, -12336189, 29606851, 197379011, -473709629, -3158064189, 7579354051, 50529027011, -121269664829, -808464432189, 1940314637251, 12935430915011, -31045034196029, -206966894640189

Offset: 0

Andrey Zabolotskiy, Feb 06 2017

Related to base i-1 representation of integers (Khmelnik encoding): presumably a(0) is the most common first difference of A066321 (occurs with density 1/2), a(1) is the second most common difference (density 1/4), a(2) has density 1/8, and so on; in particular, A066322 consists entirely of the terms a(n) with n>3.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-16,16).

Cf. A066321, A066322, A218723.

LinearRecurrence[{0,0,0,257,0,0,0,-256}, {1, 11, -29, -189, 451, 3011, -7229, -48189}, 24]
LinearRecurrence[{1, -16, 16}, {1, 11, -29}, 24]

Vec((1 - 2*x)*(1 + 12*x) / ((1 - x)*(1 + 16*x^2)) + O(x^30)) \\ Colin Barker, Feb 07 2017

print([[1, 11, -29, -189][n%4] + [450, 3000, -7200, -48000][n%4]*(256**(n//4)-1)//255 for n in range(24)])

a(k+8) - 257 * a(k+4) + 256 * a(k) = 0, for k >= 0. - Altug Alkan, Feb 07 2017
G.f.: (24*x^2-10*x-1)/(16*x^3-16*x^2+x-1).
From Colin Barker, Feb 07 2017: (Start)
a(n) = (-13 + (15+25*i)*(-4*i)^n + (15-25*i)*(4*i)^n) / 17 where i=sqrt(-1).
a(n) = a(n-1) - 16*a(n-2) + 16*a(n-3) for n>2.
(End)

cp's OEIS Frontend

A218748 a(n) = (45^n - 1)/44.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A218749 a(n) = (46^n - 1)/45.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A218751 a(n) = (48^n - 1)/47.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A282137 Expansion of (24x^2-10x-1)/(16x^3-16x^2+x-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula