A133853 -id:A133853 - 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

A364072 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)63^(n-d-k), with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 65, 1, 1, 4161, 192, 1, 1, 266305, 28545, 382, 1, 1, 17043521, 3891520, 101125, 635, 1, 1, 1090785345, 511266561, 23105270, 261780, 951, 1, 1, 69810262081, 66021638592, 4901267861, 89335610, 562296, 1330, 1, 1, 4467856773185, 8454558363265, 997262532182, 27503177191, 267021146, 1066366, 1772, 1

Offset: 0

Stefano Spezia, Jul 04 2023

T(n, k) is the number of 64-subgroups of R^n which have dimension k, where R^n is a near-vector space over a proper nearfield R.

			The triangle begins:
  1;
  1,          1;
  1,         65,         1;
  1,       4161,       192,        1;
  1,     266305,     28545,      382,      1;
  1,   17043521,   3891520,   101125,    635,   1;
  1, 1090785345, 511266561, 23105270, 261780, 951, 1;
  ...

Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023. See pp. 7-9.

Cf. A000012 (k=0), A133853 (k=1), A364069 (row sums), A364071, A364073.

T[n_,k_]:=Sum[Binomial[n,d]StirlingS2[n-d,k]63^(n-d-k),{d,0,n-k}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten

A350054 a(n) = (4^(3*n+2) - 7)/9.

Original entry on oeis.org

113, 7281, 466033, 29826161, 1908874353, 122167958641, 7818749353073, 500399958596721, 32025597350190193, 2049638230412172401, 131176846746379033713, 8395318191768258157681, 537300364273168522091633, 34387223313482785413864561

Offset: 1

Wolfdieter Lang, Jan 20 2022

Bisection of A350053, namely the even part. The odd part is given in A228871.

Winston de Greef, Table of n, a(n) for n = 1..551
Index entries for linear recurrences with constant coefficients, signature (65,-64).

Cf. A133853, A228871, A350053.

Table[(4^(3*n + 2) - 7)/9, {n, 1, 14}] (* Amiram Eldar, Jan 21 2022 *)

a(n) = (4^(3*n+2) - 7)/9 \\ Winston de Greef, Jan 27 2024

a(n) = (2^(6*n+4) - 7)/9, n >= 1.
G.f.: x*(113 - 64*x)/((1 - x)*(1 - 64*x)).
a(n) = 112*A133853(n) + 1. - Hugo Pfoertner, Jan 21 2022

cp's OEIS Frontend

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

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

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

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Magma

Mathematica

Maxima

PARI

Formula

A364072 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)*StirlingS2(n-d, k)*63^(n-d-k), with 0 <= k <= n.

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Mathematica

A350054 a(n) = (4^(3*n+2) - 7)/9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364072 Triangle read by rows: T(n, k) = Sum_{d=0..n-k} binomial(n, d)StirlingS2(n-d, k)63^(n-d-k), with 0 <= k <= n.