A164783 -id:A164783 - cp's OEIS frontend

A164784 a(n) = 6^n-5.

1, 31, 211, 1291, 7771, 46651, 279931, 1679611, 10077691, 60466171, 362797051, 2176782331, 13060694011, 78364164091, 470184984571, 2821109907451, 16926659444731, 101559956668411, 609359740010491, 3656158440062971

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m (n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m (n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (7, -6).

[6^n-5: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 24 x)/(1 - 7 x + 6 x^2), {x, 0, 30}],x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 6*a(n-1)+25 with n>1, a(1)=1. - Vincenzo Librandi, Oct 29 2009
G.f.: x*(1 + 24*x)/(1 - 7*x + 6*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 4 + (exp(5*x) - 5)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

A164785 a(n) = 5^n - 4.

Original entry on oeis.org

1, 21, 121, 621, 3121, 15621, 78121, 390621, 1953121, 9765621, 48828121, 244140621, 1220703121, 6103515621, 30517578121, 152587890621, 762939453121, 3814697265621, 19073486328121, 95367431640621, 476837158203121

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A059613.

[5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

5^Range[50]-4 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* Harvey P. Dale, Jun 07 2012 *)
CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - Harvey P. Dale, Jun 07 2012
G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009

A164786 a(n) = 8^n-7.

Original entry on oeis.org

1, 57, 505, 4089, 32761, 262137, 2097145, 16777209, 134217721, 1073741817, 8589934585, 68719476729, 549755813881, 4398046511097, 35184372088825, 281474976710649, 2251799813685241, 18014398509481977, 144115188075855865

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Index entries for linear recurrences with constant coefficients, signature (9, -8).

[8^n-7: n in [1..20]]; // Vincenzo Librandi, Aug 22 2011

8^Range[20]-7 (* or *) LinearRecurrence[{9,-8},{1,57},20] (* Harvey P. Dale, Jan 24 2013 *)

a(n) = 8*a(n-1)+49, with a(1)=1. - Vincenzo Librandi, Nov 30 2010
G.f.: x*(1+48*x)/(1-9*x+8*x^2). a(n) = 9*a(n-1)-8*a(n-2). - Colin Barker, Jan 28 2012
E.g.f.: 6 + (exp(7*x) - 7)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(7)-a(19) from Vincenzo Librandi, Oct 29 2009

A193871 Square array T(n,k) = k^n - k + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 13, 1, 1, 31, 79, 61, 21, 1, 1, 63, 241, 253, 121, 31, 1, 1, 127, 727, 1021, 621, 211, 43, 1, 1, 255, 2185, 4093, 3121, 1291, 337, 57, 1, 1, 511, 6559, 16381, 15621, 7771, 2395, 505, 73, 1, 1, 1023, 19681, 65533, 78121, 46651, 16801, 4089, 721, 91, 1

Offset: 1

Omar E. Pol, Aug 21 2011

The columns give 1^n-0, 2^n-1, 3^n-2, 4^n-3, 5^n-4, etc.

The main diagonal gives A006091, which is a sequence related to the famous "coconuts" problem.

			Array begins:
  1,   1,    1,     1,     1,    1,    1,   1,   1,   1
  1,   3,    7,    13,    21,   31,   43,  57,  73
  1,   7,   25,    61,   121,  211,  337, 505
  1,  15,   79,   253,   621, 1291, 2395
  1,  31,  241,  1021,  3121, 7771
  1,  63,  727,  4093, 15621
  1, 127, 2185, 16381
  1, 255, 6559
  1, 511
  1

M. B. Richardson, A Needlessly Complicated and Unhelpful Solution to Ben Ames Williams' Coconuts Problem, The Winnower, 3 (2016), e147175.52128. doi: 10.15200/winn.147175.52128

Row 1: A000012. Rows 2,3: A002061, A061600 but both without repetitions.
Columns 1..10: A000012, positives A000225, A058481, A141725, A164785, A164784, A164783, A164786, A177095, A170955.
Cf. A276135.

Table[k^# - k + 1 &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Nov 16 2016 *)

A253210 a(n) = 7^n + 6.

Original entry on oeis.org

7, 13, 55, 349, 2407, 16813, 117655, 823549, 5764807, 40353613, 282475255, 1977326749, 13841287207, 96889010413, 678223072855, 4747561509949, 33232930569607, 232630513987213, 1628413597910455, 11398895185373149, 79792266297612007, 558545864083284013

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226819.

Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. similar sequences listed in A253208.

Cf. A164783, A226819.

Magma
```
[7^n+6: n in [0..30]];
```
Mathematica
```
Table[7^n + 6, {n, 0, 30}]
```

G.f.: (7 - 43*x) / ((1 - x)*(1 - 7*x)).

a(n) = 8*a(n-1) - 7*a(n-2) for n>1.

cp's OEIS Frontend

A164784 a(n) = 6^n-5.

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A164785 a(n) = 5^n - 4.

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A164786 a(n) = 8^n-7.

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A193871 Square array T(n,k) = k^n - k + 1 read by antidiagonals.

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A253210 a(n) = 7^n + 6.

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Magma

Mathematica

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