A058481 -id:A058481 - cp's OEIS frontend

A059515 Square array T(k,n) by antidiagonals, where T(k,n) is number of ways of placing n identifiable nonnegative intervals with a total of exactly k starting and/or finishing points.

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 7, 1, 0, 0, 0, 12, 25, 1, 0, 0, 0, 6, 138, 79, 1, 0, 0, 0, 0, 294, 1056, 241, 1, 0, 0, 0, 0, 270, 5298, 7050, 727, 1, 0, 0, 0, 0, 90, 12780, 70350, 44472, 2185, 1, 0, 0, 0, 0, 0, 16020, 334710, 817746, 273378, 6559, 1, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Jan 19 2001

See A300729 for a triangular version of this array. - Peter Bala, Jun 13 2019

			Rows are: 1,0,0,0,0,..., 0,1,1,0,0,..., 0,1,7,12,6,..., 0,1,25,138,294,..., etc. T(1,1)=1 since if a is starting point of interval and A is end point then only possibility is aA (zero length). T(2,1)=1 since possibility is a-A (positive length). T(3,2)=12 since possibilities are: aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB.

IBM Ponder This, Jan 01 2001

Sum of rows gives A059516. Columns include A000007, A057427, A058481, A059117. Final positive number in each row is A000680.

Cf. A300729.

T(k, n) = T(k - 2, n - 1) * k * (k - 1)/2 + T(k - 1, n - 1) * k^2 + T(k, n - 1) * k * (k + 1)/2 with T(0, 0) = 1 = lambda(k, n) + lambda(k + 1, n) where lambda is A059117(k, n).

A132753 a(n) = 2^(n+1) - n + 1.

Original entry on oeis.org

3, 4, 7, 14, 29, 60, 123, 250, 505, 1016, 2039, 4086, 8181, 16372, 32755, 65522, 131057, 262128, 524271, 1048558, 2097133, 4194284, 8388587, 16777194, 33554409, 67108840, 134217703, 268435430, 536870885, 1073741796, 2147483619

Offset: 0

Gary W. Adamson, Aug 28 2007

Apart from a(0): Row sums of triangle A132752 (old name).

Apart from a(0): Binomial transform of [1, 3, 0, 4, 0, 4, 0, 4, ...].

			a(3) = 14 = sum of row 3 terms of triangle A132752: (3 + 5 + 5 + 1).
a(3) = 14 = (1, 3, 3, 1) dot (1, 3, 0, 4) = (1 + 9 + 0 + 4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A132752.

Cf. A003462, A007051, A024023, A029858, A034472, A052548, A058481, A067771, A079004, A100774, A115099, A134931. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008

[2^(n+1) -n+1: n in [0..40]]; // G. C. Greubel, Feb 16 2021

A132753:= n-> 2^(n+1) -n+1; seq(A132753(n), n=0..40) # G. C. Greubel, Feb 16 2021

Table[2^(n+1) -n+1, {n, 0, 30}] (* Bruno Berselli, Aug 31 2013 *)

PARI
```
a(n)=2^(n+1)-n+1
```

Vec( (3-8*x+6*x^2)/((1-x)^2*(1-2*x)) + O(x^40)) \\ Colin Barker, Mar 14 2014

[2^(n+1) -n+1 for n in (0..40)] # G. C. Greubel, Feb 16 2021

From Colin Barker, Mar 14 2014: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (3 - 8*x + 6*x^2)/((1-x)^2 * (1-2*x)). (End)
E.g.f.: (1-x)*exp(x) + 2*exp(2*x). - G. C. Greubel, Feb 16 2021

More terms Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Changed first member, and better name from Ralf Stephan, Aug 31 2013

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A058482 Number of 3 X n binary matrices with no zero rows or columns.

Original entry on oeis.org

1, 25, 265, 2161, 16081, 115465, 816985, 5745121, 40294561, 282298105, 1976795305, 13839692881, 96884227441, 678208723945, 4747518463225, 33232801429441, 232630126566721, 1628412435648985, 11398891698588745, 79792255837258801, 558545832702224401

Offset: 1

Vladeta Jovovic, Nov 26 2000

Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Cf. A055602, A024206, A055609 (unlabeled case), A058481, column 3 of A183109 and A218695.

LinearRecurrence[{11,-31,21},{1,25,265},30] (* Harvey P. Dale, Aug 15 2014 *)

a(n) = 7^n-3*3^n+3 \\ Charles R Greathouse IV, Feb 10 2017

Number of m X n binary matrices with no zero rows or columns is Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
a(n) = 7^n-3*3^n+3.
a(n) = 11*a(n-1)-31*a(n-2)+21*a(n-3). G.f.: -x*(21*x^2+14*x+1) / ((x-1)*(3*x-1)*(7*x-1)). - Colin Barker, Jul 10 2013

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

More terms from Colin Barker, Jul 10 2013

A080798 Largest prime factor of 3^n-2.

Original entry on oeis.org

7, 5, 79, 241, 727, 23, 937, 19681, 431, 499, 4703, 8093, 40193, 2869781, 483671, 94747, 4657, 232452293, 498112057, 2812679, 31381059607, 3765727153, 1364071, 44594137339, 125231, 13170403, 5353801183, 4159349, 46050353857, 294487079, 26892769, 29178816413, 3533781113

Offset: 2

Hugo Pfoertner, Mar 25 2003

nonn

Hugo Pfoertner, Table of n, a(n) for n = 2..210

Cf. A014224, A014232, A080443, A080892.

Cf. A006530, A058481.

[Max(PrimeDivisors(3^n-2)):n in [2..30]]; // Marius A. Burtea, Jul 12 2019

FactorInteger[#][[-1,1]]&/@(3^Range[2,30]-2) (* Harvey P. Dale, Apr 07 2022 *)

a(n) = vecmax(factor(3^n-2)[,1]); \\ Michel Marcus, Jul 12 2019

a(n) = A006530(A058481(n)). - Michel Marcus, Jul 12 2019

A137410 a(n) = (5^n - 3)/2.

Original entry on oeis.org

-1, 1, 11, 61, 311, 1561, 7811, 39061, 195311, 976561, 4882811, 24414061, 122070311, 610351561, 3051757811, 15258789061, 76293945311, 381469726561, 1907348632811, 9536743164061, 47683715820311, 238418579101561, 1192092895507811, 5960464477539061, 29802322387695311, 149011611938476561

Offset: 0

Ctibor O. Zizka, Apr 15 2008

Sequence is a(n) = a(n;5,3,1) where a(n;A,B,r) = (A^n - B^r)/(A - B) for arbitrary integers A, B, r with A != B.
Primes of this form are sometimes of interest, examples:
A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,
A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,
A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,
A=4, B=1, r=1 gives A002450,
A=4, B=2, r=1 gives A083420,
A=4, B=2, r=2 gives A002446,
A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,
A=5, B=2, r=1 gives A037577.
Sum of n-th row of triangle of powers of 5: 1; 5 1 5; 25 5 1 5 25; 125 25 5 1 5 25 125; ... (cf. Examples). - Philippe Deléham, Feb 24 2014
Integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n (see Campbell and Zujev). - Michel Marcus, Mar 02 2016

			From _Philippe Deléham_, Feb 24 2014: (Start)
a(1) = 1;
a(2) = 5 + 1 + 5 = 11;
a(3) = 25 + 5 + 1 + 5 + 25 = 61;
a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 0..100
Geoffrey B Campbell and Aleksander Zujev, On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, arXiv:1603.00080 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000040, A000225, A001348, A002446, A002450, A003462, A003463, A004061, A014224, A024049, A028491, A037577, A058481, A083420, A125831.

[(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

LinearRecurrence[{6, -5}, {1, 11}, 25] (* Vincenzo Librandi, Mar 02 2016 *)
(5^Range[30]-3)/2 (* Harvey P. Dale, Feb 24 2017 *)

a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016

a(n) = (5^n - 3)/2.
From Colin Barker, May 01 2012: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)
a(n) = 5*a(n-1) + 6, a(1) = 1. - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = A024049(n)/2 - 1 = A125831(n) - 1.
E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

More terms from Michel Marcus, Mar 02 2016

Edited and missing term a(0) inserted by M. F. Hasler, Jul 10 2018

A168589 a(n) = (2 - 3^n)*(-1)^n.

Original entry on oeis.org

1, 1, -7, 25, -79, 241, -727, 2185, -6559, 19681, -59047, 177145, -531439, 1594321, -4782967, 14348905, -43046719, 129140161, -387420487, 1162261465, -3486784399, 10460353201, -31381059607, 94143178825, -282429536479

Offset: 0

Klaus Brockhaus, Nov 30 2009

A signed version of A058481 preceded by 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (-4, -3).

Cf. A058481 (3^n-2).

Magma
```
[ (2-3^n)*(-1)^n: n in [0..25] ];
```

Table[(2 - 3^n)*(-1)^n, {n,0,50}] (* G. C. Greubel, Jul 26 2016 *)

a(n)=(2-3^n)*(-1)^n \\ Charles R Greathouse IV, Jul 26 2016

a(n) = -4*a(n-1) - 3*a(n-2) for n > 1; a(0) = 1, a(1) = 1.
G.f.: (1 + 5*x)/((1+x)*(1+3*x)).
E.g.f.: 2*exp(-x) - exp(-3*x). - G. C. Greubel, Jul 26 2016

A002000 a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 7.

Original entry on oeis.org

7, 322, 33385282, 37210469265847998489922, 51522323599677629496737990329528638956583548304378053615581043535682

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..6
E. B. Escott, Rapid method for extracting a square root, Amer. Math. Monthly, 44 (1937), 644-646.

Cf. A000032, A001999, A006267, A219161, A271223.

[n eq 1 select 7 else Self(n-1)^3 - 3*Self(n-1): n in [1..6]]; // Vincenzo Librandi, Feb 09 2017

a := proc(n) option remember; if n = 0 then 7 else a(n-1)^3 - 3*a(n-1) end if; end;
seq(a(n), n = 0..4); # Peter Bala, Nov 15 2022

RecurrenceTable[{a[0] == 7, a[n] == a[n - 1]^3 - 3 a[n - 1]}, a, {n, 0, 8}]
(* Vincenzo Librandi, Feb 09 2017 *)
NestList[#(#^2-3)&,7,4] (* Harvey P. Dale, Aug 11 2021 *)

From Peter Bala, Feb 01 2017: (Start)
a(n) = ((7 + sqrt(45))/2)^(3^n) + ((7 - sqrt(45))/2)^(3^n).
a(n) = 2*T(3^n,7/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.
Product_{n >= 0} (1 + 2/(a(n) - 1)) = 3*sqrt(5)/5.
Cf. A001999 and A219161. (End)
From Peter Bala, Nov 15 2022: (Start)
a(n) = Lucas(4*(3^n)).
a(n+1) == a(n) (mod 3^(n+1)) (a particular case of the Gauss congruences for the Lucas numbers).
Conjecture: a(n+1) == a(n) (mod 3^(n+r+2)) for n >= r.
The least positive residue of a(n) mod(3^n) = 3^n - 2 = A058481(n). In the ring of 3-adic integers the limit_{n -> oo} a(n) exists and is equal to -2.
Product_{k = 0..n} (a(k) - 1) = (1/3)*Lucas(6*(3^n)). (End)

A108470 Table read by antidiagonals: T(n,k) = number of labeled partitions of (n,k) into pairs (i,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 15, 1, 1, 31, 79, 79, 31, 1, 1, 63, 241, 339, 241, 63, 1, 1, 127, 727, 1351, 1351, 727, 127, 1, 1, 255, 2185, 5235, 6721, 5235, 2185, 255, 1, 1, 511, 6559, 20119, 31831, 31831, 20119, 6559, 511, 1, 1, 1023, 19681, 77379

Offset: 1

Christian G. Bower, Jun 03 2005

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object.

			1 1 1 1 1 ...
1 3 7 15 31 ...
1 7 25 79 241 ...
1 15 79 339 1351 ...
1 31 241 1351 6721 ...

Cf. A108461. Columns 1-3: A000012, A000225, A058481. Main diagonal: A023997.

T(n,k):=sum(m!*stirling2(k,m)*stirling2(n-k+1,m),m,1,min(k,n-k+1)); /* Vladimir Kruchinin, Apr 11 2015 */

antidiag(nn) = {for (n=1, nn, for (k=1, n, print1(sum(m=1, min(k, n-k+1), m!*stirling(k, m, 2)*stirling(n-k+1, m, 2)), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

tabl(nn) = {default(seriesprecision, nn); for (n=1, nn, for (k=1, nn, print1(k!*polcoeff(polcoeff(n!*exp((exp(x)-1)*(exp(y)-1))+O(x^(n+1)), n, x), k, y), ", "); ); print(););} \\ Michel Marcus, Apr 11 2015

Double e.g.f.: exp((exp(x)-1)*(exp(y)-1)).

T(n,k) = Sum{m=1..min(k,n-k+1)} m!*stirling2(k,m)*stirling2(n-k+1,m). - Vladimir Kruchinin, Apr 11 2015

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 *)

cp's OEIS Frontend

A059515 Square array T(k,n) by antidiagonals, where T(k,n) is number of ways of placing n identifiable nonnegative intervals with a total of exactly k starting and/or finishing points.

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A132753 a(n) = 2^(n+1) - n + 1.

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A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

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A058482 Number of 3 X n binary matrices with no zero rows or columns.

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Mathematica

PARI

Formula

Extensions

A080798 Largest prime factor of 3^n-2.

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Magma

Mathematica

PARI

Formula

A137410 a(n) = (5^n - 3)/2.

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Magma

Mathematica

PARI

Formula

Extensions

A168589 a(n) = (2 - 3^n)*(-1)^n.

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