A005061 -id:A005061 - cp's OEIS frontend

A167936 a(n) = 2^n - A108411(n).

0, 1, 1, 5, 7, 23, 37, 101, 175, 431, 781, 1805, 3367, 7463, 14197, 30581, 58975, 124511, 242461, 504605, 989527, 2038103, 4017157, 8211461, 16245775, 33022991, 65514541, 132623405, 263652487, 532087943, 1059392917, 2133134741, 4251920575, 8546887871

Offset: 0

Paul Curtz, Nov 15 2009

The binomial transform of (0 followed by A077917).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A005061, A077917, A083658, A085350, A108411, A167762, A167784.

I:=[0,1,1]; [n le 3 select I[n] else 2*Self(n-1) +3*Self(n-2) -6*Self(n-3): n in [1..40]]; // G. C. Greubel, Sep 10 2023

LinearRecurrence[{2,3,-6}, {0,1,1}, 50] (* G. C. Greubel, Jul 01 2016 *)

def A167936(n): return (1<>1) # Chai Wah Wu, Nov 14 2023

def A167936(n): return 2^n - ((n+1)%2)*3^(n//2) - (n%2)*3^((n-1)//2)
[A167936(n) for n in range(41)] # G. C. Greubel, Sep 10 2023

a(n) = A167762(n+1) - A167762(n).
a(n+1) - a(n) = A167784(n).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: x*(1-x)/((1-2*x)*(1-3*x^2)).
a(2n) = A005061(n), a(2n+1) = A085350(n).
a(n) - 2*a(n-1) = (-1)^(n+1)*A083658(n+1).
From G. C. Greubel, Sep 10 2023: (Start)
a(n) = (1/2)*(2^(n+1) - (1+(-1)^n)*3^(n/2) - (1-(-1)^n)*3^((n-1)/2)).
E.g.f.: exp(2*x) - cosh(sqrt(3)*x) - (1/sqrt(3))*sinh(sqrt(3)*x). (End)

Edited and extended by R. J. Mathar, Feb 27 2010

A016197 a(n) = 12^n - 11^n.

Original entry on oeis.org

0, 1, 23, 397, 6095, 87781, 1214423, 16344637, 215622815, 2801832661, 35979939623, 457696700077, 5777672071535, 72470493235141, 904168630965623, 11229773405170717, 138934529031464255, 1713164078241143221

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (23, -132).

Cf. k^n-(k-1)^n: A000225 (k=2), A001047 (k=3), A005061 (k=4), A005060 (k=5), A005062 (k=6), A016169 (k=7), A016177 (k=8), A016185 (k=9), A016189 (k=10), A016195 (k=11), this sequence (k=12).

[12^n-11^n: n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

f[n_]:= 12^n - 11^n; f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
CoefficientList[Series[x/((1 - 11 x) (1 - 12 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 03 2014 *)

a(n)=12^n-11^n \\ Charles R Greathouse IV, Aug 03 2014

G.f.: x/((1-11x)(1-12x)).
E.g.f.: e^(12*x)-e^(11*x). - Mohammad K. Azarian, Jan 14 2009
a(0)=0, a(n)=12*a(n-1)+11^(n-1). - _Vincenzo Librandi-, Feb 09 2011
a(0)=0, a(1)=1, a(n)=23*a(n-1)-132*a(n-2). - Vincenzo Librandi, Feb 09 2011

A051589 Number of 5xn binary matrices such that any 2 rows have a common 1.

Original entry on oeis.org

0, 1, 63, 3367, 167835, 7803391, 339133803, 13887495007, 541044196875, 20237096702431, 732455240043243, 25820836854042847, 891331324715015115, 30260208833985800671, 1013882831306569043883, 33620617443978687281887, 1105857774681062127612555

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..670
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (in Russian), Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.

Cf. A005061, A051587, A051588.

List([0..20], n-> 32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n); # G. C. Greubel, Nov 12 2019

[32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n: n in [0..20]]; // Vincenzo Librandi, Sep 18 2018

A051589(n):=32^n -10*24^n +30*20^n -5*18^n +5*17^n -70*16^n -30*15^n + 135*14^n +30*13^n -140*12^n -2*11^n +130*10^n -110*9^n +45*8^n -10*7^n +6^n; seq(A051589(n), n=0..20); # G. C. Greubel, Nov 12 2019

Table[32^n -10*24^n +30*20^n -5*18^n +5*17^n -70*16^n -30*15^n +135*14^n +30*13^n -140*12^n -2*11^n +130*10^n -110*9^n +45*8^n -10*7^n +6^n, {n, 0, 30}] (* Vincenzo Librandi, Sep 18 2018 *)

vector(21, n, m=n-1; 32^m -10*24^m +30*20^m -5*18^m +5*17^m -70*16^m -30*15^m +135*14^m +30*13^m -140*12^m -2*11^m +130*10^m -110*9^m +45*8^m -10*7^m +6^m) \\ G. C. Greubel, Nov 12 2019

[32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = 32^n - 10*24^n + 30*20^n - 5*18^n + 5*17^n - 70*16^n - 30*15^n + 135*14^n + 30*13^n - 140*12^n - 2*11^n + 130*10^n - 110*9^n + 45*8^n - 10*7^n + 6^n.

G.f.: x*(933561925632000*x^14 -1286309121638400*x^13 +786606914672640*x^12 -287219252934144*x^11 +70324589076096*x^10 -12248067009984*x^9 +1568017231256*x^8 -150181430252*x^7 +10834851518*x^6 -587198697*x^5 +23594853*x^4 -684354*x^3 +13636*x^2 -169*x +1) / ((6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(12*x -1)*(13*x -1)*(14*x -1)*(15*x -1)*(16*x -1)*(17*x -1)*(18*x -1)*(20*x -1)*(24*x -1)*(32*x -1)). - Colin Barker, Feb 22 2013

Revised Aug 03 2000

A069378 Number of n X 3 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

7, 37, 197, 1041, 5503, 29089, 153769, 812849, 4296863, 22713981, 120070149, 634712209, 3355201895, 17736195433, 93756691401, 495614587553, 2619907077991, 13849295944501, 73209847696773

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Column 3 of A359576.

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5))); // G. C. Greubel, Apr 22 2018

Rest[CoefficientList[Series[x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4 +2*x^5), {x,0,50}], x]] (* G. C. Greubel, Apr 22 2018 *)

x='x+O('x^30); Vec(x*(7-12*x+x^2+2*x^3-2*x^4)/(1 -7*x+9*x^2 +x^3- 4*x^4+2*x^5)) \\ G. C. Greubel, Apr 22 2018

G.f.: x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5). - Vladeta Jovovic, Jul 02 2003

A069379 Number of n X 4 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

15, 175, 1985, 22193, 247759, 2764991, 30856705, 344356289, 3842988975, 42887577455, 478623939553, 5341429762353, 59610217019311, 665248512113343, 7424156719466465, 82853403589520257, 924641917817567951

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Column 4 of A359576.

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A280208 Numbers m such that 4^m - 3^m is not squarefree, but 4^d - 3^d is squarefree for every proper divisor d of m.

Original entry on oeis.org

4, 14, 55, 78, 111, 253, 342, 355

Offset: 1

Juri-Stepan Gerasimov, Dec 28 2016

Where numbers m such that 4^m - 3^m is not squarefree: numbers of the form i*a(j) for i >= 1.
The smallest squares of 4^m - 3^m as defined above are 25, 49, 121, 169, 1369, 529, 361, 5041. - Robert Price, Mar 07 2017
431 <= a(9) <= 1081. 1081, 3403 are terms. - Chai Wah Wu, Jul 20 2020

			4 is in this sequence because all 4^1 - 3^1 = 1, 4^2 - 3^2 = 7 are squarefrees where 1, 2 are proper divisors of 4 and 4^4 - 3^4 = 175 = 7*5^2 is not squarefree;
14 is in this sequence because all 4^1 = 3^2 = 1, 4^2 - 3^2 = 7, 4^7 - 3^7 = 14197 are squarefrees where 1, 2, 7 are proper divisors of 14 and 4^14 - 3^14 = 263652487 = 7^2*3591*14197 is not squarefree.

Cf. A005061.

Cf. Numbers m such that (k+1)^m - k^m is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of m: A237043 (k = 1), A280203 (k = 2), this sequence (k = 3), A280209 (k = 4), A280307 (k = 6).

Function[s, DeleteCases[#, 0] &@ MapIndexed[#1 Boole[! AnyTrue[Take[s, First@ #2 - 1], Function[k, Divisible[#1, k]]]] &, s]]@ Select[Range@ 80, ! SquareFreeQ[4^# - 3^#] &] (* Michael De Vlieger, Dec 30 2016 *)

a(6)-a(8) from Jinyuan Wang, May 15 2020

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 12, 97, 660, 4081, 23772, 133057, 724260, 3863761, 20308332, 105558817, 544039860, 2785713841, 14192221692, 72020501377, 364354427460, 1838822866321, 9262446387852, 46585947584737

Offset: 0

N. J. A. Sloane

As (0,0,1,12,97,...) this is the fourth binomial transform of cosh(x)-1. It is the binomial transform of A016269, when this has two leading zeros. Its e.g.f. is then exp(4x)cosh(x) - exp(4x). - Paul Barry, May 13 2003
This gives the third column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. below, and A193685 for comments on the general case. - Wolfdieter Lang, Oct 08 2011
From Kevin Long, Mar 25 2017: (Start)
In the power set poset 2^(n+2), a(n) gives the number of size 3 subposets {A,B,C} such that A subset of C, B subset of C, and A||B. By symmetry, it also counts the size 3 subposets {A,B,C} such that C subset of A, C subset of B, and A||B.
By the power set poset, I mean the subsets of [n+2] ordered by inclusion. A||B means A and B are incomparable.
The result can be proved by showing that the formula holds. 5^n counts triples (A,B,C) of subsets of [n] where A subset of  C and B subset of C, since for each x in [n], it is either in C only, in A and C, in B and C, in all three, or in none. However, this also counts the cases where A subset of B and where B subset of A, and we want A||B.
Each case can be counted by 4^n, since if A subset of B⊆C, then each element x of [n] is either in all three, in B and C, in only C, or in none. Hence we subtract 2*4^n from 5^n. These two cases intersect, however, when A = B subset of C, which can be counted by 3^n, since each element x of [n] can be either in all three sets, in only C, or in none.
For the purposes of inclusion-exclusion, we add these sets back in to get 5^n-2*4^n+3^n to count all triples (A,B,C) where A subset of C, B subset of C, and A||B. We want sets, not triples, so this double-counts the sets since interchanging A and B give the same set, so we divide this by 2. Hence the formula for a(n) counts these subposets for 2^(n+2).  (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-47,60).

Cf. A000244, A005061, A016753, A132495, A193685, A245019.

[(5^(n+2) - 2*4^(n+2) + 3^(n+2))/2: n in [0..30]]; // G. C. Greubel, Sep 15 2018

CoefficientList[ Series[ 1/((1 - 3x)(1 - 4x)(1 - 5x)), {x, 0, 25} ], x ]
LinearRecurrence[{12,-47,60}, {1, 12, 97}, 30] (* G. C. Greubel, Sep 15 2018 *)

Vec(1/((1-3*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 5^(n+2)/2 - 4^(n+2) + 3^(n+2)/2. - Paul Barry, May 13 2003
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,3), (n >= 2). - Milan Janjic, Apr 26 2009
a(n) = 9*a(n-1) - 20*a(n-2) + 3^n, n >= 2. - Vincenzo Librandi, Mar 20 2011
O.g.f.: 1/((1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: (d^2/dx^2) (exp(3*x)*((exp(x)-1)^2)/2!). - Wolfdieter Lang, Oct 08 2011
a(n) = A245019(n+2)/2. - Kevin Long, Mar 24 2017

A069380 Number of n X 5 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

31, 781, 18621, 433809, 10056959, 232824241, 5388274121, 124693133113, 2885579381831, 66776768695477, 1545323639404349, 35761396310047985, 827579980089997079, 19151628770974955241, 443201843190147840905

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Column 5 of A359576.

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

cp's OEIS Frontend

A167936 a(n) = 2^n - A108411(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Python

SageMath

Formula

Extensions

A016197 a(n) = 12^n - 11^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A051589 Number of 5xn binary matrices such that any 2 rows have a common 1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A069378 Number of n X 3 binary arrays with a path of adjacent 1's from top row to bottom row.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A069379 Number of n X 4 binary arrays with a path of adjacent 1's from top row to bottom row.

Views

Author

Keywords

Links

Crossrefs

A280208 Numbers m such that 4^m - 3^m is not squarefree, but 4^d - 3^d is squarefree for every proper divisor d of m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A016753 Expansion of 1/((1-3*x)*(1-4*x)*(1-5*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).