A000400 -id:A000400 - cp's OEIS frontend

A367247 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 6.

0, 7, 131, 1609, 16415, 150817, 1296191, 10641169, 84520175, 654958177, 4980233951, 37312922929, 276288797135, 2026564724737, 14750977566911, 106695818055889, 767748717541295, 5500729672814497, 39270143125479071, 279511731951144049, 1984459091985376655, 14059238393314971457

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366963.

Index entries for linear recurrences with constant coefficients, signature (18,-107,210).

Cf. A366963.
Cf. A000351, A000400, A000420.
Cf. A367243, A367244, A367245, A367246, A367248, A367249, A367250.

LinearRecurrence[{18,-107,210},{0,7,131},22]

a(n) = 27*7^(n-1) - 47*6^(n-1) + 4*5^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(7 + 5*x)/((1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
E.g.f.: (162*exp(7*x) - 329*exp(6*x) + 168*exp(5*x) - 1)/42.

A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7.

Original entry on oeis.org

0, 5, 111, 1601, 19095, 204545, 2045511, 19508081, 179752215, 1613908385, 14202967911, 123028446161, 1052237271735, 8907026785025, 74758478722311, 623053865857841, 5162154289325655, 42558224511290465, 349394287423788711, 2858263098464575121, 23311522539676521975

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366964.

Index entries for linear recurrences with constant coefficients, signature (21,-146,336).

Cf. A366964.
Cf. A000400, A000420, A001018.
Cf. A367243, A367244, A367245, A367246, A367247, A367249, A367250.

LinearRecurrence[{21,-146,336},{0,5,111},21]

a(n) = 23*8^(n-1) - 41*7^(n-1) + 3*6^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(5 + 6*x)/((1 - 6*x)*(1 - 7*x)*(1 - 8*x)).
E.g.f.: (161*exp(8*x) - 328*exp(7*x) + 168*exp(6*x) - 1)/56.

A047852 a(n) = A047848(4, n).

Original entry on oeis.org

1, 2, 9, 58, 401, 2802, 19609, 137258, 960801, 6725602, 47079209, 329554458, 2306881201, 16148168402, 113037178809, 791260251658, 5538821761601, 38771752331202, 271402266318409, 1899815864228858, 13298711049602001, 93090977347214002, 651636841430498009, 4561457890013486058

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000400(n-1) for n >= 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000400, A047848.

[(7^n +5)/6: n in [0..40]]; // G. C. Greubel, Jan 12 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=7*a[n-1]+1 od: seq(a[n]+1, n=0..19); # Zerinvary Lajos, Mar 20 2008

(7^Range[0,40] +5)/6 (* G. C. Greubel, Jan 12 2025 *)

def A047852(n): return (pow(7, n) + 5)//6
print([A047852(n) for n in range(41)]) # G. C. Greubel, Jan 12 2025

a(n) = (7^n + 5)/6. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 06 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 8*a(n-1) - 7*a(n-2) for n > 1.
G.f.: (1 - 6*x)/(1 - 8*x + 7*x^2). (End)
a(n) = 7*a(n-1) - 5, with a(0)=1. - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(6*x) + 5)/6. - Elmo R. Oliveira, Aug 29 2024

a(20)-a(23) from Elmo R. Oliveira, Aug 29 2024

A056547 a(n) = 6na(n-1) + 1 with a(0)=1.

Original entry on oeis.org

1, 7, 85, 1531, 36745, 1102351, 39684637, 1666754755, 80004228241, 4320228325015, 259213699500901, 17108104167059467, 1231783500028281625, 96079113002205966751, 8070645492185301207085, 726358094296677108637651

Offset: 0

Henry Bottomley, Jun 20 2000

			a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.

Harvey P. Dale, Table of n, a(n) for n = 0..346

Cf. A000522, A010844, A010845, A056545, A056546 for analogs. A056547/(A000142*A000400) is an increasingly good approximation to 6th root of e.

nxt[{n_,a_}]:={n+1,6a(n+1)+1}; NestList[nxt,{0,1},20][[;;,2]] (* Harvey P. Dale, Jul 17 2024 *)

a(n) = floor(e^(1/6)*6^n*n!).
a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - Philippe Deléham, Mar 14 2004
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.
a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).
The sequence b(n) := 6^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = 6^n*n!*( 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/(6*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/6) = 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/((6*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 10, 12, 9, 1, 6, 21, 44, 63, 54, 27, 1, 8, 36, 104, 214, 312, 324, 216, 81, 1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243, 1, 12, 78, 340, 1095, 2712, 5284, 8136, 9855, 9180, 6318, 2916, 729, 1, 14, 105, 532, 2009, 5922, 13993, 26840, 41979

Offset: 0

Paul D. Hanna, Jun 01 2003

			Triangle begins:
  1;
  1,  2,  3;
  1,  4, 10,  12,   9;
  1,  6, 21,  44,  63,   54,   27;
  1,  8, 36, 104, 214,  312,  324,  216,   81;
  1, 10, 55, 200, 530, 1052, 1590, 1800, 1485, 810, 243;

Alois P. Heinz, Rows n = 0..100, flattened

Cf. A000079, A000244, A000400, A002426, A084600 - A084606, A006139, A084609 - A084615, A101822, A212697.

a084608 n = a084608_list !! n
a084608_list = concat $ iterate ([1,2,3] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

A084608:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*2^(k-2*j)*3^j: j in [0..k]]) >;
[A084608(n,k): k in [0..2*n], n in [0..13]]; // G. C. Greubel, Mar 27 2023

f:= proc(n) option remember; expand((1+2*x+3*x^2)^n) end:
T:= (n,k)-> coeff(f(n), x, k):
seq(seq(T(n, k), k=0..2*n), n=0..10);  # Alois P. Heinz, Apr 03 2011

row[n_] := (1+2x+3x^2)^n + O[x]^(2n+1) // CoefficientList[#, x]&; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+2*x+3*x^2)^n,k,x); print1(t",")); print(" "))

def A084608(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*2^(k-2*j)*3^j for j in range(k//2+1))
flatten([[A084608(n,k) for k in range(2*n+1)] for n in range(14)]) # G. C. Greubel, Mar 27 2023

From G. C. Greubel, Mar 27 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*2^(k-2*j)*3^j.
T(n, n) = A084609(n).
T(n, 2*n-1) = A212697(n), n >= 1.
T(n, 2*n) = A000244(n).
Sum_{j=0..2*n} T(n, k) = A000400(n).
Sum_{k=0..2*n} (-1)^k*T(n, k) = A000079(n).
Sum_{k=0..n} T(n-k, k) = A101822(n). (End)

A100309 Modulo 2 binomial transform of 6^n.

Original entry on oeis.org

1, 7, 37, 259, 1297, 9079, 47989, 335923, 1679617, 11757319, 62145829, 435020803, 2178463249, 15249242743, 80603140213, 564221981491, 2821109907457, 19747769352199, 104381066575909, 730667466031363, 3658979549971729

Offset: 0

Paul Barry, Dec 06 2004

6^n may be retrieved through 6^n = Sum_{k=0..n} (-1)^A010060(n-k) * mod(binomial(n,k), 2) * a(k).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, arXiv:1011.6083 [math.NT], 2010-2012; J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29.

Cf. A000120, A000400, A001316, A001317, A010060, A038183.

Cf. A100307, A100308, A100310, A100311.

[(&+[6^k*(Binomial(n,k) mod 2): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Feb 02 2023

a[n_]:= a[n]= Sum[6^k*Mod[Binomial[n,k], 2], {k,0,n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Feb 02 2023 *)

def A100309(n): return sum((bool(~n&n-k)^1)*6**k for k in range(n+1)) # Chai Wah Wu, May 03 2023

def A100309(n): return sum(6^k*(binomial(n, k)%2) for k in range(n+1))
[A100309(n) for n in range(41)] # G. C. Greubel, Feb 02 2023

a(n) = Sum_{k=0..n} mod(binomial(n, k), 2) * 6^k.
From Vladimir Shevelev, Dec 26-27 2013: (Start)
Sum_{n>=0} 1/a(n)^r = Product_{k>=0} (1 + 1/(6^(2^k)+1)^r),
Sum_{n>=0} (-1)^A000120(n)/a(n)^r = Product_{k>=0} (1 - 1/(6^(2^k)+1)^r), where r>0 is a real number.
In particular,
Sum_{n>=0} 1/a(n) = Product_{k>=0} (1 + 1/(6^(2^k)+1)) = 1.1746508...;
Sum_{n>=0} (-1)^A000120(n)/a(n) = 5/6.
a(2^n) = 6^(2^n) + 1, n>=0.
Note that analogs of Stephan's limit formulas (see Shevelev link) reduce to the relations a(2^t*n+2^(t-1)) = 35*(6^(2^(t-1)+1))/(6^(2^(t-1))-1) * a(2^t*n+2^(t-1)-2), t>=2. In particular, for t=2,3,4, we have the following formulas:
a(4*n+2) = 37 * a(4*n);
a(8*n+4) = 1297/37 * a(8*n+2);
a(16*n+8)= 1679617/47989 * a(16*n+6), etc. (End)

A109354 a(n) = 6^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 6, 216, 46656, 60466176, 470184984576, 21936950640377856, 6140942214464815497216, 10314424798490535546171949056, 103945637534048876111514866313854976, 6285195213566005335561053533150026217291776, 2280250319867037997421842330085227917956272625811456

Offset: 0

Philippe Deléham, Aug 25 2005

nonn

Sequence given by the Hankel transform (see A001906 for definition) of A078018 = {1, 1, 7, 55, 469, 4237, 39907, 387739, ...}; example: det([1, 1, 7, 55; 1, 7, 55, 469; 7, 55, 469, 4237; 55, 469, 4237, 39907]) = 6^6 = 46656.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 6 types of edge. - Mark Stander, Apr 11 2019

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000400, A006125, A047656, A053763, A053764, A109345.

Table[6^((n^2-n)/2),{n,0,10}] (* Harvey P. Dale, May 28 2013 *)

a(n) = 6^((n^2 - n)/2); \\ Michel Marcus, Apr 12 2019

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(6i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(6*x). - Ilya Gutkovskiy, Jun 04 2020

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2020

A160869 a(n) = sigma(6^(n-1)).

Original entry on oeis.org

1, 12, 91, 600, 3751, 22932, 138811, 836400, 5028751, 30203052, 181308931, 1088123400, 6529545751, 39179682372, 235085301451, 1410533397600, 8463265086751, 50779784492892, 304679288612371, 1828077476115000, 10968470088963751, 65810836228506612

Offset: 1

N. J. A. Sloane, Nov 15 2009

Colin Barker, Table of n, a(n) for n = 1..1000
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Row 6 of array in A160870.

Cf. A000203, A000400, A059387.

[(2^n-1)*(3^n-1)/2: n in [1..50]]; // G. C. Greubel, Apr 30 2018

Table[(2^n-1)*(3^n-1)/2,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{12,-47,72,-36}, {1, 12, 91, 600}, 50] (* G. C. Greubel, Apr 30 2018 *)

Vec(-x*(6*x^2-1)/((x-1)*(2*x-1)*(3*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Nov 24 2014

for(n=1, 50, print1((2^n-1)*(3^n-1)/2, ", ")) \\ G. C. Greubel, Apr 30 2018

a(n) = A059387(n)/2. - Vladimir Joseph Stephan Orlovsky, Apr 28 2010
a(n) = 12*a(n-1)-47*a(n-2)+72*a(n-3)-36*a(n-4). - Colin Barker, Nov 24 2014
G.f.: -x*(6*x^2-1) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Nov 24 2014
a(n) = A000203(A000400(n-1)). - Michel Marcus, Sep 18 2018

More terms from Vladimir Joseph Stephan Orlovsky, Apr 28 2010
More terms from Colin Barker, Nov 24 2014
Better definition from Altug Alkan, Oct 06 2015

A175434 (Digit sum of 2^n) mod n.

Original entry on oeis.org

0, 0, 2, 3, 0, 4, 4, 5, 8, 7, 3, 7, 7, 8, 11, 9, 14, 1, 10, 11, 5, 3, 18, 13, 4, 14, 8, 15, 12, 7, 16, 26, 29, 27, 24, 28, 19, 29, 32, 21, 9, 4, 13, 14, 17, 24, 21, 25, 16, 26, 29, 27, 24, 28, 37, 29, 23, 12, 18, 22, 13, 23, 26, 24, 21, 43, 43, 35, 20, 0, 15, 37, 37, 56, 50, 30, 27, 22, 31, 32, 26, 42, 39, 34, 43, 26, 20, 27, 24, 28, 55, 47, 32, 57, 45, 31, 40, 14, 8, 15

Offset: 1

N. J. A. Sloane, Dec 03 2010

			For n = 1,2,3,4,5,6, the digit-sum of 2^n is 2,4,8,7,5,10, so
a(1) through a(6) are 0,0,2,3,0,4. - _N. J. A. Sloane_, Aug 12 2014

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Table[Mod[Total[IntegerDigits[2^n]],n],{n,100}] (* Harvey P. Dale, Aug 12 2014 *)

Offset changed to 1 at the suggestion of Harvey P. Dale, Aug 12 2014

A240840 Floor(6^n/(1+1/(2cos(5Pi/11)))^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 17, 22, 30, 40, 53, 71, 95, 126, 168, 223, 297, 395, 525, 698, 928, 1234, 1640, 2180, 2899, 3854, 5123, 6811, 9055, 12038, 16003, 21275, 28282, 37599, 49984, 66448, 88336, 117433, 156115

Offset: 0

Kival Ngaokrajang, Apr 13 2014

nonn

a(n) is the perimeter (rounded down) of a hendecaflake after n iterations, let a(0) = 1. The total number of sides is 11*A000400(n). The total number of holes is A016123(n), n >=1. 1/(2*cos(5*Pi/11)) = A231186.

Kival Ngaokrajang, Illustration of hendecaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A016123, A231186, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake), A240841 (tridecaflake).

A240840:=n->floor(6^n/(1+1/(2*cos(5*Pi/11)))^n); seq(A240840(n), n=0..50); # Wesley Ivan Hurt, Apr 13 2014

Table[Floor[6^n/(1 + 1/(2*Cos[5*Pi/11]))^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 13 2014 *)

{a(n)=floor(6^n/(1+1/(2*cos(5*Pi/11)))^n)}
       for (n=0, 100, print1(a(n), ", "))

cp's OEIS Frontend

A367247 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 6.

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A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7.

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A047852 a(n) = A047848(4, n).

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A056547 a(n) = 6*n*a(n-1) + 1 with a(0)=1.

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A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+3*x^2)^n.

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A100309 Modulo 2 binomial transform of 6^n.

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A109354 a(n) = 6^((n^2 - n)/2).

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A056547 a(n) = 6na(n-1) + 1 with a(0)=1.

A084608 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+3x^2)^n.

A240840 Floor(6^n/(1+1/(2cos(5Pi/11)))^n).