A001019 -id:A001019 - cp's OEIS frontend

A223576 T(n,k)=Number of nXk 0..2 arrays with antidiagonals unimodal.

3, 9, 9, 27, 81, 27, 81, 729, 729, 81, 243, 6561, 16038, 6561, 243, 729, 59049, 352836, 352836, 59049, 729, 2187, 531441, 7762392, 16230456, 7762392, 531441, 2187, 6561, 4782969, 170772624, 746600976, 746600976, 170772624, 4782969, 6561, 19683

Offset: 1

R. H. Hardin Mar 22 2013

Table starts
.....3..........9.............27.................81...................243
.....9.........81............729...............6561.................59049
....27........729..........16038.............352836...............7762392
....81.......6561.........352836...........16230456.............746600976
...243......59049........7762392..........746600976...........64207683936
...729.....531441......170772624........34343644896.........5521860818496
..2187....4782969.....3756997728......1579807665216.......474880030390656
..6561...43046721....82653950016.....72671152599936.....40839682613596416
.19683..387420489..1818386900352...3342873019597056...3512212704769291776
.59049.3486784401.40004511807744.153772158901464576.302050292610159092736

			Some solutions for n=3 k=4
..0..1..2..2....0..2..0..1....0..1..2..2....0..0..1..2....0..1..0..2
..0..1..2..2....1..1..2..0....1..1..0..1....1..2..1..2....0..2..1..0
..1..2..1..1....1..2..0..1....1..0..2..2....1..0..2..0....0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1000

Column 1 is A000244

Column 2 is A001019

Let U(z) = (z^4+6*z^3+23*z^2+18*z+24)/24

T(n,k) = U(min(n,k))^(max(n,k)-min(n,k)+1) * product{ U(i)^2 , i=1..(min(n,k)-1) }

A239014 Exponents m such that the decimal expansion of 9^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 12, 13, 14, 17, 26, 34, 68, 406, 926, 2227, 3379, 3824, 26364, 197564, 9669757, 11470439, 15754533, 18945654, 25742286, 38175282, 237545304, 320907073, 2928221215, 3653563322, 5788579994, 25722005323, 30228962873, 137527721034, 217558664165, 523648850797

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030705.
Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.
Not just two time A001019.

Except for its second term, A030705 is a subsequence.

Cf. A001019, A020665, A031142, A239008, A239009, A239010, A239011, A239012, A239013, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[9, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 10000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(27)-a(31) from Bert Dobbelaere, Jan 21 2019

a(32)-a(36) from Chai Wah Wu, Jan 13 2020

A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

Original entry on oeis.org

0, 3, 79, 1323, 18175, 223323, 2555119, 27828363, 292407775, 2990349243, 29943991759, 294872615403, 2864776362175, 27525734996763, 262061152909999, 2475899571994443, 23240879960425375, 216963121865909883, 2015960236625789839, 18656492902684557483, 172056837889322101375

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366965.

Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A366965.
Cf. A000420, A001018, A001019.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367250.

LinearRecurrence[{24,-191,504},{0,3,79},21]

a(n) = 17*9^(n-1) - 31*8^(n-1) + 2*7^n.
a(n) = 24*a(n-1) - 191*a(n-2) + 504*a(n-3) for n > 3.
O.g.f.: x^2*(3 + 7*x)/((1 - 7*x)*(1 - 8*x)*(1 - 9*x)).
E.g.f.: (136*exp(9*x) - 279*exp(8*x) + 144*exp(7*x) - 1)/72.

A367250 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 9.

Original entry on oeis.org

0, 1, 35, 703, 11231, 158311, 2062655, 25466743, 302423471, 3487593511, 39314599775, 435241463383, 4748453693711, 51186327429511, 546278900354495, 5781325731101623, 60750456603203951, 634502309615150311, 6592506388026870815, 68188442304165981463, 702543059232886986191

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366966.

Index entries for linear recurrences with constant coefficients, signature (27,-242,720).

Cf. A366966.
Cf. A001018, A001019, A011557.
Cf. A367243, A367244, A367245, A367246, A367247, A367248, A367249.

LinearRecurrence[{27,-242,720},{0,1,35},21]

a(n) = 9*10^(n-1) - 17*9^(n-1) + 8^n.
a(n) = 27*a(n-1) - 242*a(n-2) + 720*a(n-3) for n > 3.
O.g.f.: x^2*(1 + 8*x)/((1 - 8*x)*(1 - 9*x)*(1 - 10*x)).
E.g.f.: (81*exp(10*x) - 170*exp(9*x) + 90*exp(8*x) - 1)/90.

A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456, 480, 489, 480, 456

Offset: 0

M. F. Hasler, Jun 17 2012

The n-th row also yields the number of ways to get a total of n, n+1,..., 9n, when summing n integers ranging from 1 to 9.
The row sums equal 9^n = A001019(n).
The row lengths are 1+8n = A017077(n).

			The triangle starts:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1; (row sum = 9, row length = 9)
(row n=2) 1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1; (sum = 81, length = 17)
(row n=3) 1,3,6,10,15,21,28,36,45,52,57,60,61,60,... (sum = 729, length = 25)
(row n=4) 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456,... (sum = 9^4; length = 33),
etc.

Seiichi Manyama, Rows n = 0..49, flattened

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Cf. A001019, A017077.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 9-nomials as a table
r := 9:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do; # Peter Bala, Sep 07 2013

concat(vector(5,k,Vec(sum(j=0,8,x^j)^(k-1))))

T(n,k) = Sum_{i=0..floor(k/9)} (-1)^i*binomial(n,i)*binomial(n+k-1-9*i,n-1) for n >= 0 and 0 <= k <= 8*n. - Peter Bala, Sep 07 2013

A055260 Sums of two powers of 9.

Original entry on oeis.org

2, 10, 18, 82, 90, 162, 730, 738, 810, 1458, 6562, 6570, 6642, 7290, 13122, 59050, 59058, 59130, 59778, 65610, 118098, 531442, 531450, 531522, 532170, 538002, 590490, 1062882, 4782970, 4782978, 4783050, 4783698, 4789530, 4842018, 5314410, 9565938, 43046722

Offset: 0

Henry Bottomley, Jun 22 2000

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001019, A052216.

t = 9^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[9^Range[0,10],2]//Union (* Harvey P. Dale, Jul 03 2019 *)

def valuation(n, b):
  v = 0
  while n > 1: n //= b; v += 1
  return v
def aupto(lim):
  pows = [9**i for i in range(valuation(lim-1, 9) + 1)]
  sum_pows = sorted([a+b for i, a in enumerate(pows) for b in pows[i:]])
  return [s for s in sum_pows if s <= lim]
print(aupto(43046722)) # Michael S. Branicky, Feb 10 2021

from math import isqrt
def A055260(n): return 9**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+9**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

a(n) = 9^(n-trinv(n))+9^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)

Regarded as a triangle T(n, k) = 9^n + 9^k, so as a sequence a(n) = 9^A002262(n) + 9^A003056(n).

A127796 a(n) = nextprime(9^n) - 9^n.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 16, 2, 26, 10, 8, 4, 2, 2, 26, 4, 70, 34, 2, 8, 118, 4, 8, 68, 56, 28, 50, 28, 62, 158, 16, 122, 92, 28, 20, 110, 140, 70, 28, 44, 20, 124, 316, 38, 8, 44, 136, 58, 110, 2, 148, 170, 116, 170, 40, 2, 182, 10, 46, 254, 56, 14, 8, 2, 190, 148, 382, 10, 56, 10

Offset: 0

Artur Jasinski, Jan 29 2007

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A013597, A013598, A013602, A013599, A013600, A013601, A127796, A033873, A127797, A127798, A127799.

Cf. A013632, A001019.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[k = NextPrime[9^x] - 9^x; AppendTo[a, k], {x, 0, 100}]; a

a(n) = A013632(A001019(n)). - Michel Marcus, Nov 18 2019

A211866 (9^n - 5) / 4.

Original entry on oeis.org

1, 19, 181, 1639, 14761, 132859, 1195741, 10761679, 96855121, 871696099, 7845264901, 70607384119, 635466457081, 5719198113739, 51472783023661, 463255047212959, 4169295424916641, 37523658824249779, 337712929418248021, 3039416364764232199, 27354747282878089801

Offset: 1

Reinhard Zumkeller, Feb 12 2013

(2*n, a(n)) are the solutions of Diophantine equation 3^x = 4*y + 5.
Second bisection of A080926. - Bruno Berselli, Feb 12 2013
Sum of n-th row of triangle of powers of 9: 1; 9 1 9; 81 9 1 9 81; 729 81 9 1 9 81 729; ... - Philippe Deléham, Feb 24 2014

			a(1) = 1;
a(2) = 9 + 1 + 9 = 19;
a(3) = 81 + 9 + 1 + 9 + 81 = 181;
a(4) = 729 + 81 + 9 + 1 + 9 + 81 + 729 = 1639; etc. - _Philippe Deléham_, Feb 24 2014

Jiri Herman, Radan Kucera and Jaromir Simsa, Equations and Inequalities, Springer (2000), p. 225 (5.3).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A000792, A001019, A080926, A084222.

a211866 = (flip div 4) . (subtract 5) . (9 ^)

I:=[1,19]; [n le 2 select I[n] else 10*Self(n-1)-9*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 26 2014

A211866:=n->(9^n-5)/4; seq(A211866(n), n=1..50); # Wesley Ivan Hurt, Nov 13 2013

(9^Range[25] - 5)/4 (* Bruno Berselli, Feb 12 2013 *)
CoefficientList[Series[(1 + 9 x)/((1 - x) (1 - 9 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

makelist((9^n-5)/4,n,1,30); /* Martin Ettl, Feb 12 2013 */

a(n)=(9^n-5)/4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.:  x*(1+9*x)/((1-x)*(1-9*x)). - Bruno Berselli, Feb 12 2013
a(n)-a(n-1) = A000792(6n-4). - Bruno Berselli, Feb 12 2013
a(n) = 9*a(n-1) + 10, a(1) = 1. - Philippe Deléham, Feb 24 2014
a(n) = -A084222(2*n). - Philippe Deléham, Feb 24 2014

A217999 T(n,k)=Number of nXk arrays of the minimum or maximum value of corresponding elements and their horizontal and vertical neighbors in a random 0..2 nXk array.

Original entry on oeis.org

3, 9, 9, 25, 81, 25, 81, 729, 729, 81, 241, 6561, 19665, 6561, 241, 719, 59049, 531143, 531143, 59049, 719, 2181, 531441, 14345273, 43046713, 14345273, 531441, 2181, 6543, 4782969, 387332417, 3486751263, 3486751263, 387332417, 4782969, 6543

Offset: 1

R. H. Hardin Oct 17 2012

Table starts
.....3........9..........25...........81..........241..........719
.....9.......81.........729.........6561........59049.......531441
....25......729.......19665.......531143.....14345273....387332417
....81.....6561......531143.....43046713...3486751263.282429513233
...241....59049....14345273...3486751263.847279307713
...719...531441...387332417.282429513233
..2181..4782969.10458051407
..6543.43046721
.19623

			Some solutions for n=3 k=4
..0..2..2..0....0..2..2..2....0..2..1..1....0..0..0..2....0..1..0..2
..1..0..2..0....0..2..0..0....1..2..0..1....0..1..1..0....0..0..1..2
..1..0..0..0....0..2..2..0....1..1..0..1....2..2..0..0....2..0..2..1

R. H. Hardin, Table of n, a(n) for n = 1..49

Column 2 is 9^n, A001019

A308656 Number of ways to write n as (2^a9^b)^2 + c(2c+1) + d*(3d+1), where a and b are nonnegative integers, and c and d are integers.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 3, 2, 3, 1, 4, 2, 1, 4, 3, 4, 3, 5, 4, 3, 6, 2, 2, 4, 3, 6, 2, 4, 5, 3, 6, 4, 4, 4, 4, 4, 4, 1, 4, 5, 5, 2, 3, 3, 2, 8, 3, 4, 5, 3, 5, 3, 3, 5, 3, 7, 1, 3, 5, 4, 6, 3, 6, 2, 2, 6, 5, 4, 6, 6, 7, 3, 4, 9, 5, 4, 5, 3, 4, 4, 11, 5, 5, 12, 5, 7, 5, 4, 10, 2, 7, 8, 4, 8, 7, 12, 5, 5, 5, 5

Offset: 1

Zhi-Wei Sun, Jun 14 2019

nonn

Note that {x*(2x+1): x is an integer} = {n*(n+1)/2: n = 0,1,2,...}.
Conjecture 1: a(n) > 0 for all n > 0.
Conjecture 2: If f(x) is one of the polynomials x*(4x+1), x*(5x+2), x*(5x+4), x*(7x+3)/2 and x(7x+5)/2, then any positive integer n can be written as (2^a*9^b)^2 + f(c) + d*(3d+1)/2, where a and b are nonnegative integers, and c and d are integers.
Conjecture 3: Let r be 1 or 2. Then any positive integer n can be written as (2^a*7^b)^2 + c*(2c+1) + d*(3d+r), where a and b are nonnegative integers, and c and d are integers.
Conjecture 4: If g(x) is one of the polynomials x*(x+1), x*(4x+3), x*(7x+1)/2, x*(7x+3)/2 and x*(7x+5)/2, then any positive integer n can be written as (2^a*7^b)^2 + g(c) + d*(3d+1)/2, where a and b are nonnegative integers, and c and d are integers.
We have verified a(n) > 0 for all n = 1..10^8, and Conjectures 2-4 for all n = 1..10^6.
See also A308640, A308641, and A308644 for similar conjectures.
Jiao-Min Lin (a student at Nanjing University) has found a counterexample to Conjecture 1: a(2109982225) = 0. - Zhi-Wei Sun, Jul 30 2022

			a(13) = 1 with 13 = (2^0*9^0)^2 + 2*(2*2+1) + (-1)*(3*(-1)+1).
a(3515) = 1 with 3515 = (2^0*9^1)^2 + 0*(2*0+1) + (-34)*(3*(-34)+1).
a(124076) = 1 with 124076 = (2^3*9^1)^2 + 206*(2*206+1) + 106*(3*106+1).
a(141518) = 1 with 141518 = (2^1*9^2)^2 + (-188)*(2*(-188)+1) + 122*(3*122+1).
a(345402) = 1 with 345402 = (2^7*9^0)^2 + 18*(2*18+1) + (-331)*(3*(-331)+1).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

A000079, A000217, A000420, A001019, A001318, A308566, A308584, A308621, A308623, A308640, A308641, A308644.

PQ[n_]:=PQ[n]=IntegerQ[Sqrt[12n+1]];
tab={};Do[r=0;Do[If[PQ[n-81^a*4^b-x(2x+1)],r=r+1],{a,0,Log[81,n]},{b,0,Log[4,n/81^a]},{x,-Floor[(Sqrt[8(n-81^a*4^b)+1]+1)/4],(Sqrt[8(n-81^a*4^b)+1]-1)/4}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A223576 T(n,k)=Number of nXk 0..2 arrays with antidiagonals unimodal.

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A239014 Exponents m such that the decimal expansion of 9^m exhibits its first zero from the right later than any previous exponent.

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A367249 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 8.

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A367250 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 9.

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A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

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A055260 Sums of two powers of 9.

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A127796 a(n) = nextprime(9^n) - 9^n.

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A211866 (9^n - 5) / 4.

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A308656 Number of ways to write n as (2^a9^b)^2 + c(2c+1) + d*(3d+1), where a and b are nonnegative integers, and c and d are integers.