A008600 -id:A008600 - cp's OEIS frontend

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).

0, 0, 2, 0, 6, 4, 0, 10, 12, 6, 0, 14, 20, 18, 8, 0, 18, 28, 30, 24, 10, 0, 22, 36, 42, 40, 30, 12, 0, 26, 44, 54, 56, 50, 36, 14, 0, 30, 52, 66, 72, 70, 60, 42, 16, 0, 34, 60, 78, 88, 90, 84, 70, 48, 18, 0, 38, 68, 90, 104, 110, 108, 98, 80, 54, 20, 0, 42, 76, 102

Offset: 0

Philippe Deléham, Dec 05 2013

Row sums are A006331(n).
Diagonal sums are A212964(n+1).
T(2n,n)=A002943(n).

			Triangle begins:
  0
  0, 2
  0, 6, 4
  0, 10, 12, 6
  0, 14, 20, 18, 8
  0, 18, 28, 30, 24, 10

Cf. columns: A000004, A016825, A017113, A017593, A051062, 10*A005408, A073762.

Cf. diagonals: A005843, A008588, A008592, A008596, A008600, A008604.

T(n+k,k) = A005843(k)*A005408(n).

Sum_{k=0..n} T(n,k) = n*(n+1)*(2*n+1)/3 = A006331(n).

A237274 a(n) = A236283(n) mod 9.

Original entry on oeis.org

2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1, 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1

Offset: 0

Paul Curtz, Feb 05 2014

nonn

(Conjecture) This has period 18: repeat 2, 1, 4, 5, 1, 4, 2, 7, 7, 5, 7, 7, 2, 4, 1, 5, 4, 1.
The first 19 terms and the following 17 are palindromes.
The sorted terms in the conjectured period are 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 5, 5, 5, 7, 7, 7, 7.
Via the extended differences of A236283(n+1) and A236283(n+18) - A236283(n) which is A008600(n+9)=162, 180,... ,it is easy to see that A236283(0)=2.
A236283(-n) = A236283(n).
A236283(n) difference table:
2,  1,  4,  5, 10, 13, 20,  25, 34, 41,...
-1, 3,  1,  5,  3,  7,  5,   9,  7, 11,... = A097062(n+1)
4, -2,  4, -2,  4, -2,  4,  -2,  4, -2,...
-6, 6, -6,  6, -6,  6, -6,   6, -6,  6,... .
A097062(n+1) mod 9 = (a(n+1) -a(n)) mod 9 =
period 18: repeat 8, 3, 1, 5, 3, 7, 5, 0, 7, 2, 0, 4, 2, 6, 4, 8, 6, 1 =b(n).    b(n) + b(18-n)= 9, 9, 9, 9, 9, 9, 9, 0, 9.
Ordered b(n)=
period 18: repeat 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8.

a(n) = A236283(n) mod 9.

A240022 Total number of digits in palindromes with n digits.

Original entry on oeis.org

10, 18, 270, 360, 4500, 5400, 63000, 72000, 810000, 900000, 9900000, 10800000, 117000000, 126000000, 1350000000, 1440000000, 15300000000, 16200000000, 171000000000, 180000000000, 1890000000000, 1980000000000, 20700000000000, 21600000000000

Offset: 1

Arkadiusz Wesolowski, Mar 30 2014

Let f(1) = g(1) = 10 and f(2) = 1; d(n) denotes the number of digits in f(n) and for n >= 3, f(n) = 10*f(n-1) + 5*10^(d(n-1)-1) if n is odd, otherwise f(n) = f(n-1) + 10^(d(n-1)-1)/2. Let g(n) = 18*f(n) for n > 1. It gives g(2) = 18, g(3) = 270, g(4) = 360, g(5) = 4500, .... In fact g(n) produces a different sequence than a(n).

			There are nine 2-digit palindromes, so a(2) = 2*9 = 18.

Index entries for sequences related to palindromes
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-100).

Cf. A008600, A070252.

print1("10, 18, "); m=9; for(n=3, 24, if(bitand(n, 1), m=10*m); print1(m*n, ", "));

Vec(2*x*(50*x^4+35*x^2+9*x+5)/(10*x^2-1)^2 + O(x^100)) \\ Colin Barker, Mar 31 2014

a(n) = n*A070252(n).

a(n) = 20*a(n-2)-100*a(n-4) for n>5. G.f.: 2*x*(50*x^4+35*x^2+9*x+5) / (10*x^2-1)^2. - Colin Barker, Mar 31 2014

A242570 a(n) = 252 * n.

Original entry on oeis.org

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

Offset: 0

Derek Orr, May 17 2014

As lcm(1,2,3,...,9) = 2520, 10*a(n) + k is divisible by each k from 1 through 9.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

Cf. A044102, A135628.

252*Range[0, 49] (* Alonso del Arte, May 17 2014 *)
LinearRecurrence[{2,-1},{0,252},50] (* Harvey P. Dale, Mar 25 2025 *)

PARI
```
for(n=0,50,print(252*n))
```

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 252*x/(x-1)^2.
E.g.f.: 252*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

A269222 Period 4: repeat [1,9,8,9].

Original entry on oeis.org

1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9

Offset: 1

Peter M. Chema, Jul 11 2016

Digital root of Fib(18*n).

Decimal expansion of 221/1111.

			For n=2, a(2) = digital root of Fibonacci(18*2) or 14930352; therefore, a(2) = 9, since the digital root of 14930352 = 9.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A008600, A010888.

Table[NestWhile[Total@ IntegerDigits@ # &, Fibonacci[18 n], IntegerLength@ # > 1 &], {n, 120}] (* Michael De Vlieger, Jul 11 2016 *)
PadRight[{},100,{1,9,8,9}] (* Harvey P. Dale, Sep 24 2021 *)

a(n)=[9,1,9,8][n%4+1] \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A010888(Fibonacci(18*n)).
From Wesley Ivan Hurt, Sep 03 2022: (Start)
a(n) = a(n-4) for n >= 5.
a(n) = (9/4)*(3+(-1)^n)-7*sin(n*Pi/2)/2. (End)

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A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2*k*(2*n+1).

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A237274 a(n) = A236283(n) mod 9.

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A240022 Total number of digits in palindromes with n digits.

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A242570 a(n) = 252 * n.

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A269222 Period 4: repeat [1,9,8,9].

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A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).