A164123 -id:A164123 - cp's OEIS frontend

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 1

Klaus Brockhaus, Jul 03 2009, Jul 05 2009

Interleaving of A000244 and 3*A000244.
Unsigned version of A128019.
Partial sums are in A164123.
Apparently a(n) = A056449(n-1) for n > 1. a(n) = A108411(n) for n >= 1.
Binomial transform is A026150 without initial 1, second binomial transform is A001834, third binomial transform is A030192, fourth binomial transform is A161728, fifth binomial transform is A162272.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000244 (powers of 3), A128019 (expansion of (1-3x)/(1+3x^2)), A164123, A026150, A001834, A030192, A161728, A162272.

Essentially the same as A056449 (3^floor((n+1)/2)) and A108411 (powers of 3 repeated).

[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..35] ];

CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]] (* or *) With[{c= 3^Range[20]},Join[{1},Riffle[c,c]]](* Harvey P. Dale, Feb 17 2012 *)

a(n)=3^(n>>1) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = 3^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 3*x)/(1 - 3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

G.f. corrected, formula simplified, comments added by Klaus Brockhaus, Sep 18 2009

A331787 T(b,n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 0, 2, 4, 14, 0, 1, 2, 7, 30, 0, 2, 4, 6, 16, 62, 0, 1, 4, 3, 14, 25, 126, 0, 2, 2, 6, 8, 14, 52, 254, 0, 1, 4, 5, 4, 13, 30, 79, 510, 0, 2, 4, 6, 8, 10, 28, 62, 160, 1022, 0, 1, 2, 3, 8, 5, 22, 23, 62, 241, 2046, 0, 2, 4, 6, 8, 10, 12, 34, 48, 126, 484, 4094

Offset: 2

Jianing Song, Jan 25 2020

Write n = (b-1)*s + t, 1 <= t <= b-1. The smallest N_0 such that none of S(N_0), S(N_0+1), ..., S(N_0+m-1) is divisible by n is given by N_0 = b^(u_0) - b^s*(t-gcd(t,b-1)+1) + 1, where u_0 is the smallest nonnegative solution to (b-1)*u == -gcd(t,b-1) (mod n). See my link below for more detailed information.

			Table begins
  b\n  1  2  3   4   5   6    7    8    9    10
   2   0  2  6  14  30  62  126  254  510  1022
   3   0  1  4   7  16  25   52   79  160   241
   4   0  2  2   6  14  14   30   62   62   126
   5   0  1  4   3   8  13   28   23   48    73
   6   0  2  4   6   4  10   22   34   46    34
   7   0  1  2   5   8   5   12   19   26    47
   8   0  2  4   6   8  10    6   14   30    46
   9   0  1  4   3   8   9   12    7   16    25
  10   0  2  2   6   8   8   12   14    8    18

Jianing Song, Table of n, a(n) for n = 2..7627 (Note: T(b,n) occurs at the ((n+b-2)*(n+b-1)/2-b+3)-th place)
Jianing Song, Proof of the formula, and the examples N_0 such that none of S(N_0), S(N_0+1), ..., S(N_0+m-1) is divisible by n.

Cf. A331789.

Cf. A000918 (row 2), A164123 (row 3), A331786 (row 10).

T(b,n) = my(s=(n-1)\(b-1), t=(n-1)%(b-1)+1); b^s*(2*t-gcd(t,b-1)+1)-2

If n = (b-1)*s + t, 1 <= t <= b-1, then T(b,n) = b^s*(2*t-gcd(t,b-1)+1) - 2. See my link for a proof of the formula.
T(b,n) = T(b,n-1) + b*T(b,n-b+1) - b*T(b,n-b) for b >= 2, n >= b+1.
T(b,n) = O(b^(n/(b-1))).

A164560 Partial sums of A164532.

Original entry on oeis.org

1, 5, 11, 35, 71, 215, 431, 1295, 2591, 7775, 15551, 46655, 93311, 279935, 559871, 1679615, 3359231, 10077695, 20155391, 60466175, 120932351, 362797055, 725594111, 2176782335, 4353564671, 13060694015, 26121388031, 78364164095

Offset: 1

Klaus Brockhaus, Aug 16 2009

Interleaving of A164559 and A024062 without initial term 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Cf. A164532, A164123 (partial sums of A162436), A164559 (6^n/3-1), A024062 (6^n-1), A026549.

T:=[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..28] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

a(n) = 6*a(n-2)+5 for n > 2; a(1) = 1, a(2) = 5.
a(n) = (3-(-1)^n)*6^(1/4*(2*n-1+(-1)^n))/2-1.
G.f.: x*(1+4*x)/((1-x)*(1-6*x^2)).
a(n) = A026549(n) - 1.

A164265 Partial sums of A162766.

Original entry on oeis.org

4, 7, 19, 28, 64, 91, 199, 280, 604, 847, 1819, 2548, 5464, 7651, 16399, 22960, 49204, 68887, 147619, 206668, 442864, 620011, 1328599, 1860040, 3985804, 5580127, 11957419, 16740388, 35872264, 50221171, 107616799, 150663520, 322850404

Offset: 1

Klaus Brockhaus, Aug 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A162766, A164123 (partial sums of A162436).

T:=[ n le 2 select 5-n else 3*Self(n-2): n in [1..33] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

LinearRecurrence[{1,3,-3},{4,7,19},40] (* Harvey P. Dale, Aug 28 2016 *)

a(n)=if(n%2,15,7)*3^(n\2)\2-3 \\ Charles R Greathouse IV, Jul 15 2011

a(n) = 3*a(n-2)+7 for n > 2; a(1) = 4, a(2) = 7.
a(n) = (11-4*(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2-7/2.
G.f.: x*(4+3*x)/((1-x)*(1-3*x^2)).

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627

Offset: 0

G. C. Greubel, Oct 31 2022

Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A052993, A062318, A164123, A254006.

I:=[3,6,11]; [1,2] cat [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..60]];

LinearRecurrence[{1,3,-3}, {1,2,3,6,11}, 61]

def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def A358027(n): return (1/3)*( 4*A254006(n) + 7*A254006(n-1) +2*int(n==0) + 2*int(n==1) - 3 )
[A358027(n) for n in (0..60)]

a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1)).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.
E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).
G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022

cp's OEIS Frontend

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

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A331787 T(b,n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending antidiagonals.

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A164560 Partial sums of A164532.

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A164265 Partial sums of A162766.

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A358027 Expansion of g.f.: (1 + x - 2*x^2 + 2*x^4)/((1-x)*(1-3*x^2)).

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A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).