A062318 -id:A062318 - cp's OEIS frontend

A181287 Numbers of the form i*5^j-1 (i=1..4, j >= 0).

0, 1, 2, 3, 4, 9, 14, 19, 24, 49, 74, 99, 124, 249, 374, 499, 624, 1249, 1874, 2499, 3124, 6249, 9374, 12499, 15624, 31249, 46874, 62499, 78124, 156249, 234374, 312499, 390624, 781249, 1171874, 1562499, 1953124, 3906249, 5859374, 7812499, 9765624, 19531249, 29296874, 39062499, 48828124, 97656249, 146484374, 195312499

Offset: 1

N. J. A. Sloane, Jan 25 2011

Row numbers of Pascal's Triangle where none of the binomial coefficients in that row is divisible by 5. - Thomas M. Green, Apr 02 2013

			For n = 7, a(7) = 14 and the binomial coefficients in the 14th row of Pascal's Triangle are 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 and none of the elements in that row is divisible by 5. - _Thomas M. Green_, Apr 05 2013

Index entries for linear recurrences with constant coefficients, signature (1,0,0,5,-5).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), this sequence (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

a(n) = a(n-1)+5*a(n-4)-5*a(n-5). G.f.: x^2*(x+1)*(x^2+1) / ((x-1)*(5*x^4-1)). [Colin Barker, Feb 01 2013]

A140576 Numbers of the form i*9^j-1 (i=1..8, j >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 26, 35, 44, 53, 62, 71, 80, 161, 242, 323, 404, 485, 566, 647, 728, 1457, 2186, 2915, 3644, 4373, 5102, 5831, 6560, 13121, 19682, 26243, 32804, 39365, 45926, 52487, 59048, 118097, 177146, 236195, 295244, 354293, 413342, 472391, 531440, 1062881

Offset: 1

N. J. A. Sloane, Jan 25 2011

A base-9 analog of A051885.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,9,-9).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), this sequence (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

G.f.: x^2*(x+1)*(x^2+1)*(x^4+1) / ((x-1)*(3*x^4-1)*(3*x^4+1)). [Colin Barker, Feb 01 2013]

A165804 Numbers of the form i*8^j-1 (i=1..7, j >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 15, 23, 31, 39, 47, 55, 63, 127, 191, 255, 319, 383, 447, 511, 1023, 1535, 2047, 2559, 3071, 3583, 4095, 8191, 12287, 16383, 20479, 24575, 28671, 32767, 65535, 98303, 131071, 163839, 196607, 229375, 262143, 524287, 786431, 1048575, 1310719, 1572863, 1835007, 2097151

Offset: 1

N. J. A. Sloane, Jan 25 2011

Numbers whose sum of digits in base 8 sets a new record. - Harvey P. Dale, Jan 10 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,8,-8).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), this sequence (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

Sort[Flatten[Table[i 8^j-1,{i,1,7},{j,0,7}]]]  (* Harvey P. Dale, Feb 03 2011 *)

G.f.: x^2*(x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)*(8*x^7-1)). [Colin Barker, Feb 01 2013]

A181288 Numbers of the form i*6^j-1 (i=1..5, j >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 11, 17, 23, 29, 35, 71, 107, 143, 179, 215, 431, 647, 863, 1079, 1295, 2591, 3887, 5183, 6479, 7775, 15551, 23327, 31103, 38879, 46655, 93311, 139967, 186623, 233279, 279935, 559871, 839807, 1119743, 1399679, 1679615, 3359231, 5038847, 6718463, 8398079, 10077695, 20155391, 30233087

Offset: 1

N. J. A. Sloane, Jan 25 2011

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), this sequence (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

Union[Flatten[Table[i*6^j-1,{j,0,20},{i,5}]]] (* Harvey P. Dale, Nov 12 2012 *)

G.f.: x^2*(x^4+x^3+x^2+x+1) / ((x-1)*(6*x^5-1)). [Colin Barker, Feb 01 2013]

A181303 Numbers of the form i*7^j-1 (i=1..6, j >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 48, 97, 146, 195, 244, 293, 342, 685, 1028, 1371, 1714, 2057, 2400, 4801, 7202, 9603, 12004, 14405, 16806, 33613, 50420, 67227, 84034, 100841, 117648, 235297, 352946, 470595, 588244, 705893, 823542, 1647085, 2470628, 3294171, 4117714, 4941257

Offset: 1

N. J. A. Sloane, Jan 25 2011

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,7,-7).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), this sequence (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

Sort[Flatten[Table[i 7^j-1,{i,1,6},{j,0,7}]]]  (* Harvey P. Dale, Feb 03 2011 *)

G.f.: x^2*(x+1)*(x^2-x+1)*(x^2+x+1) / ((x-1)*(7*x^6-1)). - Colin Barker, Feb 01 2013

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50

Offset: 2

N. J. A. Sloane, Jun 07 2011

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
G. Royle, Cages of higher valency

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

A164123 Partial sums of A162436.

Original entry on oeis.org

1, 4, 7, 16, 25, 52, 79, 160, 241, 484, 727, 1456, 2185, 4372, 6559, 13120, 19681, 39364, 59047, 118096, 177145, 354292, 531439, 1062880, 1594321, 3188644, 4782967, 9565936, 14348905, 28697812, 43046719, 86093440, 129140161, 258280324, 387420487, 774840976, 1162261465

Offset: 1

Klaus Brockhaus, Aug 10 2009

Interleaving of A058481 and A100774 without initial term 0.
Apparently a(n) = A062318(n+2) - 1.
The terms beginning with a(2) are the row numbers in Pascal's Triangle where every 3rd element in those rows is divisible by 3 and none of the other elements in those rows are divisible by 3. - Thomas M. Green, Apr 03 2013

			For n = 3, a(3) = 7. The binomial coefficients of the 7th row of Pascal's Triangle are 1 7 21 35 35 21 7 1 and every 3rd element is a multiple of 3. - _Thomas M. Green_, Apr 03 2013

Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A162436, A058481 (3^n-2), A100774 (2*(3^n - 1)), A062318, A038754, A038754.

T:=[ n le 2 select 2*n-1 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]];

Accumulate[Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]]] (* Harvey P. Dale, Feb 17 2012 *)

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

a(n) = A038754(n+1) - 2.
a(n) = 3*a(n-2) + 4 for n > 2; a(1) = 1, a(2) = 4.
a(n) = (5 - (-1)^n)*3^(1/4*(2*n - 1 + (-1)^n))/2 - 2.
G.f.: x*(1 + 3*x)/((1 - x)*(1 - 3*x^2)).
E.g.f.: 2*(cosh(sqrt(3)*x) - cosh(x)) + sqrt(3)*sinh(sqrt(3)*x) - 2*sinh(x). - Stefano Spezia, Dec 31 2022

Incorrect formula removed by Stefano Spezia, Dec 31 2022

A052993 a(n) = a(n-1) + 3a(n-2) - 3a(n-3), with a(0)=a(1)=1, a(2)=4.

Original entry on oeis.org

1, 1, 4, 4, 13, 13, 40, 40, 121, 121, 364, 364, 1093, 1093, 3280, 3280, 9841, 9841, 29524, 29524, 88573, 88573, 265720, 265720, 797161, 797161, 2391484, 2391484, 7174453, 7174453, 21523360, 21523360, 64570081, 64570081, 193710244

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1069
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A062318.

I:=[1,1,4]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 21 2018

spec := [S,{S=Prod(Sequence(Prod(Union(Z,Z,Z),Z)),Sequence(Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

(3^(1+Floor[(Range@40-1)/2])-1)/2 (* Federico Provvedi, Nov 22 2018 *)
LinearRecurrence[{1,3,-3}, {1,1,4}, 30] (* or *) RecurrenceTable[{a[n + 2] == 3*a[n] + 1, a[0] == 1, a[1] == 1}, a, {n,0,30}] (* G. C. Greubel, Nov 21 2018 *)

x='x+O('x^30); Vec(1/((1-3*x^2)*(1-x))) \\ G. C. Greubel, Nov 21 2018

s=(1/((1-3*x^2)*(1-x))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 21 2018

G.f.: 1/((1-3*x^2)*(1-x)).
a(n+2) = 3*a(n) + 1, where a(0) = a(1) = 1.
a(n) = -1/2 + Sum((1/4)*(1+3*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1 + 3*_Z^2)).
a(n) = Sum{k=0..n} 3^(k/2)*(1-(-1)^k)/(2*sqrt(3)). - Paul Barry, Jul 28 2004
a(n) = (3^(1+floor((n-1)/2)) - 1)/2. - Federico Provvedi, Nov 22 2018
a(n)-a(n-1) = A254006(n). - R. J. Mathar, Feb 27 2019

More terms from James Sellers, Jun 06 2000

A152714 Triangle read by rows: T(n,k) = 3^min(k, n-k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 9, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 27, 9, 3, 1, 1, 3, 9, 27, 27, 9, 3, 1, 1, 3, 9, 27, 81, 27, 9, 3, 1, 1, 3, 9, 27, 81, 81, 27, 9, 3, 1, 1, 3, 9, 27, 81, 243, 81, 27, 9, 3, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 11 2008

			Triangle begins
  {1},
  {1, 1},
  {1, 3, 1},
  {1, 3, 3,  1},
  {1, 3, 9,  3,  1},
  {1, 3, 9,  9,  3,   1},
  {1, 3, 9, 27,  9,   3,  1},
  {1, 3, 9, 27, 27,   9,  3,  1},
  {1, 3, 9, 27, 81,  27,  9,  3, 1},
  {1, 3, 9, 27, 81,  81, 27,  9, 3, 1},
  {1, 3, 9, 27, 81, 243, 81, 27, 9, 3, 1}

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A004197, A144464, A152716, A152717, A062318 (row sums).

[[3^(Min(k,n-k)): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Sep 01 2018

Clear[a, k, m]; k = 3; a[0] = {1}; a[1] = {1, 1};
a[n_] := a[n] = Join[{1}, k*a[n - 2], {1}];
Table[a[n], {n, 0, 10}];
Flatten[%]
Table[3^(Min[k, n - k]), {n, 0, 100}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2018 *)

for(n=0,15, for(k=0,n, print1(3^(min(k,n-k)), ", "))) \\ G. C. Greubel, Sep 01 2018

T(n,k) = 3^min(k, n-k) = 3^A004197(n,k). - Philippe Deléham, Feb 25 2014

Better name by Philippe Deléham, Feb 25 2014

A060647 Number of alpha-beta evaluations in a tree of depth n and branching factor b=3.

Original entry on oeis.org

1, 3, 5, 11, 17, 35, 53, 107, 161, 323, 485, 971, 1457, 2915, 4373, 8747, 13121, 26243, 39365, 78731, 118097, 236195, 354293, 708587, 1062881, 2125763, 3188645, 6377291, 9565937, 19131875, 28697813, 57395627, 86093441, 172186883, 258280325, 516560651, 774840977

Offset: 0

Frank Ellermann, Apr 17 2001

			a(2n+1) = 2*a(2n) + 1, a(15) = a(2*7+1) = 2*a(14) + 1 = 2*4373 + 1 = 8747.

P. H. Winston, Artificial Intelligence, Addison-Wesley, 1977, pp. 115-122, (alpha-beta technique).

Harry J. Smith, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (1, 3, -3).

For b=2 see A052955.

Cf. A068911.

A060647 := proc(n,b) option remember: if n mod 2 = 0 then RETURN(2*b^(n/2)-1) else RETURN(b^((n-1)/2) +b^((n+1)/2)-1) fi: end: for n from 0 to 60 do printf(`%d,`, A060647(n,3)) od:
a[0]:=1:a[1]:=3:for n from 2 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=0..33); # Zerinvary Lajos, Mar 17 2008

f[n_] := Simplify[Sqrt[3]^n(1 + 2/Sqrt[3]) + (1 - 2/Sqrt[3])(-Sqrt[3])^n - 1]; Table[ f[n], {n, 0, 34}] (* or *)
f[n_] := If[ EvenQ[n], 2(3^(n/2)) - 1, 3^((n - 1)/2) + 3^((n + 1)/2) - 1]; Table[ f[n], {n, 0, 34}] (* or *)
CoefficientList[ Series[(1 + 2x - x^2)/((1 - x)(1 - 3x^2)), {x, 0, 35}], x] (* Robert G. Wilson v, Nov 17 2005 *)

a(n) = { if (n%2==0, 2*(3^(n/2)) - 1, my(m=(n - 1)/2); 3^m + 3^(m + 1) - 1) } \\ Harry J. Smith, Jul 09 2009

a(2n) = 2*(3^n) - 1, a(2n+1) = 3^n + 3^(n+1) - 1.
Formula for b branches: a(2n) = 2*(b^n)-1, a(2n+1) = b^n+b^(n+1)-1.
a(n) = A068911(n+1) - 1.
G.f.: (1+2*z-z^2)/((1-z)*(1-3*z^2)). - Emeric Deutsch, Nov 18 2002
a(n) = (sqrt(3))^n(1+2/sqrt(3))+(1-2/sqrt(3))(-sqrt(3))^n-1. - Paul Barry, Apr 17 2004
a(2n+1) = 3*a(2n-1) + 2;  a(2n) = (a(2n-1) + a(2n+1))/2, with a(1)= 1. See A062318 for case where a(1)= 0.
a(n) = (2^((1+(-1)^n)/2))*(b^((2*n-1+(-1)^n)/4))+((1-(-1)^n)/2)*(b^((2*n+1-(-1)^n)/4))-1, with b=3. - Luce ETIENNE, Aug 30 2014

More terms from James Sellers, Apr 19 2001

cp's OEIS Frontend

A181287 Numbers of the form i*5^j-1 (i=1..4, j >= 0).

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A140576 Numbers of the form i*9^j-1 (i=1..8, j >= 0).

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A165804 Numbers of the form i*8^j-1 (i=1..7, j >= 0).

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A181288 Numbers of the form i*6^j-1 (i=1..5, j >= 0).

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A181303 Numbers of the form i*7^j-1 (i=1..6, j >= 0).

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A191595 Order of smallest n-regular graph of girth 5.

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A164123 Partial sums of A162436.

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Magma

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A052993 a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3), with a(0)=a(1)=1, a(2)=4.

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A052993 a(n) = a(n-1) + 3a(n-2) - 3a(n-3), with a(0)=a(1)=1, a(2)=4.