A062107 -id:A062107 - cp's OEIS frontend

A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

1, 3, 11, 33, 101, 303, 911, 2733, 8201, 24603, 73811, 221433, 664301, 1992903, 5978711, 17936133, 53808401, 161425203, 484275611, 1452826833, 4358480501, 13075441503, 39226324511, 117678973533, 353036920601, 1059110761803

Offset: 0

Klaus Brockhaus, Mar 17 2003

The limit of the local minima, lim_{n->infinity} A081249(n)/n^2 = 1/10. For local maxima cf. A081251.

			11 is a term since A081249(10)/10^2 = 11/100 = 0.110, A081249(11)/11^2 = 13/121 = 0.107, A081249(12)/12^2 = 16/144 = 0.111.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A005052, A062107, A081134, A081249, A081251.

List([0..30], n-> (5*3^n +(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

[Floor(3^n*5/4): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+2 od: seq(a[n], n=0..30); # Zerinvary Lajos, Apr 28 2008

Floor[5*3^Range[0, 30]/4] (* Wesley Ivan Hurt, Mar 30 2017 *)

vector(30, n, n--; (5*3^n +(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

[(5*3^n +(-1)^n -2)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = floor(3^n*5/4).
G.f.: x*(1+x^2)/((1-x)*(1+x)*(1-3*x)).
a(n) = 3*a(n-1) + 1*a(n-2) - 3*a(n-3).
a(n) = (5*3^n + (-1)^n - 2)/4. - Paul Barry, May 19 2003
a(n) = a(n-2) + 10*3^(n-2) for n > 1.
a(n+2) - a(n) = A005052(n).
a(2*n) = Sum_{j=1..n+1} A062107(2*j).
a(2*n+1) = Sum_{j=1..n+1} A062107(2*j+1).
With a leading 0, this is a(n) = (5*3^n - 6 + 4*0^n - 3*(-1)^n)/12, the binomial transform of A084183. - Paul Barry, May 19 2003
Convolution of 3^n and {1, 0, 2, 0, 2, 0, ...}. a(n) = Sum_{k=0..n} ((1 + (-1)^k) - 0^k)*3^(n-k) = Sum_{k=0..n} ((1 + (-1)^(n-k)) - 0^(n-k))3^k. - Paul Barry, Jul 19 2004
a(n) = 2*a(n-1) + 3*a(n-2) + 2, a(0)=1, a(1)=3. - Zerinvary Lajos, Apr 28 2008

Offset changed from 1 to 0 by Vincenzo Librandi, Jun 10 2011

A005052 a(n) = 10*3^n.

Original entry on oeis.org

10, 30, 90, 270, 810, 2430, 7290, 21870, 65610, 196830, 590490, 1771470, 5314410, 15943230, 47829690, 143489070, 430467210, 1291401630, 3874204890, 11622614670, 34867844010, 104603532030, 313810596090, 941431788270, 2824295364810, 8472886094430

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3).

Except for initial terms, same as A062107.

[10*3^n: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

10*3^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

a(n)=10*3^n \\ Charles R Greathouse IV, Oct 07 2015

From Elmo R. Oliveira, Aug 14 2024: (Start)
G.f.: 10/(1-3*x).
E.g.f.: 10*exp(3*x).
a(n) = 3*a(n-1), n > 0. (End)

A062104 Square array read by antidiagonals: number of ways a black pawn (starting at any square on the second rank) can (theoretically) end at various squares on an infinite chessboard.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 3, 9, 15, 0, 1, 3, 10, 25, 40, 0, 1, 3, 10, 29, 69, 109, 0, 1, 3, 10, 30, 84, 193, 302, 0, 1, 3, 10, 30, 89, 242, 544, 846, 0, 1, 3, 10, 30, 90, 263, 698, 1544, 2390, 0, 1, 3, 10, 30, 90, 269, 774, 2016, 4406, 6796, 0, 1, 3, 10, 30, 90, 270

Offset: 0

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board.

			Array begins:
0       0       0       0       0       0       0       0       0       0       0       0 ...
1       1       1       1       1       1       1       1       1       1       1 ...
2       3       3       3       3       3       3       3       3       3 ...
6       9       10      10      10      10      10      10      10 ...
15      25      29      30      30      30      30      30 ...
40      69      84      89      90      90      90 ...
109     193     242     263     269     270 ...
302     544     698     774 ...
846     1544    2016 ...
2390    4406 ...
6796 ...

A062106 gives the left column and A062107 the diagonal of the table. A062105 is a more regular variant. Cf. also A062103. Trinv given at A054425.

[seq(CPTSeq(j),j=0..91)]; CPTSeq := n -> ChessPawnTriangle( (1+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
ChessPawnTriangle := proc(r,c) option remember; if(r < 2) then RETURN(0); fi; if(c < 1) then RETURN(0); fi; if(2 = r) then RETURN(1); fi; if(4 = r) then RETURN(1+ChessPawnTriangle(r-1,c-1)+ChessPawnTriangle(r-1,c)+ChessPawnTriangle(r-1,c+1));
else RETURN(ChessPawnTriangle(r-1,c-1)+ChessPawnTriangle(r-1,c)+ChessPawnTriangle(r-1,c+1)); fi; end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
CPTSeq[n_] := ChessPawnTriangle[(1 + (n - ((trinv[n]*(trinv[n] - 1))/2))), ((((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1)];
ChessPawnTriangle[r_, c_] := ChessPawnTriangle[r, c] = Which[r < 2, 0, c < 1, 0, 2 == r, 1, 4 == r, 1 + ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1], True, ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1]];
Table[CPTSeq[j], {j, 0, 91}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

Edited by N. J. A. Sloane, May 22 2014

A172447 a(n) = (-1 + 52^(2n + 1) - 3*n)/9.

Original entry on oeis.org

1, 4, 17, 70, 283, 1136, 4549, 18202, 72815, 291268, 1165081, 4660334, 18641347, 74565400, 298261613, 1193046466, 4772185879, 19088743532, 76354974145, 305419896598, 1221679586411, 4886718345664, 19546873382677, 78187493530730, 312749974122943, 1250999896491796

Offset: 0

Paul Curtz, Feb 03 2010

a(n) mod 10 gives the 10-periodic sequence 1, 4, 7, 0, 3, 6, 9, 2, 5, 8 (and repeat, A131579 shifted, A144468 reversed) which contains all ten digits, that has a "palindromic" symmetry: 1 + 8 = 4 + 5 = 7 + 2 = 0 + 9 = 3 + 6 = 9.

The inverse binomial transform gives 1, 3, 10, 30, 90, ... (A062107 shifted). - R. J. Mathar, Feb 11 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A072197 (first differences).

[(-1+5*2^(2*n+1)-3*n)/9: n in [0..30]]; // Vincenzo Librandi, Aug 05 2011

LinearRecurrence[{6, -9, 4}, {1, 4, 17}, 30] (* Harvey P. Dale, Mar 25 2016 *)
((-1 + 5 2^(2# + 1) - 3#)/9  &) /@ Range[0, 29] (* Alonso del Arte, Apr 25 2020 *)

a(n)=(10*4^n-3*n)\9 \\ Charles R Greathouse IV, Jul 21 2015

val powerOf2: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 2)
(0 to 29).map(n => (-1 + 5 * powerOf2(2 * n + 1) - 3 * n)/9) // Alonso del Arte, Apr 25 2020

a(n) = 6*a(n - 1) - 9*a(n - 2) + 4*a(n - 3).
a(n + 1) - 4*a(n) = n.
a(n) = A172416(2n + 1).
G.f.: (1 - 2*x + 2*x^2)/((1 - 4*x) * (x - 1)^2). - R. J. Mathar, Feb 11 2010
E.g.f.: (10*exp(4*x) - (1 + 3*x)*exp(x))/9. - G. C. Greubel, Nov 02 2018

Definition replaced by closed formula by R. J. Mathar, Feb 11 2010

A062106 Number of ways a black pawn (from any starting square on the second back rank) can (theoretically) end on the n-th square of the leftmost file counted from the back rank.

Original entry on oeis.org

0, 1, 2, 6, 15, 40, 109, 302, 846, 2390, 6796, 19426, 55767, 160668, 464305, 1345282, 3906701, 11367696, 33135987, 96740610, 282831981, 827939880, 2426431239, 7118546874, 20904025380, 61439768166, 180725813478, 532004277518

Offset: 1

Antti Karttunen, May 30 2001

nonn

The left column of table A062104. Cf. also A062107.

Maple
```
[seq(ChessPawnTriangle(j,1),j=1..40)];
```

cp's OEIS Frontend

A081250 Numbers k such that A081249(m)/m^2 has a local minimum for m = k.

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A005052 a(n) = 10*3^n.

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A062104 Square array read by antidiagonals: number of ways a black pawn (starting at any square on the second rank) can (theoretically) end at various squares on an infinite chessboard.

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Maple

Mathematica

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A172447 a(n) = (-1 + 5*2^(2*n + 1) - 3*n)/9.

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Magma

Mathematica

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A062106 Number of ways a black pawn (from any starting square on the second back rank) can (theoretically) end on the n-th square of the leftmost file counted from the back rank.

Views

Author

Keywords

Crossrefs

Programs

Maple

A172447 a(n) = (-1 + 52^(2n + 1) - 3*n)/9.