A065455 -id:A065455 - cp's OEIS frontend

A090992 Number of meaningful differential operations of the n-th order on the space R^7.

7, 13, 24, 45, 84, 158, 296, 557, 1045, 1966, 3691, 6942, 13038, 24516, 46055, 86585, 162680, 305809, 574624, 1080106, 2029680, 3814941, 7169145, 13474502, 25322375, 47592650, 89441626, 168100324, 315917527, 593742597, 1115852904, 2097145317

Offset: 1

Branko Malesevic, Feb 29 2004

Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^6. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

Also (starting 4,7,...) the number of zig-zag paths from top to bottom of a rectangle of width 8, whose color is that of the top right corner. - Joseph Myers, Dec 23 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux DirectionalDerivative, arXiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
2008/9 British Mathematical Olympiad Round 1: Thursday, 4 December 2008, Problem 1 [From _Joseph Myers_, Dec 23 2008]
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Cf. A090989-A090995.
Partial sums of pairwise sums of A065455.
Cf. A000079, A007283, A020701, A020714, A129638.

a:=[7,13,24,45];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2] - 2*a[n-3] - a[n-4]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 7; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

LinearRecurrence[{1, 3, -2, -1}, {7, 13, 24, 45}, 32] (* Jean-François Alcover, Nov 25 2017 *)

my(x='x+O('x^40)); Vec(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))) \\ G. C. Greubel, Feb 02 2019

a=(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

a(n+4) = a(n+3) + 3*a(n+2) - 2*a(n+1) - a(n).

G.f.: x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)). - Colin Barker, Mar 08 2012

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

More terms from Joseph Myers, Dec 23 2008

A065494 Number of (binary) bit strings in which no even length block of 0's is followed by an even length block of 1's.

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 57, 112, 216, 420, 815, 1580, 3069, 5950, 11552, 22408, 43487, 84378, 163725, 317700, 616444, 1196172, 2321007, 4503704, 8738921, 16956954, 32903164, 63845000, 123884479, 240384374, 466440273, 905077080, 1756205088

Offset: 0

Len Smiley, Nov 24 2001

			a(6)=64-7=57 because 000011, 001111, 001100, 001101, 100110, 010011, 110011 are forbidden.

Index entries for linear recurrences with constant coefficients, signature (0,3,2,-1).

Cf. A061279 (forbids odd block 0's - odd block 1's), A065455, A065495, A065497.

O.g.f.: (1+x)^2/(1-3x^2-2x^3+x^4)

A065495 Number of (binary) bit strings of length n in which an odd length block of 0's is followed by an odd length block of 1's.

Original entry on oeis.org

1, 2, 6, 14, 32, 72, 156, 336, 712, 1496, 3120, 6464, 13328, 27360, 55968, 114144, 232192, 471296, 954816, 1931264, 3900800, 7869312, 15858432, 31928832, 64232704, 129128960, 259431936, 520941056, 1045557248, 2097616896

Offset: 2

Len Smiley, Nov 24 2001

			a(4) = 6 because of 0100, 0101, 1010, 1101, 0111, 0001.

Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4).

Cf. A061279 [=2^n - a(n)], A065455, A065494, A065497.

G.f.: x^2/((1-2*x)*(1-2*x^2-2*x^3)).

A065497 Number of (binary) bit strings of length n having at least one even length block of 0's followed by an even length block of 1's.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 7, 16, 40, 92, 209, 468, 1027, 2242, 4832, 10360, 22049, 46694, 98419, 206588, 432132, 900980, 1873297, 3884904, 8038295, 16597478, 34205700, 70372728, 144550977, 296486538, 607301551, 1242406568, 2538762208, 5182207180

Offset: 0

Len Smiley, Nov 24 2001

			a(6)=7 because of 000011, 001100, 001101, 001111, 010011, 100110, 110011.

Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-5,2).

Cf. A065455, A065494, A065495.

CoefficientList[Series[x^4/((1 - 2 x) (1 - 3 x^2 - 2 x^3 + x^4)), {x, 0, 33}], x] (* Georg Fischer, May 15 2019 *)

G.f.: x^4/((1 - 2*x)*(1 - 3*x^2 - 2*x^3 + x^4)). [Corrected by Georg Fischer, May 15 2019]

Offset changed from 4 to 0 by Georg Fischer, May 15 2019

A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 1, 5, 10, 9, 0, 0, 0, 6, 15, 19, 9, 0, 0, 0, 6, 21, 34, 28, 0, 0, 0, 0, 0, 27, 55, 62, 28, 0, 0, 0, 0, 0, 27, 82, 117, 90, 0, 0, 0, 0, 0, 0, 0, 109, 199, 207, 90, 0, 0, 0, 0, 0, 0, 0, 109, 308, 406, 297, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, ... row n=0
1, 2, 3, 4, 5, 6, 6, 0, 0, ... row n=1
1, 3, 6, 10, 15, 21, 27, 27, 0, 0, ... row n=2
0, 3, 9, 19, 34, 55, 82, 109, 109, 0, 0, ... row n=3
0, 0, 9, 28, 62, 117, 199, 308, 417, 417, 0, 0, ... row n=4
0, 0, 0, 28, 90, 207, 406, 714, 1131, 1548, 1548, 0, 0, ... row n=5
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 6
1, 3, 6, 10, 15, 21, 27, 27
3, 9, 19, 34, 55, 82, 109, 109
9, 28, 62, 117, 199, 308, 417, 417
28, 90, 207, 406, 714, 1131, 1548, 1548
90, 297, 703, 1417, 2548, 4096, 5644, 5644
297, 1000, 2417, 4965, 9061, 14705, 20349, 20349
1000, 3417, 8382, 17443, 32148, 52497, 72846, 72846
3417, 11799, 29242, 61390, 113887, 186733, 259579, 259579
11799, 41041, 102431, 216318, 403051, 662630, 922209, 922209
...

Cf. Similar sequences: A216201, A216210, A216216, A216218, ...

T(n,n+4) = T(n,n+5) = A094829(n+2).
T(n,n+3) = A094834(n+1).
T(n,n+2) = A094833(n+1).
T(n,n+1) = A094832(n).
T(n,n) = A094831(n).
T(n+1,n) = T(n+2,n) = A094826(n).
sum(T(n-k,k), 0<=k<=n) = A065455(n).

A217733 Expansion of (1+x-x^2)/((1-x)(1-3x^2-x^3)).

Original entry on oeis.org

1, 2, 4, 8, 15, 29, 54, 103, 192, 364, 680, 1285, 2405, 4536, 8501, 16014, 30040, 56544, 106135, 199673, 374950, 705155, 1324524, 2490416, 4678728, 8795773, 16526601, 31066048, 58375577, 109724746, 206192780, 387549816, 728303087, 1368842229, 2572459078, 4834829775, 9086219464

Offset: 0

Philippe Deléham, Mar 22 2013

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Cf. A065455, A216236.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-x^2)/((1-x)*(1-3*x^2-x^3)))); // Bruno Berselli, Mar 25 2013

CoefficientList[Series[(1 + x - x^2)/((1 - x) (1 - 3 x^2 - x^3)), {x, 0, 40}], x] (* Bruno Berselli, Mar 25 2013 *)

makelist(coeff(taylor((1+x-x^2)/((1-x)*(1-3*x^2-x^3)), x, 0, n), x, n), n, 0, 40); /* Bruno Berselli, Mar 25 2013 */

G.f.: (1+x-x^2)/(1-x-3*x^2+2*x^3+x^4).
a(n) = sum( A216236(n-k,k), 0<=k<=n ).
a(n) = a(n-1)+3*a(n-2)-2*a(n-3)-a(n-4) for n>=4, a(0)=1, a(1)=2, a(2)=4, a(3)=8.
a(n+1) - a(n) = A065455(n).

A065506 Number of (binary) bit strings of length n having an even length block of 0's followed by an odd length block of 1's.

Original entry on oeis.org

1, 2, 7, 15, 39, 84, 196, 419, 928, 1965, 4227, 8871, 18742, 39032, 81481, 168606, 349011, 718371, 1477783, 3028412, 6200296, 12660171, 25823604, 52569417, 106908199, 217086287, 440402878, 892384788, 1806730377, 3654428154

Offset: 3

Len Smiley, Nov 26 2001

Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-2).

Cf. A065455.

G.f.: x^3/((1-2*x)*(1-3*x^2-x^3)).

a(n) = 2^n - A065455(n).

cp's OEIS Frontend

A090992 Number of meaningful differential operations of the n-th order on the space R^7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A065494 Number of (binary) bit strings in which no even length block of 0's is followed by an even length block of 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A065495 Number of (binary) bit strings of length n in which an odd length block of 0's is followed by an odd length block of 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A065497 Number of (binary) bit strings of length n having at least one even length block of 0's followed by an even length block of 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217733 Expansion of (1+x-x^2)/((1-x)*(1-3*x^2-x^3)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Formula

A065506 Number of (binary) bit strings of length n having an even length block of 0's followed by an odd length block of 1's.

Views

Author

Keywords

Links

Crossrefs

Formula

A217733 Expansion of (1+x-x^2)/((1-x)(1-3x^2-x^3)).