A005712 -id:A005712 - cp's OEIS frontend

A005713 Define strings S(0)=0, S(1)=11, S(n) = S(n-1)S(n-2); iterate.

1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane

a(A035336(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = 1 - A123740(n).  This can be seen as follows. Define words T(0)=0, T(1)=1, T(n) = T(n-1)T(n-2). Then T(infinity) is the binary complement of the infinite Fibonacci word A003849. Obviously S(n) is the [1->11] transform of T(n). The claim now follows from the observation (see Comments of A123740) that doubling the 0's in the infinite Fibonacci word A003849 gives A123740. - Michel Dekking, Oct 21 2018
From Michel Dekking, Oct 22 2018: (Start)
Here is a proof of Cloitre's (corrected) formula
      a(n) = abs(A014677(n+1)).
Since abs(-1) = abs(1) = 1, one has to prove that A014677(k)=0 if and only if there is an n such that AB(n) = k (using that a(n) = 1 - A123740(n)). Now A014677 is the sequences of first differences of A001468, and the 0's in A014677 occur if and only if there occurs a block 22 in A001468, which is given by
      A001468(n) = floor((n+1)*phi) - floor(n*phi), n >= 0.
But then
      A001468(n) = A014675(n-1), n > 0.
The sequence A014675 is fixed point of the morphism 1->2, 2->21, which is alphabet equivalent to the morphism 1->12, 2->1, the classical Fibonacci morphism in standard form. This implies that the 22 blocks in A001468 occur at position n+1 in if and only if 3 occurs in the fixed point A270788 of the 3-symbol Fibonacci morphism at k, which happens if and only if there is an n such that AB(n)=k (see Formula of A270788). (End)

			The infinite word is S(infinity) = 110111101101111011110110...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
R. K. Guy, Letter to N. J. A. Sloane, 1987
Jeffrey Shallit and Anatoly Zavyalov, Transduction of Automatic Sequences and Applications, arXiv:2303.15203 [cs.FL], 2023, see p. 31.

Cf. A005614, A003849.

Cf. A001468, A014675, A014677, A123740, A270788.

a005713 n = a005713_list !! n
a005713_list = 1 : 1 : concat (sibb [0] [1,1]) where
   sibb xs ys = zs : sibb ys zs where zs = xs ++ ys
-- Reinhard Zumkeller, Dec 30 2011

s[0] = {0}; s[1] = {1, 1}; s[n_] := s[n] = Join[s[n-1], s[n-2]]; s[10] (* Jean-François Alcover, May 15 2013 *)
nxt[{a_,b_}]:={b,Join[a,b]}; Drop[Nest[nxt,{{0},{1,1}},10][[1]],3] (* Harvey P. Dale, Jan 31 2019 *)

a(n,f1,f2)=local(f3); for(i=3,n,f3=concat(f2,f1); f1=f2; f2=f3); f2

printp(a(10,[ 0 ],[ 1,1 ])) \\ Would give S(10). Sequence is S(infinity).

From Benoit Cloitre, Apr 21 2003: (Start)
For n > 1, a(n-1) = floor(phi*ceiling(n/phi)) - ceiling(phi*floor(n/phi)) where phi=(1+sqrt(5))/2.
For n >= 0, a(n) = abs(A014677(n+1)). (End)

Corrected by Michael Somos

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.

Original entry on oeis.org

1, 4, 2, 9, 7, 3, 17, 14, 10, 4, 28, 25, 19, 13, 5, 43, 39, 33, 24, 16, 6, 62, 58, 50, 41, 29, 19, 7, 86, 81, 73, 61, 49, 34, 22, 8, 115, 110, 100, 88, 72, 57, 39, 25, 9, 150, 144, 134, 119, 103, 83, 65, 44, 28, 10, 191, 185, 173, 158, 138, 118, 94, 73, 49, 31

Offset: 1

Clark Kimberling, Jun 22 2012

Principal diagonal: A213782.
Antidiagonal sums: A005712.
row 1,  (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.
row 2,  (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).
row 3,  (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...4....9....17...28...43....62
2...7....14...25...39...58....81
3...10...19...33...50...73....100
4...13...24...41...61...88....119
5...16...29...49...72...103...138
6...19...34...57...83...118...157
7...22...39...65...94...133...176

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500, A213778.

b[n_] := Floor[(n + 2)/2]; c[n_] := n;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213781 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005712 *)

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + x - (2*n - 1)*x^2 + (n -1)*x^3) and g(x) = (1 + x)(1 - x)^4.

A005719 Quadrinomial coefficients.

Original entry on oeis.org

2, 12, 40, 101, 216, 413, 728, 1206, 1902, 2882, 4224, 6019, 8372, 11403, 15248, 20060, 26010, 33288, 42104, 52689, 65296, 80201, 97704, 118130, 141830, 169182, 200592, 236495, 277356, 323671, 375968, 434808, 500786, 574532, 656712, 748029, 849224, 961077

Offset: 2

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

a(n)= A008287(n, 5), n >= 2 (sixth column of quadrinomial coefficients).

A005719:=(2-2*z**2+z**3)/(z-1)**6; [Conjectured by Simon Plouffe in his 1992 dissertation.]

a(n)= binomial(n, 2)*(n^3+11*n^2+46*n-24)/60, n >= 2.
G.f.: (x^2)*(2-2*x^2+x^3)/(1-x)^6. (numerator polynomial is N4(5, x) from A063421.)
a(n) = 2*binomial(n,2) + 6*binomial(n,3) + 4*binomial(n,4) + binomial(n,5) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A005720 Quadrinomial coefficients.

Original entry on oeis.org

1, 10, 44, 135, 336, 728, 1428, 2598, 4455, 7282, 11440, 17381, 25662, 36960, 52088, 72012, 97869, 130986, 172900, 225379, 290444, 370392, 467820, 585650, 727155, 895986, 1096200, 1332289, 1609210, 1932416, 2307888, 2742168, 3242393, 3816330, 4472412, 5219775

Offset: 2

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

a(n)= A008287(n, 6), n >= 2 (seventh column of quadrinomial coefficients).

A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by Simon Plouffe in his 1992 dissertation.]

LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,10,44,135,336,728,1428},40] (* or *) Table[Binomial[n+1,3] (n^3+15n^2+86n-120)/120,{n,2,41}] (* Harvey P. Dale, Jun 23 2011 *)

a(n)=(n^6 + 15*n^5 + 85*n^4 - 135*n^3 - 86*n^2 + 120*n)/720 \\ Charles R Greathouse IV, Jun 23 2011

a(n)= binomial(n+1, 3)*(n^3+15*n^2+86*n-120)/120, n >= 2.
G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7. (numerator polynomial is N4(6, x) from A063421).
a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 23 2011
a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A005726 Quadrinomial coefficients.

Original entry on oeis.org

1, 2, 6, 20, 65, 216, 728, 2472, 8451, 29050, 100298, 347568, 1208220, 4211312, 14712960, 51507280, 180642391, 634551606, 2232223626, 7862669700, 27727507521, 97884558992, 345891702456, 1223358393120, 4330360551700

Offset: 1

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. K. Guy, Letter to N. J. A. Sloane, 1987

for n from 1 to 40 do printf(`%d,`,coeff(expand(sum(x^j, j=0..3)^n), x, n-1)) od:
F := (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1);  G := t/((t-1)*(t^2+1)); Ginv := RootOf(numer(G-x),t);  ogf := series(eval(F,t=Ginv),x=0,20); # Mark van Hoeij, Oct 30 2011

Table[Sum[Binomial[n,k]Binomial[n,2k+1],{k,0,Floor[n/2]}],{n,30}] (* Harvey P. Dale, Oct 19 2013 *)

a(n) = Sum_{k=0..floor(n/2)}, C(n,k) C(n,2k+1). - Paul Barry, May 15 2003
a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-4k,n-1],{k,0,Floor[(n-1)/4]}]. - David Callan, Jul 03 2006
G.f.: F(G^(-1)(x)) where F(t) = (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1) and G(t) = t/((t-1)*(t^2+1)). - Mark van Hoeij, Oct 30 2011
Conjecture: 2*(n-1)*(2*n+1)*(13*n-14)*a(n) +(-143*n^3+297*n^2-148*n+12) *a(n-1) -4*(n-1)*(26*n^2-41*n+9)*a(n-2) -16*(n-1)*(n-2)*(13*n-1) *a(n-3)=0. - R. J. Mathar, Nov 13 2012
a(n) = A008287(n,n-1). - Sean A. Irvine, Aug 15 2016

More terms from James Sellers, Aug 21 2000

A005723 Quadrinomial coefficients.

Original entry on oeis.org

1, 12, 155, 2128, 30276, 440484, 6506786, 97181760, 1463609356, 22187304112, 338118529539, 5175023913008, 79492847013100, 1224838471521240, 18922450356489780, 293003808610433280, 4546150487318508156, 70662280030419277200, 1100069396653853657564

Offset: 1

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987

Bisection of A005190.

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 19, 10, 1, 0, 1, 15, 45, 45, 15, 1, 0, 1, 21, 90, 141, 90, 21, 1, 0, 1, 28, 161, 357, 357, 161, 28, 1, 0, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 0, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 0, 1, 55, 615, 2850, 6765, 8953

Offset: 0

Philippe Deléham, Oct 30 2006

Subtriangle (1 <= k <= n) is in A056241.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  6,   6,    1;
  0, 1, 10,  19,   10,    1;
  0, 1, 15,  45,   45,   15,    1;
  0, 1, 21,  90,  141,   90,   21,    1;
  0, 1, 28, 161,  357,  357,  161,   28,    1;
  0, 1, 36, 266,  784, 1107,  784,  255,   36,   1;
  0, 1, 45, 414, 1554, 2907, 2907, 1554,  414,  45,  1;
  0, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55, 1;

Columns: A000007, A000012, A000217, A005712, A005714, A005716.

Cf. A027907, A056241.

T(2*k-1,k) = A082758(k-1)for k >= 1.
Sum_{k=0..n} T(n,k) = A124302(n); see also A007051.
Sum_{k=0..n} (-1)^(n-k)*T(n,k) = A117569(n).
G.f.: (1-x*(y+2)+x^2)/(1-2x*(1+y)+(1+y+y^2)*x^2). - Philippe Deléham, Oct 30 2011

A368881 a(n) = binomial(n+3, 4) + binomial(n+1, 3) + 1.

Original entry on oeis.org

1, 2, 7, 20, 46, 91, 162, 267, 415, 616, 881, 1222, 1652, 2185, 2836, 3621, 4557, 5662, 6955, 8456, 10186, 12167, 14422, 16975, 19851, 23076, 26677, 30682, 35120, 40021, 45416, 51337, 57817, 64890, 72591, 80956, 90022, 99827, 110410, 121811, 134071

Offset: 0

Joshua Swanson, Jan 08 2024

The number of bigrassmannian permutations in the type B hyperoctahedral group of order 2^n*n!, i.e., those with a unique left and right type B descent or the identity. This can be characterized by avoiding 18 signed permutation patterns.

			For n=2, all eight 2 X 2 signed permutation matrices are bigrassmannian except the negative of the identity matrix, or equivalently the one with window notation [-1 -2], so a(2) = 7.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Joshua Swanson, Bigrassmannians and pattern avoidance notes.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A050407.

It appears that this is equal to {A005712}+1, also ({A212039}+2)/3 .

Table[Binomial[n + 3, 4] + Binomial[n + 1, 3] + 1, {n, 0, 20}]
LinearRecurrence[{5,-10,10,-5,1},{1,2,7,20,46},50] (* Harvey P. Dale, Jan 21 2025 *)

def A368881(n): return 1+(n*(n*(n*(n + 10) + 11) + 2))//24 # Chai Wah Wu, Jan 27 2024

a(n) = (1/24)*(n^4 + 10*n^3 + 11*n^2 + 2*n + 24).
G.f.: (x^4 - 5x^3 + 7x^2 - 3x + 1)/(1-x)^5.
E.g.f.: exp(x)*(24 + 24*x + 48*x^2 + 16*x^3 + x^4)/24. - Stefano Spezia, Jan 09 2024

A005724 Quadrinomial coefficients.

Original entry on oeis.org

3, 40, 546, 7728, 112035, 1650792, 24608948, 370084832, 5603730876, 85316186400, 1304770191802, 20029132137840, 308437355259930, 4762695514958640, 73716196036213800, 1143325208566357440, 17765127399780725316, 276484586847524844768, 4309270265307160983144

Offset: 1

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987

Cf. A008287.

seq(coeff((1+x+x^2+x^3)^(2*n),x,3*n-1), n=1..30); # Robert Israel, Aug 15 2016

a(n) = A008287(2 * n, 3 * n - 1). - Sean A. Irvine, Aug 15 2016

Offset changed by Robert Israel, Aug 15 2016

cp's OEIS Frontend

A005713 Define strings S(0)=0, S(1)=11, S(n) = S(n-1)S(n-2); iterate.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A213781 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A005719 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

A005720 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A005726 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Programs

Maple

Mathematica

Formula

Extensions

A005723 Quadrinomial coefficients.

Views

Author

Keywords

References

Links

Crossrefs

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A368881 a(n) = binomial(n+3, 4) + binomial(n+1, 3) + 1.

Views

Author

Keywords

Comments

A213781 Rectangular array: (row n) = bc, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and = convolution.