A001628 -id:A001628 - cp's OEIS frontend

A152881 Positions of those 1's that are followed by a 0, summed over all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

0, 1, 5, 15, 40, 95, 213, 455, 940, 1890, 3720, 7194, 13710, 25805, 48055, 88665, 162272, 294865, 532395, 955795, 1707110, 3034836, 5372400, 9473700, 16646700, 29155225, 50908793, 88644915, 153952120, 266726195, 461066385, 795320159

Offset: 1

Emeric Deutsch, Jan 04 2009

a(n) = Sum(k*A119469(n+1,k),k>=0).

For n>1, a(n-1) is the n-th antidiagonal sum of A213777. [Clark Kimberling, Jun 21 2012]

			a(4)=15 because the Fibonacci binary words of length 4 are 1110, 1111, 1101, 1010, 1011, 0110, 0111, 0101 and the positions of those 1's that are followed by a 0 are 3, 2, 1, 3, 1, 3 and 2; their sum is 15.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A119469.

G := z^2*(1+2*z)/(1-z-z^2)^3: Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 1 .. 34);

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

a(n) = A001628(n-1) + 2*A001628(n-2), n>1, a(0)=0, a(1)=1. [Vladimir Kruchinin, Apr 26 2011]

A268400 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly twice.

Original entry on oeis.org

1, 5, 23, 99, 413, 1691, 6842, 27464, 109631, 435887, 1728018, 6835668, 26996393, 106486529, 419639903, 1652533719, 6504159137, 25589302163, 100646529977, 395775842389, 1556107102849, 6117771240251, 24050813530815, 94550689834203, 371715533473021, 1461430355605367, 5746128800657639, 22594839306797223

Offset: 3

Ran Pan, Feb 03 2016

nonn

This sequence is related to paired pattern P_2 in Pan and Remmel's link.

G. C. Greubel, Table of n, a(n) for n = 3..1000
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A000108, A001628, A033184.

Rest[Rest[Rest[CoefficientList[Series[-((-1 + Sqrt[1 - 4 x])^3 x (-1 + Sqrt[1-4 x] + 2 x))/(2 (1 - Sqrt[1 - 4 x] + (-5 + Sqrt[1 - 4 x]) x)^3), {x, 0, 40}], x]]]] (* Vincenzo Librandi, Feb 28 2016 *)

a(n):=((sum((m+2)*(sum((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k),k,0,m/2)) *binomial(2*n-m-5,n-m-3),m,1,n-3))+2*binomial(2*n-4,n-2))/(2*n-2); /* Vladimir Kruchinin, Feb 28 2016 */

G.f.: -((-1 + f(x))^3*x*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^3), where f(x) = sqrt(1 - 4*x).
a(n) = ((Sum_{m=1..n-3}((m+2)*(Sum_{k=0..m/2}((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k)))*binomial(2*n-m-5,n-m-3)))+2*binomial(2*n-4,n-2))/(2*n-2), n>2. - Vladimir Kruchinin, Feb 28 2016
a(n) ~ 7*2^(2*n+2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 28 2016
D-finite with recurrence (n-1)*(9*n-56)*a(n) +(-31*n^2+203*n-16)*a(n-1) +(-193*n^2+1371*n-2138)*a(n-2) +3*(217*n^2-1497*n+2274)*a(n-3) +2*(41*n-94)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

A057280 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057995.

Original entry on oeis.org

2, 17, 5, 225, 120, 15, 4080, 3050, 700, 50, 94440, 89225, 28625, 3775, 175, 2666880, 3006000, 1208975, 223175, 19225, 625, 89016480, 115299900, 54824650, 12689800, 1537100, 93500, 2250, 3430929600, 4973077800, 2695596850, 737744125

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) =( p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057995(k,m).

			k=2: F2(n)=((16+5*n)*(n+1)*F0(n+1)+(17+5*n)*(n+2)*F0(n))/50, cf. A001628.

Cf. A000045, A037027, A057995.

A057995 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057280.

Original entry on oeis.org

1, 16, 5, 300, 160, 20, 6840, 4850, 1075, 75, 186120, 159650, 48175, 6100, 275, 5916240, 5846700, 2168650, 379700, 31550, 1000, 215717040, 238437900, 103057800, 22426825, 2605175, 153875, 3625, 8888140800, 10772348400

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of F0(n) := A000045(n+1) n >= 0, (Fibonacci starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057280).

			k=2: F2(n)=((16+5*n)*(n+1)*F0(n+1)+(17+5*n)*(n+2)*F0(n))/50, cf. A001628.

Cf. A000045, A037027, A057280.

A067988 Row sums of triangle A067330; also of triangle A067418.

Original entry on oeis.org

1, 3, 10, 25, 60, 133, 284, 585, 1175, 2310, 4464, 8502, 15995, 29775, 54920, 100487, 182556, 329555, 591550, 1056405, 1877821, 3323868, 5860800, 10297500, 18033925, 31487643, 54824854, 95211205, 164948700, 285121105, 491804144, 846631137, 1454746355, 2495275650

Offset: 0

Wolfdieter Lang, Feb 15 2002

a(n) is the sum of the positions of the 0's in all Fibonacci binary words of length n+1. A Fibonacci binary word is a binary word having no 00 subword. Example: a(3)=25 because the Fibonacci binary words of length 4 are 1110, 1111, 1101, 1010, 1011, 0110, 0111 and 0101, the positions of the 0's being 4, 3, 2, 4, 2, 1, 4, 1, 1 and 3 (their sum is 25). - Emeric Deutsch, Jan 04 2009

Cf. A000045, A001628, A067330, A067418.

a:=n->sum(binomial(n-j,j)*n*j/2,j=0..n): seq(a(n), n=2..30); # Zerinvary Lajos, Oct 19 2006

Table[((n+2)((3n+5)Fibonacci[n+1]+(n+1)Fibonacci[n]))/10,{n,0,30}] (* Harvey P. Dale, Feb 02 2020 *)

a(n) = (n+2)*((3*n+5)*F(n+1)+(n+1)*F(n))/10, with F(n) := A000045(n) (Fibonacci).
G.f.: (1+x^2)/(1-x-x^2)^3.
Sum_{j=0..n} binomial(n-j,j)*n*j/2. - Zerinvary Lajos, Oct 19 2006
E.g.f.: exp(x/2)*(5*(10 + 18*x + 7*x^2)*cosh(sqrt(5)*x/2) + sqrt(5)*(14 + 46*x + 15*x^2)*sinh(sqrt(5)*x/2))/50. - Stefano Spezia, Aug 30 2025

a(29)-a(33) from Stefano Spezia, Aug 30 2025

A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

Original entry on oeis.org

2, 7, 21, 53, 124, 273, 577, 1181, 2358, 4614, 8880, 16854, 31612, 58691, 108003, 197203, 357596, 644463, 1155059, 2059897, 3656988, 6465660, 11388480, 19990140, 34976870, 61019071, 106160481, 184228193, 318948124, 550962717, 949781269, 1634103701, 2806342578

Offset: 3

N. J. A. Sloane, May 09 2012

Apparently the number of Dyck n-paths that have n-2 peaks after changing each valley to a peak by the transformation DU -> UD. E.g., the Dyck 3-paths UUUDDD and UUDUDD have 1 peak after changing DU to UD so a(3) = 2. - David Scambler, Sep 03 2012

Robert P. P. McKone, Table of n, a(n) for n = 3..5000
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. Column 2 of A091533. Partial sums of A036681.

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 7, 21, 53}, {3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
A212338[1, n2_] = 0; A212338[n1_, 1] = 0; A212338[2, n2_] = 0; A212338[n1_, 2] = 0; A212338[3, 3] = 1; A212338[n1_, n2_] := A212338[n1, n2] = A212338[n1 - 1, n2] + A212338[n1, n2 - 1] + A212338[n1 - 1, n2 - 1] + A212338[n1 - 2, n2] + A212338[n1, n2 - 2]; Table[A212338[5, y], {y, 3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
QQQ2[t,x]=2/(1 + (t*x - t)*(1 + t) +((1 + (t*x - t)*(1 + t))^2 - 4*t*x)^(1/2)); CoefficientList[Coefficient[Series[QQQ2[t, x], {t, 0, 22}], x],t] (* Robert Price, Jun 05 2012 *)

g.f. -x^3*(2+x) / (x^2+x-1)^3, i.e., a(n) = 2*A001628(n-3) + A001628(n-4). - R. J. Mathar, Jun 27 2012

a(n) = a(n-1) + a(n-2) + A067331(n-3). E.g., a(5) = 21 = 7 + 2 + 12. - David Scambler, Sep 03 2012

A245961 Number of 4-cycles in the Lucas cube Lambda(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 15, 35, 80, 171, 355, 715, 1410, 2730, 5208, 9810, 18280, 33745, 61785, 112309, 202840, 364245, 650705, 1157015, 2048532, 3612900, 6349200, 11121300, 19421150, 33820061, 58740915, 101777495, 175945280, 303516015, 522541903, 897942115

Offset: 0

Emeric Deutsch, Aug 10 2014

The vertex set of the Lucas cube Lambda(n) is the set of all binary strings of length n without consecutive 1's and without a 1 in the first and the last bit. Two vertices of the Lucas cube are adjacent if their strings differ in exactly one bit.

			a(3)=0 because the Lucas cube Lambda(3) is the star-tree on 4 vertices.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
Eric Weisstein's World of Mathematics, Lucas Cube Graph
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A000045, A000032, A001628, A001629.

Cf. A364605 (number of 6-cycles).

[((n-n^2)*Fibonacci(n) + (3*n^2 - 5*n)*Fibonacci(n-1))/10: n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

with(combinat): a := proc (n) options operator, arrow: (1/10)*n*(1-n)*fibonacci(n)+(1/10)*n*(3*n-5)*fibonacci(n-1) end proc: seq(a(n), n = 0 .. 35);

Table[((n - n^2) Fibonacci[n] + (3 n^2 - 5 n) Fibonacci[n - 1])/10, {n, 0, 50}] (* Vincenzo Librandi, Aug 11 2014 *)
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 0, 2, 5, 15}, 20] (* Eric W. Weisstein, Jul 29 2023 *)
CoefficientList[Series[(-2 + x) x^3/(-1 + x + x^2)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 29 2023 *)

concat([0,0,0,0], Vec(x^4*(x-2)/(x^2+x-1)^3 + O(x^100))) \\ Colin Barker, Aug 13 2014

a(n) = ((n-n^2)*F(n) + (3n^2 - 5n)*F(n-1))/10, where F(n) = A000045(n), the Fibonacci numbers. Formula follows from Eq. (4) of the Klavzar 2005 reference and from the first formula on p. 511 of the Klavzar 2013 reference.
a(n) = Sum(L(i)*b(n-3-i), i=0..n-4), where L(i) = A000032(i) are the Lucas numbers and b(j) = A001629(j+1) is the number of edges in the Fibonacci cube Gamma(j) (see Prop. 9 of the Klavzar 2005 reference).
a(n) = 3*a(n-1)-5*a(n-3)+3*a(n-5)+a(n-6). G.f.: x^4*(x-2) / (x^2+x-1)^3. - Colin Barker, Aug 11 2014
a(n) = 2*A001628(n-4)-A001628(n-5). - R. J. Mathar, Jul 24 2022
G.f.: (-2+x)*x^4/(-1+x+x^2)^3. - Eric W. Weisstein, Jul 29 2023

A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

Original entry on oeis.org

1, 3, 12, 37, 111, 315, 864, 2307, 6027, 15471, 39132, 97755, 241606, 591636, 1437078, 3465748, 8305161, 19788957, 46910232, 110686101, 260064912, 608684490, 1419591546, 3300027546, 7648265728, 17676484410, 40747630332, 93704299336, 214999206831, 492262973433

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (3,3,-8,-12,3,17,15,6,1).

Cf. A002478, A362126.

Cf. A000217, A001628, A382614.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^2)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, n-k));

a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,n-k).
a(n) = 3*a(n-1) + 3*a(n-2) - 8*a(n-3) - 12*a(n-4) + 3*a(n-5) + 17*a(n-6) + 15*a(n-7) + 6*a(n-8) + a(n-9).
G.f.: 1/(1-x-2*x^2-x^3)^3. - Vincenzo Librandi, Apr 10 2025

A006479 From variance of Fibonacci search.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 52, 134, 318, 713, 1531, 3180, 6432, 12732, 24756, 47417, 89665, 167694, 310628, 570562, 1040226, 1883953, 3391799, 6073848, 10824096, 19204536, 33936456

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

K. J. Overholt, Efficiency of the Fibonacci search method, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 92-96.
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 (4,-3,-5,5,3,-2,-1).

Cf. A006478.

A006479:=z**3*(1+z+z**2)/(z-1)/(z**2+z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

(Conjecture equivalent to Plouffe g.f.): a(n) = -3 - 2*A001629(n+2) - 3*A001629(n+1) + 2*A001628(n-1) + A020701(n+1). - R. J. Mathar, Dec 06 2010

A057282 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057281.

Original entry on oeis.org

2, 5, 17, 15, 120, 225, 50, 700, 3050, 4080, 175, 3775, 28625, 89225, 94440, 625, 19225, 223175, 1208975, 3006000, 2666880, 2250, 93500, 1537100, 12689800, 54824650, 115299900, 89016480, 8125, 438250, 9670750, 112454500, 737744125

Offset: 1

Wolfdieter Lang, Sep 13 2000

The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A057281(k,m).
a(k,0)= A020876(k), k >= 0.

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n) := A000045(n); see A001628.
2; 5,17; 15,120,225; 50,700,3050,4080; 175,3775,28625,89225,94440; ...

Cf. A000045, A037027, A057995, A057280, A057281.

Row sums: A151615.

cp's OEIS Frontend

A152881 Positions of those 1's that are followed by a 0, summed over all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

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A268400 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly twice.

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A057280 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057995.

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A057995 Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057280.

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A067988 Row sums of triangle A067330; also of triangle A067418.

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A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

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A245961 Number of 4-cycles in the Lucas cube Lambda(n).

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A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

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