A091002 -id:A091002 - cp's OEIS frontend

A091000 Number of closed walks of length n on the Petersen graph rooted at a given vertex.

1, 0, 3, 0, 15, 12, 99, 168, 759, 1764, 6315, 16896, 54783, 156156, 484851, 1421784, 4330887, 12861588, 38846907, 116016432, 349097871, 1045196460, 3139783683, 9410962440, 28249664535, 84715439172, 254213426379, 762506061408

Offset: 0

Paul Barry, Dec 12 2003

If p >= 7 is a prime, then p divides a(p) (provable by easy application of Fermat's Little Theorem). - Adam P. Goucher, Sep 11 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

List([0..30], n -> (3^n+(-2)^(n+2)+5)/10); # G. C. Greubel, Feb 01 2019

[(3^n+(-2)^(n+2)+5)/10: n in [0..30]]; // G. C. Greubel, Feb 01 2019

Table[{1,0,0}.MatrixPower[{{0,3,0},{1,0,2},{0,1,2}},n].{1,0,0},{n,1,100}] (* Adam P. Goucher, Sep 11 2013 *)
LinearRecurrence[{2,5,-6}, {1,0,3}, 30] (* G. C. Greubel, Feb 01 2019 *)

vector(30, n, n--; (3^n+(-2)^(n+2)+5)/10) \\ G. C. Greubel, Feb 01 2019

[(3^n+(-2)^(n+2)+5)/10 for n in (0..30)] # G. C. Greubel, Feb 01 2019

G.f.: (1-2*x-2*x^2)/((1-x)*(1+2*x)*(1-3*x)).
a(n) = (3^n + (-2)^(n+2) + 5)/10.
a(n) = (A000244(n) + 9*A001045(n+1)(-1)^n + 6*A001045(n)(-1)^(n+1))/10.
3^n = a(n) + 3*A091001(n) + 6*A091002(n)
E.g.f.: (exp(3*x) + 4*exp(-2*x) + 5*exp(x))/10. - G. C. Greubel, Feb 01 2019

A091001 Number of walks of length n between adjacent nodes on the Petersen graph.

Original entry on oeis.org

0, 1, 0, 5, 4, 33, 56, 253, 588, 2105, 5632, 18261, 52052, 161617, 473928, 1443629, 4287196, 12948969, 38672144, 116365957, 348398820, 1046594561, 3136987480, 9416554845, 28238479724, 84737808793, 254168687136, 762595539893

Offset: 0

Paul Barry, Dec 12 2003

N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 20.
F. Harary, Graph Theory, Addison-Wesley, 1969, p. 89.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

List([0..30], n -> (3^(n+1)+(-2)^(n+3)+5)/30); # G. C. Greubel, Feb 01 2019

[(3^(n+1)+(-2)^(n+3)+5)/30: n in [0..30]]; // G. C. Greubel, Feb 01 2019

Table[(3^(n+1)+(-2)^(n+3)+5)/30, {n,0,30}] (* or *) LinearRecurrence[{2, 5,-6}, {0,1,0}, 30] (* G. C. Greubel, Feb 01 2019 *)

vector(30, n, n--; (3^(n+1)+(-2)^(n+3)+5)/30) \\ G. C. Greubel, Feb 01 2019

[(3^(n+1)+(-2)^(n+3)+5)/30 for n in (0..30)] # G. C. Greubel, Feb 01 2019

G.f.: x*(1-2*x)/((1-x)*(1+2*x)*(1-3*x)).
a(n) = (3^(n+1) + (-2)^(n+3) + 5)/30.
3^n = A091000(n) + 3*a(n) + 6*A091002(n).
a(n) = (A000244(n) - A001045(n+1)*(-1)^n - 6*A001045(n)*(-1)^n)/10.
a(n) = A091002(n+1) - 2*A091002(n). - R. J. Mathar, Oct 30 2014
E.g.f.: (3*exp(3*x) - 8*exp(-2*x) +5*exp(x))/30. - G. C. Greubel, Feb 01 2019

A091005 Expansion of x^2/((1-2x)(1+3*x)).

Original entry on oeis.org

0, 0, 1, -1, 7, -13, 55, -133, 463, -1261, 4039, -11605, 35839, -105469, 320503, -953317, 2876335, -8596237, 25854247, -77431669, 232557151, -697147165, 2092490071, -6275373061, 18830313487, -56482551853, 169464432775, -508359743893, 1525146340543

Offset: 0

Paul Barry, Dec 13 2003

Inverse binomial transform of A091002.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,6).

Cf. A015441.

Concatenation([0], List([1..30], n -> (3*2^n + 2*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019

[0] cat [(3*2^n + 2*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019

a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Join[{0, 0}, LinearRecurrence[{-1, 6}, {1, -1}, 30]] (* G. C. Greubel, Feb 01 2019 *)
CoefficientList[Series[x^2/((1-2x)(1+3x)),{x,0,30}],x] (* Harvey P. Dale, Apr 30 2022 *)

vector(30, n, n--; (3*2^n + 2*(-3)^n - 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019

[0] + [(3*2^n + 2*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

2^n = A091003(n) + 3*A091004(n) + 6*a(n).
a(n) = (3*2^n + 2*(-3)^n - 5*0^n)/30.
E.g.f.: (3*exp(2*x) + 2*exp(-3*x) - 5)/30. - G. C. Greubel, Feb 01 2019

A249992 Expansion of 1/((1+x)(1+2x)(1-3x)).

Original entry on oeis.org

1, 0, 7, 6, 49, 84, 379, 882, 3157, 8448, 27391, 78078, 242425, 710892, 2165443, 6430794, 19423453, 58008216, 174548935, 522598230, 1569891841, 4705481220, 14124832267, 42357719586, 127106713189, 381253030704, 1143893309839, 3431411494062, 10294771353097

Offset: 0

Alex Ratushnyak, Dec 27 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,7,6).

Cf. A249993.
Cf. A000392: expansion of x^3/((1-x)*(1-2*x)*(1-3*x)).
Cf. A091002: expansion of x^2/((1-x)*(1+2*x)*(1-3*x)).
Cf. A094705: expansion of   x/((1+x)*(1-2*x)*(1-3*x)).

[(3^(n+2) + (-1)^n*(2^(n+4) - 5))/20: n in [0..50]]; // G. C. Greubel, Jul 21 2022

seq((9/20)*3^n+(4/5)*(-2)^n-(1/4)*(-1)^n, n=0 .. 100); # Robert Israel, Dec 28 2014

LinearRecurrence[{0, 7, 6}, {1, 0, 7}, 29] (* Jean-François Alcover, Oct 05 2017 *)
CoefficientList[Series[1/((1+x)(1+2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, May 26 2020 *)

Vec(1/((1+x)*(1+2*x)*(1-3*x)) + O(x^50)) \\ Michel Marcus, Dec 28 2014

[(3^(n+2) +(-1)^n*(2^(n+4) -5))/20 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: 1/((1+x)*(1+2*x)*(1-3*x)).
a(n) = ( 3^(n+2) + (2^(n+4) - 5)*(-1)^n )/20. - Colin Barker, Dec 28 2014
a(n) = 7*a(n-2) + 6*a(n-3). - Colin Barker, Dec 28 2014
E.g.f.: (9/20)*exp(3*x) + (4/5)*exp(-2*x) - (1/4)*exp(-x). - Robert Israel, Dec 28 2014

A341091 Triangle read by rows: Coefficients for calculation of the sum of all the finite differences from order zero to order k. Sum_{n=0..k} T(n, k)*b(n) = b(0) + b(1) + ... + b(k) + (b(1) - b(0)) + ... + (b(k) - b(k-1)) + ((b(2) - b(1)) - (b(1) - b(0))) + ... .

Original entry on oeis.org

1, 0, 2, 1, -1, 3, 0, 3, -3, 4, 1, -2, 7, -6, 5, 0, 4, -8, 14, -10, 6, 1, -3, 13, -21, 25, -15, 7, 0, 5, -15, 35, -45, 41, -21, 8, 1, -4, 21, -49, 81, -85, 63, -28, 9, 0, 6, -24, 71, -129, 167, -147, 92, -36, 10, 1, -5, 31, -94, 201, -295, 315, -238, 129, -45, 11

Offset: 0

Thomas Scheuerle, Feb 13 2022

If we want to calculate the sum of finite differences for a sequence b(n):
  b(0)*T(0, n) + ... + b(n)*T(n, n) = b(0) + b(1) + ... + b(n) + (b(1) - b(0)) + ... + (b(n) - b(n-1)) + ((b(2) - b(1)) - (b(1) - b(0))) + ... This sum includes the sequence b(n) itself. This defines an invertible linear sequence transformation with a deep connection to Bernoulli numbers and other interesting sequences of rational numbers.
From Thomas Scheuerle, Apr 29 2024: (Start)
These are the coefficients of the polynomials defined by the recurrence: P(k, x) = P(k - 1, x) + (x^2 - x)*P(k - 2, x) + 1, with P(-1, x) = 0 and P(0, x) = 1. This can also be expressed as P(k, x) = Sum_{m=1..k+1} binomial(k+2 - m, m)*(x^2 - x)^(m - 1) = Sum_{n=0..k} T(n, k)*x^(k-n). If we would evaluate P(k, t) as sequence for some fixed t then we get the expansion of 1/((1 - x)*(1+(t-1)*x)*(1 - t*x)).
We may replace (x^2 - x) by (x^(-2) - x^(-1)) to get the coefficients in reverse order: x^k*Sum_{m=1..k+1} binomial(k+2 - m, m)*(x^(-2) - x^(-1))^(m - 1) = Sum_{n=0..k} T(n, k)*x^n = F(k, x). If we would evaluate F(k, t) as sequence for some fixed t then we get the expansion of 1/((1 - x)*(1 - (t-1)*x)*(1 - t*x)). (End)

			Triangle begins with T(n, k):
   n=   0,  1,   2,   3,   4,   5,   6,   7,   8
  k=0   1
  k=1   0,  2
  k=2   1, -1,   3
  k=3   0,  3,  -3,   4
  k=4   1, -2,   7,  -6,   5
  k=5   0,  4,  -8,  14, -10,   6
  k=6   1, -3,  13, -21,  25, -15,   7
  k=7   0,  5, -15,  35, -45,  41, -21,   8
  k=8   1, -4,  21, -49,  81, -85,  63, -28,   9
  ...

Cf. A000027, A000032, A000045, A000110, A000217.
Cf. A001045, A004006, A032346, A051744, A052875.
Cf. A084416, A130595, A145766.
Cf. A027642, A164555 (Numerators and denominators of Bernoulli numbers).
Cf. A001008, A002805 (Numerators and denominators of harmonic numbers).
Cf. A344037, A372245.
Sequences below will be obtained by evaluation of the associated polynomials:
Cf. A000295, A000392, A000975, A016208.
Cf. A016218, A016228, A016241, A016249.
Cf. A016256, A016261, A249997.

A341091(n, k) = sum(m=n, k,(-1)^(m+n)*binomial(m+1, n))

A341091(n, k) = (1/2)*(-1)^n*(2*(-1)^k*binomial(2+k, n)*hypergeom([1,k+3],k+3-n,-1)+(-1/2)^n*(2^(n+1)-1)) \\ Thomas Scheuerle, Apr 29 2024

b(0)*T(0, m) + b(1)*T(1, m) + ... + b(m)*T(m, m)
  = Sum_{j=0..m} Sum_{n=0..m-j} Sum_{k=0..n} (-1)^k*binomial(n, k)*b(j+n-k)
  = Sum_{n=0..m} b(n)*Sum_{j=n..m}(-1)^(j+n)*binomial(j+1, n).
T(n, k) = Sum_{m=n..k}(-1)^(m+n)*binomial(m+1, n).
T(n, k) = (1/2)*(-1)^n*(2*(-1)^k*binomial(2+k, n)*Hypergeometric2F1(1, k+3, k+3-n, -1)+(-1/2)^n*(2^(n+1) - 1)), where Hypergeometric2F1 is the Gaussian hypergeometric function 2F1 as defined in Mathematica. - Thomas Scheuerle, Apr 29 2024
T(k, k) = A000027(k+1) The positive integers.
|T(k-1, k)| = A000217(k) The triangular numbers.
T(k-2, k) = A004006(k).
|T(k-3, k)| = A051744(k).
T(0, k*2) = 1.
T(0, k*2 + 1) = 0.
T(1, k*2 + 1) = k + 2.
T(1, k*2 + 2) = -(k + 1).
T(n, k) with constant n and variable k, a linear recurrence relation with characteristic polynomial (x-1)*(x+1)^(n+1).
Sum_{n=0..k} T(n, k)*B_n = 1. B_n is the n-th Bernoulli number with B_1 = 1/2. B_n = A164555(n)/A027642(n).
Sum_{n=0..k} T(n, k)*(1 - B_n) = k.
Sum_{n=0..k} T(n, k)*(2*n - 3+3*B_n) = k^2.
Sum_{n=0..k} T(n, k)*A032346(n) = A032346(k+1).
From Thomas Scheuerle, Apr 29 2024: (Start)
Sum_{n=0..k} T(n, k)*A000110(n+1) = A000110(k+2) - 1.
Sum_{n=0..k} T(n, k)*(1/(1+n)) = H(1+floor(k/2)), where H(k) is the harmonic number A001008(k)/A002805(k). (End)
Sum_{n=0..k} T(n, k)*c(n) = c(k). C(k) = {-1, 0, 1/2, 1/2, 1/8, -7/20, ...} this sequence of rational numbers can be defined recursively: c(0) = -1, c(m) = (-c(m-1) + Sum_{k=0..m-1} A130595(m+1, k)*c(k))/m.
  c(m) is an eigensequence of this transformation, all eigensequences are c(m) multiplied by any factor.
Sum_{n=0..k} T(n, k)*A000045(n) = 2*(A000045(2*floor((k+1)/2) - 1) - 1). A000045 are the Fibonacci numbers.
Sum_{n=0..k} T(n, k)*A000032(n) = A000032(2*floor(k/2)+2) - 2. A000032 are the Lucas numbers.
Sum_{n=0..k} T(n, k)*A001045(n) = A145766(floor((k+1)/2)). A001045 is the Jacobsthal sequence.
This sequence acting as an operator onto a monomial n^w:
Sum_{n=0..k} T(n, k)*n^w = (1/(w+1))*k^(w+1) + Sum_{v=1..w} ((v+B_v)*(w)_v/v!)*k^(w+1-v) - A052875(w) + O_k(w) (w)_v is the falling factorial. If k > w-1 then O_k(w) = 0. If k <= w-1 then O_k(w) is A084416(w, 2+k), the sequence with the exponential generating function: (e^x-1)^(2+k)/(2-e^x).
From Thomas Scheuerle, Apr 29 2024: (Start)
This sequence acting by its inverse operator onto a monomial k^w:
Sum_{n=0..k} T(n, k)*( Sum_{m=0..k} ((-1)^(1+m+k)*binomial(k, m)*(2^(k-m) - 1)*n^m + A344037(m)*B_n) ) = k^w - A372245(w, k+3), note that A372245(w, k+3) = 0 if k+3 > w. B_n is the n-th Bernoulli number with B_1 = 1/2.
How this sequence will act as an operator onto a Dirichlet series may be developed by the formulas below:
Sum_{n=0..k} T(n, k)*2^n = A000295(k+2).
Sum_{n=0..k} T(n, k)*3^n = A000392(k+3).
Sum_{n=0..k} T(n, k)*4^n = A016208(k).
Sum_{n=0..k} T(n, k)*5^n = A016218(k).
Sum_{n=0..k} T(n, k)*6^n = A016228(k).
Sum_{n=0..k} T(n, k)*7^n = A016241(k).
Sum_{n=0..k} T(n, k)*8^n = A016249(k).
Sum_{n=0..k} T(n, k)*9^n = A016256(k).
Sum_{n=0..k} T(n, k)*10^n = A016261(k).
Sum_{n=0..k} T(n, k)*m^n = m^2*m^k/(m-1) - (m-1)^2*(m-1)^k/(m-2) + 1/((m-1)*(m-2)), for m > 2.
Sum_{n=0..k} T(n, k)*( m*B_n + (m-1)*Sum_{t=1..m} t^n )*(1/m^2) = m^k, for m > 0. B_n is the n-th Bernoulli number with B_1 = 1/2.
Sum_{n=0..k} T(n, k) zeta(-n) = Sum_{j=0..k} (-1)^(1+j)/(2+j) = (-1)^(k+1)*LerchPhi(-1, 1, k+3) - 1 + log(2).
Sum_{n=0..k} T(k - n, k)*2^n = A000975(k+1)
Sum_{n=0..k} T(k - n, k)*3^n = A091002(k+2)
Sum_{n=0..k} T(k - n, k)*4^n = A249997(k). (End)

cp's OEIS Frontend

A140431 2*A094555(n).

Views

Author

Keywords

Links

Formula

Extensions

A091000 Number of closed walks of length n on the Petersen graph rooted at a given vertex.

Views

Author

Keywords

Comments

Links

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A091001 Number of walks of length n between adjacent nodes on the Petersen graph.

Views

Author

Keywords

References

Links

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A091005 Expansion of x^2/((1-2*x)*(1+3*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A249992 Expansion of 1/((1+x)*(1+2*x)*(1-3*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A341091 Triangle read by rows: Coefficients for calculation of the sum of all the finite differences from order zero to order k. Sum_{n=0..k} T(n, k)*b(n) = b(0) + b(1) + ... + b(k) + (b(1) - b(0)) + ... + (b(k) - b(k-1)) + ((b(2) - b(1)) - (b(1) - b(0))) + ... .

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A091005 Expansion of x^2/((1-2x)(1+3*x)).

A249992 Expansion of 1/((1+x)(1+2x)(1-3x)).