A116183 -id:A116183 - cp's OEIS frontend

A020701 Pisot sequences E(3,5), P(3,5).

3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

David W. Wilson

Number of meaningful differential operations of the (n+1)-th order on the space R^3. - Branko Malesevic, Feb 29 2004

Pisano period lengths: A001175. - R. J. Mathar, Aug 10 2012

			Meaningful second-order differential operations appear in the form of five compositions as follows: 1. div grad f 2. curl curl F 3. grad div F 4. div curl F (=0) 5. curl grad f (=0)
Meaningful third-order differential operations appear in the form of eight compositions as follows: 1. grad div grad f 2. curl curl curl F 3. div grad div F 4. div curl curl F (=0) 5. div curl grad f (=0) 6. curl curl grad f (=0) 7. curl grad div F (=0) 8. grad div curl F (=0)

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Branko 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.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A020695 and hence A000045. See A008776 for definitions of Pisot sequences.

Cf. A039834, A020695, A071679, A116183.

[Fibonacci(n-4): n in [8..80]]; // Vincenzo Librandi, Jul 12 2015

CoefficientList[Series[(-2 z - 3)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{3,5},40] (* Harvey P. Dale, Apr 22 2013 *)

a(n)=fibonacci(n+4) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = Fib(n+4). a(n) = a(n-1) + a(n-2).
a(n) = A020695(n+1). - R. J. Mathar, May 28 2008
G.f.: (3+2*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-7+3*sqrt(5))+(1+sqrt(5))^n*(7+3*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016
E.g.f.: (7*sqrt(5)*sinh(sqrt(5)*x/2) + 15*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, Jun 05 2016

A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

Original entry on oeis.org

3, 6, 16, 26, 84, 126, 424, 610, 2068, 2936, 9816, 13884, 45608, 64750, 208336, 297570, 938676, 1351492, 4181752, 6071028, 18454648, 27023598, 80796336, 119300636, 351331464, 522981328, 1518742384, 2278188504, 6531607248, 9869753934, 27963677600, 42547990626

Offset: 1

Jonathan Vos Post, Apr 09 2007, Jun 08 2007

nonn

			a(1) = 3 = A020701(1) is number of meaningful differential operations of the first order on the space R^3, namely {div, grad, curl}.
a(2) = 6 = A090989(2) is number of meaningful differential operations of the 2nd order on the space R^4 (some of them are identically zero though).
a(3) = 16 = A090990(3) is number of meaningful differential operations of the 3rd order on the space R^5.

R. Bott, L. W. Tu, Differential forms in algebraic topology, New York: Springer, 1982.

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33; arXiv:0704.0750 [math.DG], 2007.

Main diagonal of A116183.

Cf. A020701, A090989-A090995, A129638, A129639.

r[n_] := Table[Boole[j == i + 1 || i + j == n + 1], {i, n}, {j, n}];
Table[Total@Total@If[n == 1, IdentityMatrix[3], MatrixPower[r[n+2], n-1]], {n, 10}]
(* Andrey Zabolotskiy, Apr 30 2021 *)

Corrected from 8th term onwards. It appears the 8th and 9th terms listed were incorrectly taken from A000045 with two numbers concatenated together, whereas the 8th, 9th and 10th terms should have been the 8th term of A090995, the 9th of A129638 and the 10th of A129639. Joseph Myers, Dec 23 2008

Name and examples corrected, terms a(11) and beyond added by Andrey Zabolotskiy, Apr 30 2021

cp's OEIS Frontend

A020701 Pisot sequences E(3,5), P(3,5).

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