login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A139714 a(n) = Sum_{ k >= 0} binomial(n,5*k+2). 12
0, 0, 1, 3, 6, 10, 15, 22, 36, 72, 165, 385, 859, 1807, 3614, 6995, 13380, 25773, 50559, 101118, 204820, 416405, 843756, 1698458, 3396916, 6765175, 13455325, 26789257, 53457121, 106914242, 214146295, 429124630, 859595529, 1720537327, 3441074654 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From Gary W. Adamson, Mar 14 2009: (Start)

M^n * [1,0,0,0,0] = [A139398(n), A139761(n), A139748(n), a(n), A133476(n)]

where M = a 5 X 5 matrix [1,1,0,0,0; 0,1,1,0,0; 0,0,1,1,0; 0,0,0,1,1; 1,0,0,0,1]

Sum of terms = 2^n. Example: M^6 = [7, 15, 20, 15, 7], sum = 2^6 = 64. (End)

{A139398, A133476, A139714, A139748, A139761} is the difference analog of the hyperbolic functions of order 5, {h_1(x), h_2(x), h_3(x), h_4(x), h_5 (x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 18 2017

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

LINKS

Table of n, a(n) for n=0..34.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

FORMULA

G.f.: -x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = round((2/5)*(2^(n-1)+phi^n*cos(Pi*(n-4)/5))), where phi is the golden ratio, round(x) is the integer nearest to x. - Vladimir Shevelev, Jun 18 2017

a(n+m) = a(n)*H_1(m) + H_2(n)*H_2(m) + H_1(n)*a(m) + H_5(n)*H_4(m) + H_4(n)*H_5(m), where H_1=A139398, H_2=A133476, H_4=A139748, H_5=A139761. - Vladimir Shevelev, Jun 18 2017

MAPLE

a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[4, 1]:

seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2015

MATHEMATICA

CoefficientList[Series[x^2 (x - 1)^2/((1 - 2 x) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

PROG

(PARI) a(n) = sum(k=0, n\5, binomial(n, 5*k+2)); \\ Michel Marcus, Dec 21 2015

(PARI) x='x+O('x^100); concat([0, 0], Vec(-x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)))) \\ Altug Alkan, Dec 21 2015

(MAGMA) [n le 5 select (n-2)*(n-1)/2 else 5*Self(n-1)- 10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 21 2015

CROSSREFS

Cf. A049016, A133476.

Sequence in context: A209231 A137358 A143963 * A063542 A262927 A122554

Adjacent sequences:  A139711 A139712 A139713 * A139715 A139716 A139717

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 13 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 21 00:36 EDT 2017. Contains 290855 sequences.