A055270 -id:A055270 - cp's OEIS frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A055272 First differences of 7^n (A000420).

Original entry on oeis.org

1, 6, 42, 294, 2058, 14406, 100842, 705894, 4941258, 34588806, 242121642, 1694851494, 11863960458, 83047723206, 581334062442, 4069338437094, 28485369059658, 199397583417606, 1395783083923242, 9770481587462694

Offset: 0

Barry E. Williams, May 28 2000

Partial sum of A055270.
Conjecture in "Introduction à la théorie des nombres" by J. M. Deconinck and Armel Mercier: this is the period length of the fraction 1/7^n. For example 1/7^2=0.0204081632653061224489795918367346938775510204....with a period of 42 digits =6*7=a(2). The period of 1/7^3 has exactly 294=a(3) digits. - Benoit Cloitre, Feb 02 2002
Also phi(7^n), where phi is Euler's totient function. - Alonso del Arte, May 08 2006
For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3,4,5,6,7} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n) is the number of compositions of n when there are 6 types of each part. - Milan Janjic, Aug 13 2010
Apart from the first term, number of monic squarefree polynomials over F_7 of degree n. - Charles R Greathouse IV, Feb 07 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Jean-Marie De Koninck and Armel Mercier, Introduction à la théorie des nombres, Collection Universitaire de Mathématiques, Modulo, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A000420, A055270.

1, seq(6*7^(n-1), n=1..20); # G. C. Greubel, Mar 16 2020

Table[EulerPhi[7^n], {n, 0, 19}] (* Alonso del Arte, May 08 2006 *)

a(n)=round(7^n*6/7) \\ Charles R Greathouse IV, Feb 07 2012

[1]+[6*7^(n-1) for n in (1..20)] # G. C. Greubel, Mar 16 2020

G.f.: (1-x)/(1-7*x).
G.f.: 1/( 1 - 6*Sum(k>=1, x^k) ).
a(n) = 6*7^(n-1), a(0)=1.
E.g.f.: (1 + 6*exp(7*x))/7. - G. C. Greubel, Mar 16 2020

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A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

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