A143025 -id:A143025 - cp's OEIS frontend

A173435 Inverse binomial transform of A143025, assuming offset zero there.

8, -6, 12, -25, 52, -106, 212, -420, 832, -1656, 3312, -6640, 13312, -26656, 53312, -106560, 212992, -425856, 851712, -1703680, 3407872, -6816256, 13632512, -27264000, 54525952, -109049856, 218099712, -436203520, 872415232, -1744838656, 3489677312, -6979338240

Offset: 0

Paul Curtz, Feb 18 2010

sign

Inverse binomial transform of 8, 2, 8, 1, 8, 2 ,8, 1,... with a(0)=8, a(1)=2 etc.

Index entries for linear recurrences with constant coefficients, signature (-4, -6, -4).

Join[{8},LinearRecurrence[{-4,-6,-4},{-6,12,-25},40]] (* Harvey P. Dale, Sep 25 2013 *)

a(n)= -4*a(n-1) -6*a(n-2) -4*a(n-3), n>3. G.f.: (26*x+36*x^2+19*x^3+8)/( (2*x+1) * (2*x^2+2*x+1)). [R. J. Mathar, Mar 10 2010]

a(n+1) +2*a(n) = (-1)^(n+1)*A009545(n-1), n > 0.

Extended by R. J. Mathar, Mar 10 2010

A243883 Numerator of circle radius r(n) at constant unit length sagitta and chord length = n.

Original entry on oeis.org

5, 1, 13, 5, 29, 5, 53, 17, 85, 13, 125, 37, 173, 25, 229, 65, 293, 41, 365, 101, 445, 61, 533, 145, 629, 85, 733, 197, 845, 113, 965, 257, 1093, 145, 1229, 325, 1373, 181, 1525, 401, 1685, 221, 1853, 485, 2029, 265, 2213, 577, 2405, 313, 2605, 677, 2813, 365, 3029

Offset: 1

Kival Ngaokrajang, Jun 13 2014

Denominator of circle radius r(n) is A143025(n+2). The integral radius appearing at n = 2, 6, 10, 14, ..., = 1, 5, 13, 25, ..., respectively which is A001844(n/4 - 1/2). Floor (r(n)) = A001971(n). For the case of sagitta = n and chord length = 1, the numerator and the denominator will be A053755(n) and A008590(n) respectively. See illustration in links.

Colin Barker, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration for n = 1..5
Wikipedia, Sagitta

Cf. A143025, A001844, A001971.

PARI
```
a(n) = numerator(n^2/8+1/2);
```

a(n) = numerator(n^2/8 + 1/2).

Empirical g.f.: -x*(x^11 +5*x^10 +x^9 +13*x^8 +2*x^7 +14*x^6 +2*x^5 +14*x^4 +5*x^3 +13*x^2 +x +5) / ((x -1)^3*(x +1)^3*(x^2 +1)^3). - Colin Barker, Jan 17 2015

A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

Original entry on oeis.org

9, 9, 81, 18, 225, 81, 441, 72, 729, 225, 1089, 162, 1521, 441, 2025, 288, 2601, 729, 3249, 450, 3969, 1089, 4761, 648, 5625, 1521, 6561, 882, 7569, 2025, 8649, 1152, 9801, 2601, 11025, 1458, 12321, 3249, 13689, 1800, 15129, 3969, 16641, 2178, 18225

Offset: 1

Paul Curtz, Nov 20 2008

nonn

The associated terms of the n-th main series of the Hydrogen energy spectrum are A000290(3), A061038(6), A061040(9), A061042(12), A061044(15), A061046(18), A061048(21), A061050(24), etc.

All numbers are multiples of 9.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1).

Cf. A143025 with a similar principle of construction.

Cf. A291050.

Denominator/@(8/(9Range[50]^2))  (* Harvey P. Dale, Mar 15 2011 *)

Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Sep 14 2022

Stratified definition, corrected indices, extended, R. J. Mathar, Dec 10 2008

A143024 Triangle read by rows: T(n,k) is the number of non-crossing connected graphs on n nodes on a circle having root (a distinguished node) of degree 1 and having k edges (n >= 2, 1 <= k <= 2n-4).

Original entry on oeis.org

1, 0, 2, 0, 0, 7, 2, 0, 0, 0, 30, 20, 4, 0, 0, 0, 0, 143, 156, 65, 10, 0, 0, 0, 0, 0, 728, 1120, 720, 224, 28, 0, 0, 0, 0, 0, 0, 3876, 7752, 6783, 3192, 798, 84, 0, 0, 0, 0, 0, 0, 0, 21318, 52668, 58520, 36960, 13860, 2904, 264, 0, 0, 0, 0, 0, 0, 0, 0, 120175, 354200, 478170

Offset: 2

Emeric Deutsch, Jul 31 2008

Row n contains 2n-4 terms, the first n-2 of which are 0.
Row sums yield A089436.
T(n,n-1) = A006013(n-2).
Sum_{k=2..2n-4} k*T(n,k) = A143025.

			T(3,2)=2 because we have {AB,BC} and {AC, BC} (A is the root).
Triangle starts:
  1;
  0,   2;
  0,   0,   7,   2;
  0,   0,   0,  30,  20,   4;
  0,   0,   0,   0, 143, 156,  65,  10;

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Cf. A007297, A089436, A006013, A143025.

T:=proc(n,k) options operator, arrow: 2*binomial(k-2,n-3)*binomial(3*n-5,2*n-k-4)/(n-2) end proc: 1; for n from 3 to 10 do 0, seq(T(n,k),k=2..2*n-4) end do; % yields sequence in triangular form

T(n,k) = 2*binomial(k-2, n-3)*binomial(3n-5, 2n-k-4)/(n-2) (n >= 3, 2 <= k <= 2n-4); T(2,1)=1; T(2,k)=0 (k >= 2).

The trivariate g.f. G=G(t,s,z) for non-crossing connected graphs on nodes on a circle, with respect to number of nodes (marked by z), number of edges (marked by t) and degree of root (marked by s) is G=z + tszg^2/[z-ts(g - z + g^2)], where g=g(t,z) satisfies tg^3 + tg^2 - (1 + 2t)zg +(1 + t)z^2 = 0 (see Domb & Barrett, Eq. (47); Flajolet & Noy, Eq. (18)).

A173559 a(n)= +2a(n-2) +4a(n-3), n>3.

Original entry on oeis.org

1, -6, -13, -27, -50, -106, -208, -412, -840, -1656, -3328, -6672, -13280, -26656, -53248, -106432, -213120, -425856, -851968, -1704192, -3407360, -6816256, -13631488, -27261952, -54528000, -109049856, -218103808, -436211712, -872407040, -1744838656

Offset: 0

Paul Curtz, Feb 21 2010

Generated by scanning the diagonal of the table generated by A143025 in the top row followed by higher order differences in the other rows:
1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8,...
7, -6, 6, -7, 7, -6, 6, -7, 7, -6, 6, -7, 7,...
-13, 12, -13, 14, -13, 12, -13, 14, -13, 12,..
25, -25, 27, -27, 25, -25, 27, -27, 25, -25,..
-50, 52, -54, 52, -50, 52, -54, 52, -50, 52, ...
102, -106, 106, -102, 102, -106, 106, -102,...

Index entries for linear recurrences with constant coefficients, signature (0,2,4).

LinearRecurrence[{0,2,4},{1,-6,-13,-27},30] (* Harvey P. Dale, Jan 27 2019 *)

a(n) = ( -13*2^n-2*A009116(n))/4, n>0.
a(n+1)-2*a(n) = -A137429(n-2), n>1.
G.f.: (6*x+15*x^2+19*x^3-1)/( (2*x-1) *(2*x^2+2*x+1)).

cp's OEIS Frontend

A173435 Inverse binomial transform of A143025, assuming offset zero there.

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A243883 Numerator of circle radius r(n) at constant unit length sagitta and chord length = n.

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A152018 Denominator of 1/n^2-1/(3n)^2 or of 8/(9n^2).

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A143024 Triangle read by rows: T(n,k) is the number of non-crossing connected graphs on n nodes on a circle having root (a distinguished node) of degree 1 and having k edges (n >= 2, 1 <= k <= 2n-4).

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A173559 a(n)= +2*a(n-2) +4*a(n-3), n>3.

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A173559 a(n)= +2a(n-2) +4a(n-3), n>3.