A016153 -id:A016153 - cp's OEIS frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.

A102902 a(n) = 9a(n-1) - 16a(n-2), with a(0) = 1, a(1) = 9.

Original entry on oeis.org

1, 9, 65, 441, 2929, 19305, 126881, 833049, 5467345, 35877321, 235418369, 1544728185, 10135859761, 66507086889, 436390025825, 2863396842201, 18788331166609, 123280631024265, 808912380552641, 5307721328585529

Offset: 0

Paul Barry, Jan 17 2005

Indranil Ghosh, Table of n, a(n) for n = 0..1221
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5, Journal of Integer Sequences, Vol. 17 (2014).
Index entries for linear recurrences with constant coefficients, signature (9,-16).

Cf. A002450, A004187, A016153, A099459.

[4^n*Evaluate(ChebyshevSecond(n+1), 9/8): n in [0..30]]; // G. C. Greubel, Dec 09 2022

LinearRecurrence[{9,-16},{1,9},20] (* Harvey P. Dale, Jul 28 2016 *)

[lucas_number1(n,9,16) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

G.f.: 1/(1-9*x+16*x^2).
a(n) = Sum_{k=0..n} binomial(2*n-k+1, k)*4^k.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-16)^k*9^(n-2*k).
a(n) = 4^n * ChebyshevU(n, 9/8). - G. C. Greubel, Dec 09 2022
From Peter Bala, Jul 23 2025: (Start)
a(n) := ((9 + sqrt(17))^(n+1) - (9 - sqrt(17))^(n+1))/(2^(n+1)*sqrt(17)).
The following products telescope:
Product_{k >= 1} 1 + 4^k/a(k) = (1 + sqrt(17))/2.
Product_{k >= 1} 1 - 4^k/a(k) = (1 + sqrt(17))/18.
Product_{k >= 1} 1 + (-4)^k/a(k) = (17 + sqrt(17))/34.
Product_{k >= 1} 1 - (-4)^k/a(k) = (17 + sqrt(17))/18. (End)

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005

Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 1, 1, 1, n].
The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
  1
  3 + 2*k
  13 + 9*k + 3*k^2
  51 + 52*k + 18*k^2 + 4*k^3
  ...

			Array begins:
  0, 1,  3, 13,  51,  205, ...
  0, 1,  5, 25, 125,  625, ...
  0, 1,  7, 43, 259, 1555, ...
  0, 1,  9, 67, 477, 3355, ...
  0, 1, 11, 97, 803, 6505, ...
  ...

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M
for(k=0,10, for(i=0,10,print1((MM(5,k)^i)[1,2],","));print())

p(n,k)=((n+4)^k-(n-1)^k)/5
for(k=0,10, for(i=0,10,print1(p(k,i),","));print())

for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print())

cp's OEIS Frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

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A102902 a(n) = 9a(n-1) - 16a(n-2), with a(0) = 1, a(1) = 9.

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A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

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PARI

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cp's OEIS Frontend

A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A102902 a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

PARI

A102902 a(n) = 9a(n-1) - 16a(n-2), with a(0) = 1, a(1) = 9.