A022088 -id:A022088 - cp's OEIS frontend

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

1, 2, 4, 3, 7, 6, 5, 11, 9, 10, 8, 18, 15, 17, 12, 13, 29, 24, 27, 19, 14, 21, 47, 39, 44, 31, 23, 16, 34, 76, 63, 71, 50, 37, 25, 20, 55, 123, 102, 115, 81, 60, 41, 33, 22, 89, 199, 165, 186, 131, 97, 66, 53, 35, 26, 144, 322, 267, 301, 212, 157, 107, 86, 57, 43, 28

Offset: 1

Casey Mongoven, Nov 07 2011

The rows of the array can be seen to have the form A(n, k) = p(n)*Fibonacci(k) + q(n)*Fibonacci(k+1) where p(n) is the sequence {0, 1, 3, 3, 3, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, ...}{n >= 1} and q(n) is the sequence {1, 3, 3, 7, 2, 9, 9, 13, 13, 17, 17, 19, 19, 23, 23, 25, ...}{n >= 1}. - G. C. Greubel, Jun 23 2022

			The even first column stolarsky array (EFC array), northwest corner:
  1......2.....3.....5.....8....13....21....34....55....89...144 ... A000045;
  4......7....11....18....29....47....76...123...199...322...521 ... A000032;
  6......9....15....24....39....63...102...165...267...432...699 ... A022086;
  10....17....27....44....71...115...186...301...487...788..1275 ... A022120;
  12....19....31....50....81...131...212...343...555...898..1453 ... A013655;
  14....23....37....60....97...157...254...411...665..1076..1741 ... A000285;
  16....25....41....66...107...173...280...453...733..1186..1919 ... A022113;
  20....33....53....86...139...225...364...589...953..1542..2495 ... A022096;
  22....35....57....92...149...241...390...631..1021..1652..2673 ... A022130;
Antidiagonal rows (T(n, k)):
   1;
   2,   4;
   3,   7,   6;
   5,  11,   9,  10;
   8,  18,  15,  17, 12;
  13,  29,  24,  27, 19, 14;
  21,  47,  39,  44, 31, 23, 16;
  34,  76,  63,  71, 50, 37, 25, 20;
  55, 123, 102, 115, 81, 60, 41, 33, 22;

Clark Kimberling, The first column of an interspersion, Fibonacci Quarterly 32 (1994), pp. 301-314.

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.
Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.
Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

From G. C. Greubel, Jun 23 2022: (Start)
T(n, 1) = A000045(n+1).
T(n, 2) = A000032(n+1), n >= 2.
T(n, 3) = A022086(n) = A097135(n), n >= 3.
T(n, 4) = A022120(n-2), n >= 4.
T(n, 5) = A013655(n-1), n >= 5.
T(n, 6) = A000285(n-2), n >= 6.
T(n, 7) = A022113(n-4), n >= 7.
T(n, 8) = A022096(n-4), n >= 8.
T(n, 9) = A022130(n-6), n >= 9.
T(n, 10) = A022098(n-5), n >= 10.
T(n, 11) = A022095(n-7), n >= 11.
T(n, 12) = A022121(n-8), n >= 12.
T(n, 13) = A022388(n-10), n >= 13.
T(n, 14) = A022122(n-10), n >= 14.
T(n, 15) = A022097(n-10), n >= 15.
T(n, 16) = A022088(n-10), n >= 16.
T(n, 17) = A022390(n-14), n >= 17.
T(n, n) = A199536(n).
T(n, n-1) = A199537(n-1), n >= 2. (End)

More terms added by G. C. Greubel, Jun 23 2022

A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.

Original entry on oeis.org

3, 1, 4, 1, 6, 4, 0, 7, 8, 6, 4, 9, 9, 8, 7, 3, 8, 1, 7, 8, 4, 5, 5, 0, 4, 2, 0, 1, 2, 3, 8, 7, 6, 5, 7, 4, 1, 2, 6, 4, 3, 7, 1, 0, 1, 5, 7, 6, 6, 9, 1, 5, 4, 3, 4, 5, 6, 2, 5, 3, 8, 3, 4, 7, 2, 4, 6, 3, 1, 2, 5, 5, 5, 3, 8, 2, 6, 8, 2, 9, 3, 9, 6, 4, 8, 6, 4, 8, 6, 4, 5, 0, 2, 7, 2, 6, 9, 3, 6, 4, 9, 8, 1, 7, 0, 4, 9, 0, 5, 6, 9, 0, 4, 6

Offset: 1

Grant Garcia, Jan 16 2011

This is an approximation to Pi.

6*(phi+1)/5 is not equal to Pi, although some have claimed this (see Dudley). - Kellen Myers, Oct 04 2013

			3.141640786499873817845504201238765741264371015766915434562538347246312555382...

Underwood Dudley, Mathematical Cranks, MAA 1992, pp. 247, 292.
Alfred S. Posamentier and Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, New York, Prometheus Books, 2007, p. 119.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hung Viet Chu, Square the Circle in One Minute, arXiv:1908.01202 [math.GM], 2019.
Futility Closet, A Surprise Visitor

Cf. A022088, A022089, A001622, A000796.

(3/10)*(1 + Sqrt(5))^2; // G. C. Greubel, Jan 17 2018

RealDigits[(6/5)GoldenRatio^2, 10, 100][[1]] (* Alonso del Arte, Apr 09 2012 *)

3*(3+sqrt(5))/5 \\ Charles R Greathouse IV, Sep 13 2013

Limit of A022089(n+2)/A022088(n) as n approaches infinity.
6*(phi + 1)/5 = 6*phi^2/5 = 3(3 + sqrt(5))/5 = 9/5 + sqrt(9/5). - Charles R Greathouse IV, Sep 13 2013
Equals 24/(5-sqrt(5))^2. - Joost Gielen, Sep 20 2013

A371843 a(n) = 5*Fibonacci(n) + (-1)^n.

Original entry on oeis.org

1, 4, 6, 9, 16, 24, 41, 64, 106, 169, 276, 444, 721, 1164, 1886, 3049, 4936, 7984, 12921, 20904, 33826, 54729, 88556, 143284, 231841, 375124, 606966, 982089, 1589056, 2571144, 4160201, 6731344, 10891546, 17622889, 28514436, 46137324, 74651761, 120789084, 195440846

Offset: 0

Paul Curtz, Apr 08 2024

			a(3) = 2*4 + 1 = 9. Also a(3) = -1 + 10*1 = 9.

Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000045, A022088, A033999, A097133.

LinearRecurrence[{0, 2, 1}, {1, 4, 6}, 50] (* Amiram Eldar, Apr 11 2024 *)

a(n) = a(n-2) + A022088(n-1).
a(n) = 2*a(n-2) + a(n-3).
a(n) = A022088(n) + A033999(n).
a(n) = - a(n-3) + 10*A000045(n-1) for n >= 3.
G.f.: (1+2*x)^2/((1+x)*(1-x-x^2)). - Joerg Arndt, Apr 13 2024

A170930 G(n,1) with n index G(n,i)=n(G(n,i-1)+G(n,i-2))=(a^i-b^i)d where d=sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

Original entry on oeis.org

0, 21, 63, 252, 945, 3591, 13608, 51597, 195615, 741636, 2811753, 10660167, 40415760, 153227781, 580930623, 2202475212, 8350217505, 31658078151, 120024886968, 455048895357, 1725221346975, 6540810726996, 24798096221913

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010

nonn

n=1 A022088 n=2 12*A002605 n=3 above n=4 ... new

			G(n,0)=0 G(n,1)=n*(n+4)

see above

a(n) = 3*a(n-1)+3*a(n-2) = 21*A030195(n). G.f.: 21*x/(1-3*x-3*x^2). [From R. J. Mathar, Feb 05 2010]

More terms from R. J. Mathar, Feb 05 2010

A172011 a(n) = 12*A002605(n).

Original entry on oeis.org

0, 12, 24, 72, 192, 528, 1440, 3936, 10752, 29376, 80256, 219264, 599040, 1636608, 4471296, 12215808, 33374208, 91180032, 249108480, 680577024, 1859371008, 5079896064, 13878534144, 37916860416, 103590789120, 283015299072, 773212176384, 2112454950912

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

The case k=2 in a family of sequences a(n)=G(k,n), G(k,0)=0, G(k,1)=k*(k+4), G(k,n)=k*G(k,n-1)+k*G(k,n-2).
The Binet formula is G(k,n) = (c^n-b^n)*d where d=sqrt(k*(k+4)); c=(k+d)/2; b=(k-d)/2.
The generating functions are k*(k+4)*x/(1-k*x-k*x^2).
The case k=1 is A022088.

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

Cf. A002605, A022088.

LinearRecurrence[{2,2},{0,12},30] (* Harvey P. Dale, Mar 06 2023 *)

Binet formula: a(n) = 2*2^n*((-1+3^(1/2))^(-n)-(-1)^n*(1+3^(1/2))^(-n))*3^(1/2) .

G.f.: 12*x/(1-2*x-2*x^2). a(n) = 2*a(n-1)+2*a(n-2).

Edited and extended by R. J. Mathar, Jan 23 2010

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 2, 1, 2, 3, 1, 4, 3, 3, 2, 6, 5, 3, 4, 3, 10, 7, 5, 4, 5, 5, 16, 11, 7, 6, 5, 7, 8, 26, 17, 11, 8, 7, 7, 10, 13, 42, 27, 17, 12, 9, 9, 10, 15, 21, 68, 43, 27, 18, 13, 11, 12, 15, 23, 34, 110, 69, 43, 28, 19, 15, 14, 17, 23, 36, 55, 178, 111, 69, 44, 29, 21

Offset: 0

Philippe Deléham, Apr 07 2013

			Square array begins:
...0....2....2....4....6...10...16...26...42...
...1....3....3....5....7...11...17...27...43...
...1....3....3....5....7...11...17...27...43...
...2....4....4....6....8...12...18...28...44...
...3....5....5....7....9...13...19...29...45...
...5....7....7....9...11...15...21...31...47...
...8...10...10...12...14...18...24...34...50...
..13...15...15...17...19...23...29...39...55...
..21...23...23...25...27...31...37...47...63...
..34...36...36...38...40...44...50...60...76...
..55...57...57...59...61...65...71...81...97...
..89...91...91...93...95...99..105..115..131...
.144..146..146..148..150..154..160..170..186...
...

Cf. A000045

T(n,0) = A000045(n).
T(1,k) = A001588(k).
T(n,1) = T(n,2) = A157725(n).
T(n,3) = A157727(n).
T(n,n)= A022086(n) = 3*A000045(n).
T(n+1,n) = A000032(n+1) = A000204(n+1).
T(n+2,n) = A000285(n).
T(n+3,n) = A013655(n+1) = A001060(n+1).
T(n+4,n) = A021120(n).
T(n+5,n) = A022088(n+2) = 5*A000045(n+2).
T(n+6,n) = A022097(n+2).
T(n+7,n) = A022122(n+2).
T(n+8,n) = 3*A013655(n+2).
T(n+9,n) = A097657(n+2).
T(n+10,n) = A022118(n+4).
T(n,n+1) = A000045(n+3).
T(n,n+2) = A013655(n+1) = A001060(n+1).
T(n,n+3) = A000032(n+3).
T(n,n+4) = A022095(n+2).
T(n,n+5) = A022120(n+2).
T(n,n+6) = A022136(n+2).
T(n,n+7) = A022098(n+4).
T(n,n+8) = A022380(n+4).
T(n,n+9) = A206419(n+6).
Sum(T(n-k,k), 0<=k<=n) = 3*A000071(n+2).

cp's OEIS Frontend

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A180251 Decimal expansion of 6*(phi+1)/5, where phi is (1 + sqrt(5))/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A371843 a(n) = 5*Fibonacci(n) + (-1)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A170930 G(n,1) with n index G(n,i)=n*(G(n,i-1)+G(n,i-2))=(a^i-b^i)*d where d=sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A172011 a(n) = 12*A002605(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Views

Author

Keywords

Examples

Crossrefs

Formula

A170930 G(n,1) with n index G(n,i)=n(G(n,i-1)+G(n,i-2))=(a^i-b^i)d where d=sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.