A199537 -id:A199537

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A199536

The first column in Clark Kimberling's even first column Stolarsky array (beginning column count at 1).

+10
3

1, 4, 6, 10, 12, 14, 16, 20, 22, 26, 28, 30, 32, 36, 38, 40, 42, 46, 48, 52, 54, 56, 58, 62, 64, 68, 70, 72, 74, 78, 80, 82, 84, 88, 90, 94, 96, 98, 100, 104, 106, 108, 110, 114, 116, 120, 122, 124, 126, 130, 132, 136, 138, 140, 142, 146, 148, 150, 152, 156

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

FORMULA

Define Phi = (1+sqrt(5))/2, then a(1) = 1, a(2*n) = 2*floor(n*Phi) + 2*n, a(2*n+1) = 2*floor(n*Phi) + 2*n + 2.

a(n) = A199535(n, n). - G. C. Greubel, Jun 22 2022

MATHEMATICA

a[n_]:= If[Mod[n, 2]==0, 2*Floor[(n/2)*GoldenRatio] +n, 2*Floor[(n-1)/2*GoldenRatio] +n+1] -Boole[n==1];

Table[a[n], {n, 80}] (* G. C. Greubel, Jun 22 2022 *)

PROG

(SageMath)

def A199536(n):

if (n==1): return 1

elif (n%2==0): return 2*floor(n*golden_ratio/2) + n

else: return 2*floor((n-1)*golden_ratio/2) +n+1

[A199536(n) for n in (1..80)] # G. C. Greubel, Jun 22 2022

CROSSREFS

Cf. A199535, A199537.

KEYWORD

nonn

AUTHOR

Casey Mongoven, Nov 07 2011

STATUS

approved

A199535

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..66.

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

FORMULA

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)

EXAMPLE

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)):

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;

CROSSREFS

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.

Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.

Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

KEYWORD

nonn,tabl

AUTHOR

Casey Mongoven, Nov 07 2011

EXTENSIONS

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

STATUS

approved

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