A191347 -id:A191347

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A191348

Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

+10
3

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0

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

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..55.

FORMULA

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019

EXAMPLE

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...

1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, ...

1, 2, 7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, ...

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...

1, 3, 14, 72, 376, 1968, 10304, 53952, 282496, 1479168, 7745024, ...

1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, ...

1, 3, 16, 90, 508, 2868, 16192, 91416, 516112, 2913840, 16450816, ...

1, 3, 17, 99, 577, 3363, 19601, 114243, 665857, 3880899, 22619537, ...

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...

1, 4, 26, 184, 1316, 9424, 67496, 483424, 3462416, 24798784, 177615776, ...

1, 4, 27, 196, 1433, 10484, 76707, 561236, 4106353, 30044644, 219825387, ...

1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, ...

1, 4, 29, 220, 1673, 12724, 96773, 736012, 5597777, 42574180, 323800109, ...

1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...

...

PROG

(PARI) T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));

matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

CROSSREFS

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.

Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.

Cf. A191347 which uses floor() in place of ceiling().

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, May 31 2011

STATUS

approved

A309852

Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = floor(sqrt(x)) and z = y-1+(y mod 2).

+10
1

2, 1, 2, 1, 1, 2, 1, 3, 3, 2, 1, 4, 9, 3, 2, 1, 7, 27, 11, 3, 2, 1, 11, 81, 36, 13, 3, 2, 1, 18, 243, 119, 45, 15, 5, 2, 1, 29, 729, 393, 161, 54, 25, 5, 2, 1, 47, 2187, 1298, 573, 207, 125, 27, 5, 2, 1, 76, 6561, 4287, 2041, 783, 625, 140, 29, 5, 2

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

OFFSET

0,1

COMMENTS

One of 4 related arrays (the others being A191347, A191348, and A309853) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309853 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

For each row n>=0 let x = 4*n+1, y = floor(sqrt(x)), T(n,0)=2, and T(n,1)=y-1+(y % 2)), then for each column k>=2: T(n, k-2)*((x-T(n, 1)^2)/4) + T(n, k-1)*T(n, 1). - Charles L. Hohn, Aug 23 2019

EXAMPLE

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ...

2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, ...

2, 3, 13, 45, 161, 573, 2041, 7269, 25889, 92205, 328393, ...

2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, ...

2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, ...

2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, ...

2, 5, 29, 155, 833, 4475, 24041, 129155, 693857, 3727595, 20025689, ...

2, 5, 31, 170, 943, 5225, 28954, 160445, 889087, 4926770, 27301111, ...

2, 5, 33, 185, 1057, 6025, 34353, 195865, 1116737, 6367145, 36302673, ...

...

PROG

(PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(x), z = y-1+(y % 2)); round(((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k);

matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019

(PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(x), z=y-1+(y % 2)); v=if(k==0, 2, k==1, z, mapget(m2, n)*((x-z^2)/4) + mapget(m1, n)*z); mapput(m2, n, if(mapisdefined(m1, n), mapget(m1, n), 0)); mapput(m1, n, v); v;

m1=Map(); m2=Map(); matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 26 2019

CROSSREFS

Row 2 is A000032, row 3 (except the first term) is A000244, row 4 is A006497, row 5 is A206776, row 6 is A172012, row 7 (except the first term) is A000351, row 8 is A087130.

Cf. A191347, A191348, and A309853.

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, Aug 20 2019

STATUS

approved

A309853

Array read by antidiagonals: ((z+sqrt(x))/2)^n + ((z-sqrt(x))/2)^n for columns n >= 0 and rows k >= 0, where x = 4*k+1 and y = ceiling(sqrt(x)) and z = y+1-(y mod 2).

+10
1

2, 1, 2, 1, 3, 2, 1, 7, 3, 2, 1, 18, 9, 5, 2, 1, 47, 27, 19, 5, 2, 1, 123, 81, 80, 21, 5, 2, 1, 322, 243, 343, 95, 23, 5, 2, 1, 843, 729, 1475, 433, 110, 25, 7, 2, 1, 2207, 2187, 6346, 1975, 527, 125, 39, 7, 2, 1, 5778, 6561, 27305, 9009, 2525, 625, 238, 41, 7

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

OFFSET

0,1

COMMENTS

One of 4 related arrays (the others being A191347, A191348, and A309852) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309852 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.

LINKS

Table of n, a(n) for n=0..64.

EXAMPLE

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, ...

2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ...

2, 5, 19, 80, 343, 1475, 6346, 27305, 117487, 505520, ...

2, 5, 21, 95, 433, 1975, 9009, 41095, 187457, 855095, ...

2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, ...

2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, ...

2, 7, 39, 238, 1471, 9107, 56394, 349223, 2162591, 13392022, ...

2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, ...

2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, ...

2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, ...

...

PROG

(PARI) T(k, n) = my(x = 4*k+1, y = ceil(sqrt(x)), z = y+1-(y % 2)); round(((z+sqrt(x))/2)^n + ((z-sqrt(x))/2)^n);

matrix(9, 9, k, n, T(k-1, n-1)) \\ Michel Marcus, Aug 22 2019

CROSSREFS

Row 2 is A005248, row 3 (except the first term) is A000244, row 4 is A228569, row 5 is A159289, row 6 is A003501, row 7 (except the first term) is A000351.

Cf. A191347, A191348, and A309852.

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, Aug 20 2019

STATUS

approved

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