A346005 -id:A346005 - cp's OEIS frontend

A346004 If n even then n otherwise ((n+1)/2)^2.

0, 1, 2, 4, 4, 9, 6, 16, 8, 25, 10, 36, 12, 49, 14, 64, 16, 81, 18, 100, 20, 121, 22, 144, 24, 169, 26, 196, 28, 225, 30, 256, 32, 289, 34, 324, 36, 361, 38, 400, 40, 441, 42, 484, 44, 529, 46, 576, 48, 625, 50, 676, 52, 729, 54, 784, 56, 841, 58, 900, 60, 961, 62, 1024, 64, 1089

Offset: 0

N. J. A. Sloane, Jul 25 2021

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be seen in the circled numbers at foot of page 63.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A125714, A346005-A346007, A346595.

A346004[n_] := If[OddQ[n], (n+1)^2/4, n]; Array[A346004, 100, 0] (* or *)
Riffle[#-2, #^2/4] & [Range[2, 100, 2]] (* Paolo Xausa, Aug 28 2024 *)

def A346004(n): return ((n+1)//2)**2 if n % 2 else n # Chai Wah Wu, Jul 25 2021

G.f.: x*(-1-2*x-x^2+2*x^3) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Aug 05 2021

a(n) = ((n^2 + 6*n + 1) - (n-1)^2*(-1)^n)/8. - Aaron J Grech, Aug 27 2024

A346007 Let b=5. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

Original entry on oeis.org

0, 1, 5, 10, 10, 5, 32, 80, 80, 40, 10, 243, 405, 270, 90, 15, 1024, 1280, 640, 160, 20, 3125, 3125, 1250, 250, 25, 7776, 6480, 2160, 360, 30, 16807, 12005, 3430, 490, 35, 32768, 20480, 5120, 640, 40, 59049, 32805, 7290, 810, 45, 100000, 50000, 10000, 1000, 50

Offset: 0

N. J. A. Sloane, Jul 25 2021

nonn

These are the numbers that would arise if the Moessner construction on page 64 of Conway-Guy's "Book of Numbers" were extended to the fifth powers.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. See pp. 63-64.

Setting b = 2, 3, or 4 gives A346004, A346005, and A346006.

Cf. A125714, A346595.

f:=proc(n,b) local i;
for i from 0 to b-1 do
if ((n+i) mod b) = 0 then return(binomial(b,i+1)*((n+i)/b)^(i+1)); fi;
od;
end;
[seq(f(n,5),n=0..80)];

from sympy import binomial
def A346007(n):
    i = (5-n)%5
    return binomial(5,i+1)*((n+i)//5)**(i+1) # Chai Wah Wu, Jul 25 2021

A346595 Successive numbers arising from the Moessner construction of the sequence A010790 (n!*(n+1)!) on pages 64, 65 of Conway-Guy's "Book of Numbers".

Original entry on oeis.org

1, 2, 5, 4, 12, 40, 51, 31, 9, 144, 564, 904, 769, 376, 106, 16, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36, 3628800, 18299520, 40691952, 52965360, 45165064, 26726896, 11323991, 3487055, 782187, 126483, 14357, 1085, 49

Offset: 1

N. J. A. Sloane, Jul 25 2021

The circled numbers 5, 31, 106, 270, 575, 1085, ... in the second row of the display at the foot of page 64 are (essentially) A212523.

This sequence can also be represented as a triangle of numbers where the rows have lengths 1, 3, 5, 7, ... - Jinyuan Wang, Aug 06 2021

			As a triangle, this is:
1,
2, 5, 4,
12, 40, 51, 31, 9,
144, 564, 904, 769, 376, 106, 16,
2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25,
86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36,
...

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be obtained by reading the successive circled numbers in the tableau at the foot of page 64.

Cf. A010790, A125714, A212523, A346004, A346005, A346006, A346007.

More terms from Jinyuan Wang, Aug 06 2021

A346006 Successive numbers arising from the Moessner construction of the sequence of fourth powers on page 64 of Conway-Guy's "Book of Numbers".

Original entry on oeis.org

0, 1, 4, 6, 4, 16, 32, 24, 8, 81, 108, 54, 12, 256, 256, 96, 16, 625, 500, 150, 20, 1296, 864, 216, 24, 2401, 1372, 294, 28, 4096, 2048, 384, 32, 6561, 2916, 486, 36, 10000, 4000, 600, 40, 14641, 5324, 726, 44, 20736, 6912, 864, 48, 28561, 8788, 1014, 52, 38416, 10976, 1176, 56, 50625, 13500, 1350, 60

Offset: 0

N. J. A. Sloane, Jul 25 2021

nonn

a(4*k+1) = (k+1)^2 for k >= 0.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be obtained by reading the successive circled numbers in the second display on page 64.

Cf. A125714, A346004, A346005, A346595.

f:=proc(n,b) local i;
for i from 0 to b-1 do
if ((n+i) mod b) = 0 then return(binomial(b,i+1)*((n+i)/b)^(i+1)); fi;
od;
end;
[seq(f(n,3),n=0..60)];

from sympy import binomial
def A346006(n):
    i = (4-n)%4
    return binomial(4,i+1)*((n+i)//4)**(i+1) # Chai Wah Wu, Jul 25 2021

Let b=4. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

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A346004 If n even then n otherwise ((n+1)/2)^2.

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A346007 Let b=5. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

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A346595 Successive numbers arising from the Moessner construction of the sequence A010790 (n!*(n+1)!) on pages 64, 65 of Conway-Guy's "Book of Numbers".

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A346006 Successive numbers arising from the Moessner construction of the sequence of fourth powers on page 64 of Conway-Guy's "Book of Numbers".

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