

A226903


Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.


8



5, 18, 53, 102, 197, 306, 491, 684, 989, 1290, 1745, 2178, 2813, 3402, 4247, 5016, 6101, 7074, 8429, 9630, 11285, 12738, 14723, 16452, 18797, 20826, 23561, 25914, 29069, 31770, 35375, 38448, 42533, 46002, 50597, 54486, 59621, 63954, 69659, 74460, 80765, 86058
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OFFSET

1,1


COMMENTS

Shiraishi's solutions to a^3 + b^3 + c^3 = d^3 are a = 3n^2; b = 6n^2  3n + 1 or 6n^2 + 3n + 1; c = 9n^3  6n^2 + 3n  1 or 9n^3 + 6n^2 + 3n, respectively, for n > 0; and d = c+1. See Smith and Mikami for a derivation.
Shiraishi's formulas show that the sequence is infinite. Hence the sequences A023042 (solutions to x^3 + y^3 + z^3 = w^3), A225908 (solutions to a^3 + b^3 = c^3  d^3), A225909 (solutions to a^3 + b^3 = (c+1)^3  c^3) and A226902 (numbers c in A225909) are also infinite.
Shiraishi's solution b = 6n^2 +/ 3n + 1 is the centered triangular numbers A005448 except 1.


REFERENCES

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.


LINKS

Table of n, a(n) for n=1..42.
David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233235.
Wikipedia (French), Shiraishi Nagatada
Wikipedia (German), Shiraishi Nagatada
Index entries for sequences related to sums of cubes


FORMULA

a(2n1) = 9n^3  6n^2 + 3n  1.
a(2n) = 9n^3 + 6n^2 + 3n.
G.f.: x*(5 + 13*x + 20*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1 + x)^3*(1  x)^4). [Bruno Berselli, Jun 22 2013]
a(n) = (18*n^3 + 27*n^2 + 27*n + 1  (3*n^2 + 3*n + 1)*(1)^n)/16. [Bruno Berselli, Jun 22 2013]


EXAMPLE

The first two terms are a(1) = 9  6 + 3  1 = 5 and a(2) = 9 + 6 + 3 = 18. Then Shiraishi's formulas give 3^3 + 4^3 + 5^3 = 6^3 and 3^3 + 10^3 + 18^3 = 19^3.


CROSSREFS

Cf. A003325, A005448, A023042, A181123, A225908, A225909, A226902.
Sequence in context: A125641 A006479 A127983 * A056782 A178684 A271771
Adjacent sequences: A226900 A226901 A226902 * A226904 A226905 A226906


KEYWORD

nonn,easy


AUTHOR

Jonathan Sondow, Jun 22 2013


STATUS

approved



