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%I #43 Jan 23 2024 12:28:40
%S 101,1233,8833,10100,990100,5882353,94122353,1765038125,2584043776,
%T 7416043776,8235038125,116788321168,123288328768,876712328768,
%U 883212321168,7681802663025,8896802846976,13793103448276,15348303604525,84651703604525,86206903448276,91103202846976
%N Numbers k whose decimal expansion can be split into two parts s and t with k = s^2 + t^2.
%C Inspired by the book "Getallentheorie - Een inleiding" from Frits Beukers, pp. 103, 104 (in Dutch).
%C Part t cannot begin with a 0 digit, so the split is k = s*10^length(t) + t.
%C For all terms except for a(1), the lengths of the parts are length(s) = floor(L/2) and length(t) = ceiling(L/2) where L = length(k).
%C Terms of the form (10^(8*(4*u+1)) + 1)/17 are a special case, being a(6) for u = 0, a(276) for u = 1, a(3102) for u = 2. These are the digits of 1/17 rounded up.
%C The corresponding right linear grammar for these is: S -> 123 T, T -> 2 8767 1232 8767 123 T | 3.
%C Most terms are either a starting point (u = 0) of an infinite list given by a regular language, or they occur later in this list of terms. Exceptions observed as standalone terms are a(1) = 101, a(4) = 10100 and a(5) = 990100.
%D Frits Beukers, "Getallen - Een inleiding" (In Dutch), Epsilon Uitgaven, Amsterdam (2015).
%H A.H.M. Smeets, <a href="/A368416/b368416.txt">Table of n, a(n) for n = 1..5841</a>
%H Steven Charlton, <a href="https://raw.githubusercontent.com/stevencharlton/squaresumcat/master/solutions.1-46.txt">List of terms, up to and including length 96 (unordered)</a>.
%H Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, <a href="https://www.researchgate.net/profile/Michael-Kleber-2/publication/225591088_A_curious_cubic_identity_and_self-similar_sums_of_squares/links/55c635f308aeca747d633576/A-curious-cubic-identity-and-self-similar-sums-of-squares.pdf">A curious cubic identity and self similar sums of squares</a>, The Mathematical Intelligencer, June 2007.
%F List of examples of regular languages that are subsets of this sequence (leading zeros must be omitted, and ^ denotes repetition of digit block(s)):
%F {(1232 8767)^(2*u) 1233 | n >= 0}; a(2) for u = 0, a(55) for u = 1, a(232) for u = 2, a(960) for u = 3, a(1320) for u = 4, a(3889) for u = 5.
%F {(8767 1232)^(2*u+1) 8768 | n >= 0}; a(14) for u = 0, a(93) for u = 1, a(395) for u = 2, a(1086) for u = 3
%F {(8832 1167)^(2*u) 8833 | n >= 0}; a(3) for u = 0, a(65) for u = 1, a(257) for u = 2, a(964) for u = 3, a(1328) for u = 4, a(4033) for u = 5.
%F {(1167 8832)^(2*u+1) 1168 | n >= 0}; a(12) for u = 0, a(85) for u = 1, a(386) for u = 2, a(1046) for u = 3
%F {(1167 8832)^(4*u+3) 1167 8833 | n >= 0}; a(230) for u = 0, a(1319) for u = 1
%F {(05882352 99117647)^(2*u) 05882353 | n >= 0}; a(6) for u = 0, a(276) for u = 1, a(3102) for u = 2.
%F {(94122352 05877647)^(2*u) 94122353 | n >= 0}; a(7) for u = 0, a(280) for u = 1, a(3122) for u = 2.
%F {(05877647 94122352)^(2*u+1) 05877648 | n >= 0}; a(76) for u = 0, a(1003) for u = 1, a(4067) for u = 2.
%F {(1765038124 8234961875)^(2*u) 1765038125 | n >= 0}; a(8) for u = 0, a(878) for u = 1, a(4493) for u = 2.
%F {(8234961875 1765038124)^(2*u+1) 8234961876 | n >= 0}; a(177) for u = 0, a(2672) for u = 1
%F {(2584043775 7415956224)^(2*u) 2584043776 | n >= 0}; a(9) for u = 0, a(886) for u = 1, a(4618) for u = 2.
%F {(7415956224 2584043775)^(2*u+1) 7415956225 | n >= 0}; a(170) for u = 0, a(2537) for u = 1.
%F {(7416043775 2583956224)^(2*u) 7416043776 | n >= 0}; a(10) for u = 0, a(924) for u = 1, a(5290) for u = 2.
%F {(2583956224 7416043775)^(2*u+1) 2583956225 | n >= 0}; a(126) for u = 0, a(1890) for u = 1.
%F {(8235038124 1764961875)^(2*u) 8235038125 | n >= 0}; a(11) for u = 0, a(932) for u = 1, a(5415) for u = 2.
%F {(1764961875 8235038124)^(2*u+1) 1764961876 | n >= 0}; a(119) for u = 0, a(1755) for u = 1.
%F {(123288328767 876711671232)^(2*u) 123288328768 | n >= 0}; a(13) for u = 0, a(1050) for u = 1.
%F {(876711671232 123288328767)^(2*u+1) 876711671233 | n >= 0}; a(254) for u = 0, a(4030) for u = 1.
%F {(1091314031180400 8908685968819599)^(2*u) 1091314031180401 | n >= 0}; a(30) for u = 0, a(3484) for u = 1.
%F {(2913840045440000 7086159954559999)^(2*u) 2913840045440001 | n >= 0}; a(34) for u = 0, a(3557) for u = 1.
%e 101 is a term since it can be split as 10^2 + 1^2 = 101. (This is so in any base.)
%e 8833 is a term since it can be split as s=88 and t=33 with 88^2 + 33^2 = 8833.
%Y Cf. A101311, A368418.
%Y Cf. A368417 (base 2).
%K nonn,base
%O 1,1
%A _A.H.M. Smeets_, Dec 23 2023