A260734 a(n) = number of steps needed to reach (n^2)-1 when starting from k = ((n+1)^2)-1 and repeatedly applying the map that replaces k with k - A002828(k), where A002828(k) = the least number of squares that add up to k.

1, 2, 2, 4, 4, 5, 5, 7, 7, 7, 8, 10, 9, 10, 10, 13, 13, 14, 13, 15, 15, 16, 17, 17, 19, 19, 19, 20, 20, 22, 22, 23, 24, 25, 24, 26, 27, 25, 28, 29, 29, 29, 30, 31, 33, 33, 33, 34, 35, 35, 37, 36, 39, 37, 38, 40, 42, 40, 42, 42, 43, 42, 45, 45, 45, 48, 45, 49, 50, 50, 48, 53, 50, 51, 54, 52, 53, 54, 56, 56, 56, 58, 59, 59, 60, 60, 60, 61, 62, 62, 62, 65, 66, 66, 65

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..1024

Formula

a(n) = A260731(((n+1)^2)-1) - A260731((n^2)-1). [The definition.]

Equally, for all n >= 1:

a(n) = A260731((n+1)^2) - A260731(n^2).

a(n) = A260732(n+1) - A260732(n).

a(n) = A260733(n+1) - A260733(n).

Mathematica

Table[Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[ First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, ((n + 1)^2) - 1, # != (n^2) - 1 &], {n, 95}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)

Prog

(Scheme, three variants, the first one utilizing memoization-macro definec)
(definec (A260734 n) (let ((end (- (A000290 n) 1))) (let loop ((k (- (A000290 (+ 1 n)) 1)) (s 0)) (if (= k end) s (loop (A255131 k) (+ 1 s))))))
(define (A260734 n) (- (A260731 (A000290 (+ 1 n))) (A260731 (A000290 n))))
(define (A260734 n) (- (A260733 (+ 1 n)) (A260733 n)))

Crossrefs

First differences of both A260732 and A260733.

Cf. A000290, A002828, A255131.

Cf. also A261224.

Sequence in context: A152850 A036714 A319398 * A265428 A035644 A288773

Adjacent sequences: A260731 A260732 A260733 * A260735 A260736 A260737

Keyword

nonn

Author

Antti Karttunen, Aug 12 2015

Status

approved