A261224 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 - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

1, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 39, 39, 40, 41, 41, 42, 42, 42, 43, 43, 44, 44, 45, 45, 46

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..6000

Formula

a(n) = A261221(((n+1)^2)-1) - A261221((n^2)-1). [The definition.]

Equally, for all n >= 1:

a(n) = A261221((n+1)^2) - A261221(n^2).

a(n) = A261222(n+1) - A261222(n).

a(n) = A261223(n+1) - A261223(n).

Mathematica

Table[-1 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, ((n + 1)^2) - 1, # != n^2 - 1 &], {n, 91}] (* Michael De Vlieger, Sep 08 2016, after Jud McCranie at A053610 *)

Prog

(Scheme, three variants, the first one utilizing memoization-macro definec)
(definec (A261224 n) (let ((end (- (A000290 n) 1))) (let loop ((k (- (A000290 (+ 1 n)) 1)) (s 0)) (if (= k end) s (loop (A260740 k) (+ 1 s))))))
(define (A261224 n) (- (A261223 (+ 1 n)) (A261223 n)))
(define (A261224 n) (- (A261221 (A000290 (+ 1 n))) (A261221 (A000290 n))))

Crossrefs

First differences of both A261222 and A261223.

Cf. A000290, A053610, A260740, A261221.

Cf. also A260734, A261229.

Sequence in context: A131411 A300068 A194202 * A125059 A369611 A029112

Adjacent sequences: A261221 A261222 A261223 * A261225 A261226 A261227

Keyword

nonn

Author

Antti Karttunen, Aug 12 2015

Status

approved