cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A234018 Values at middle points of each row of A233270: a(n) = A233270(A233268(n)).

Original entry on oeis.org

0, -1, 0, 1, 1, 3, 3, 19, 35, 67, 127, 218, 369, 660, 1267, 2476, 4863, 9453, 18078, 34173, 64374, 121515, 227965, 426603, 793638, 1482307, 2764957, 5183333, 9830514
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2013

Keywords

Comments

Please see the graph of A233270.

Crossrefs

Cf. A233268, A233270, A234019, A234020, A233274.

Programs

  • Scheme
    (define (A234018 n) (A233270 (A233268 n)))
;; Iterative version, which computes for values a(n>=4) in a single pass:
(define (A234018v2 n) (cond ((zero? n) 0) ((< n 4) (A234018 n)) (else (let* ((memosize (if (< n 8) 2 (+ 2 (expt 2 (- n 8))))) (memo (make-vector memosize 0))) (let loop ((u (- (A000079 n) 1)) (d (A000079 (- n 1))) (i 0) (j #f) (du #f)) (cond ((pow2? u) (let ((offset (- (floor->exact (/ i 2)) du))) (- (A054429 (vector-ref memo offset)) (vector-ref memo (+ offset (A000035 i)))))) ((and (< u d) (not j)) (vector-set! memo 0 u) (loop (A011371 u) (A233272 d) (+ i 1) 1 i)) (else (if (and j (< j memosize)) (vector-set! memo j u)) (loop (A011371 u) (A233272 d) (+ i 1) (and j (+ 1 j)) du))))))))
(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i))))))

Formula

a(n) = A233270(A233268(n)).

A234019 Maximum values occurring in each row of A233270: a(n) = A233270(A233268(n) - A234020(n)).

Original entry on oeis.org

0, -1, 0, 1, 2, 4, 6, 19, 38, 67, 127, 234, 419, 745, 1378, 2678, 5311, 10470, 20333
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2013

Keywords

Comments

Please see the graph of A233270.

Crossrefs

Cf. A233268, A233270, A234018, A234020.

Formula

a(n) = A233270(A233268(n) - A234020(n)).

A233270 a(n) = A233271(n) - A179016(n).

Original entry on oeis.org

0, 0, -1, 0, 0, 0, 1, 0, 0, 2, 1, 2, 0, 0, 3, 3, 4, 3, 4, 3, 3, 0, 0, 4, 4, 5, 4, 6, 5, 5, 3, 5, 5, 6, 4, 5, 4, 4, 0, 0, 5, 8, 9, 10, 13, 13, 15, 16, 17, 18, 18, 17, 17, 19, 19, 17, 17, 18, 18, 17, 16, 15, 13, 13, 10, 9, 8, 5, 0, 0, 6, 9, 14, 17, 18, 20, 22, 21
Offset: 0

Views

Author

Antti Karttunen, Dec 14 2013

Keywords

Comments

For all n>=2, a(1+A213710(n)) = n-2.
Except for a(2)=-1 (which seems to be the only negative term in the sequence), the sequences A218600 and A213710 give the positions of zeros.
Furthermore, each subrange [A213710(n)..A218600(n+1)] is palindromic. A233268 gives the middle points of those ranges, the sequence A234018 gives the values at those points, while A234019 gives the maximum term in that range in this sequence.

Examples

			This irregular table begins as:
0;
0;
-1;
0, 0;
0, 1, 0;
0, 2, 1, 2, 0;
0, 3, 3, 4, 3, 4, 3, 3, 0;
0, 4, 4, 5, 4, 6, 5, 5, 3, 5, 5, 6, 4, 5, 4, 4, 0;
...
After zero, each row n is A213709(n-1) elements long.

Links

Crossrefs

Except for a(2)=-1 (which seems to be the only negative term in the sequence), the sequences A218600 and A213710 give the positions of zeros.
Cf. A080468, A179016, A233271, A233268, A233274, A234018, A234019, A234020.

Programs

Formula

a(n) = A233271(n) - A179016(n).
a(A218602(n)) = a(n). [This is just a claim that each row is palindrome]

A234020 Offsets from the middle point of each row of A233270 to the nearest point containing a maximum value of that range.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 0, 5, 0, 0, 33, 72, 140, 267, 502, 954, 1771, 3355
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2013

Keywords

Comments

Please see the graph of A233270. Iff a(n)=0, A234018(n) = A234019(n).
Seems to grow faster than A233274. Is the ratio a(n)/A233274(n) converging to some limit?

Crossrefs

Cf. A233268, A233270, A234018, A234019, A233274.

Programs

  • Scheme
    (define (A234020 n) (let ((middle (A233268 n))) (let loop ((i middle) (m 0) (maxp middle)) (cond ((zero? (A233270 i)) (- middle maxp)) ((> (abs (A233270 i)) m) (loop (- i 1) (abs (A233270 i)) i)) (else (loop (- i 1) m maxp))))))

A233274 Relative offsets from the middle point of each row of A233271 & A218616 to the first point where the former exceeds the latter, which apart of case a(3)=-1 is always left of or at the middle point.

Original entry on oeis.org

0, 0, -1, 0, 0, 0, 0, 1, 2, 4, 8, 13, 22, 38, 68, 125, 232, 429, 786, 1428, 2578, 4645, 8364, 15064, 27145, 48990, 88736, 161813, 298001, 555451, 1048207, 1999608, 3844722, 7425094, 14356699, 27722560, 53374986
Offset: 1

Views

Author

Antti Karttunen, Jan 01 2014

Keywords

Comments

The sequence tells how many positions to the left of center of each row/subrange (of irregular tables like A233270, central point given by A233268) the sequences A233271 and A218616 cross each other (please see the linked graph).

Links

Crossrefs

Cf. A234020, A213709, A218600, A226060, A233268, A179016, A218616, A233270.

Programs

Formula

a(1)=a(2)=0, and for n > 2, a(n) = ⌈(A213709(n-1)/2)⌉ - A226060(n-2) - 1. Where ⌈x⌉ stands for ceiling(x)
Showing 1-5 of 5 results.

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