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.

A255056 Trunk of number-of-runs beanstalk: The unique infinite sequence such that a(n-1) = a(n) - number of runs in binary representation of a(n).

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 14, 18, 22, 26, 28, 30, 32, 36, 42, 46, 50, 54, 58, 60, 62, 64, 68, 74, 78, 84, 90, 94, 96, 100, 106, 110, 114, 118, 122, 124, 126, 128, 132, 138, 142, 148, 152, 156, 162, 168, 174, 180, 186, 190, 192, 196, 202, 206, 212, 218, 222, 224, 228, 234, 238, 242, 246, 250, 252, 254
Offset: 0

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Author

Antti Karttunen, Feb 14 2015

Keywords

Comments

All numbers of the form (2^n)-2 are present, which guarantees the uniqueness and also provides a well-defined method to compute the sequence, for example, via a partially reversed version A255066.
The sequence was inspired by a similar "binary weight beanstalk", A179016, sharing some general properties with it (like its partly self-copying behavior, see A255071), but also differing in some aspects. For example, here the branching degree is not the constant 2, but can vary from 1 to 4. (Cf. A255058.)

Links

Crossrefs

First differences: A255336.
Terms halved: A255057.
Cf. A255053 & A255055 (the lower & upper bound for a(n)) and also A255123, A255124 (distances to those limits).
Cf. A255327, A255058 (branching degree for node n), A255330 (number of nodes in the finite subtrees branching from the node n), A255331, A255332
Subsequence: A000918 (except for -1).
Cf. A255061, A255062, A255071, A255072, A255066, A255122.
Cf. A254113, A254114.
Cf. A255063, A255064, A255125, A255126.
Similar "beanstalk's trunk" sequences using some other subtracting map than A236840: A179016, A219648, A219666.

Programs

Formula

a(n) = A255066(A255122(n)).
Other identities and observations. For all n >= 0:
a(n) = 2*A255057(n).
A255072(a(n)) = n.
A255053(n) <= a(n) <= A255055(n).

A255125 Number of times a multiple of four is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 6, 13, 26, 47, 81, 140, 253, 482, 949, 1875, 3666, 7088, 13614, 26100, 50082, 96246, 185131, 356123, 684758, 1316197, 2530257, 4868019, 9378335, 18096921, 34974646, 67669905, 130998912, 253565649, 490501587, 947992195, 1830664188, 3533571444
Offset: 0

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Author

Antti Karttunen, Feb 18 2015

Keywords

Comments

Also the number of even numbers in range [A255062(n) .. A255061(n+1)] of A255057 (equally, in A255067). See the sum-formulas.

Examples

			For n=5 we start iterating with map m(n) = A236840(n) from the initial value (2^(5+1))-2 = 62. Thus we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36, m(36) = 32 and finally m(32) = 30, which is (2^5)-2. Of the nine numbers encountered, only 60, 36 and 32 are multiples of four, thus a(5) = 3.

Links

Crossrefs

Cf. A005811, A059841, A133872, A236840, A255056, A255057, A255061, A255062, A255067, A255071, A255126, A255332.
Similar sequences: A218542, A255063.

Programs

  • PARI
    A005811(n) = hammingweight(bitxor(n, n\2));
write_A255125_and_A255126_and_A255071(n) = { my(k, i, s25, s26); k = (2^(n+1))-2; i = 1; s25 = 0; s26 = 0; while(1, if((k%4),s26++,s25++); k = k - A005811(k); if(!bitand(k+1, k+2), break, i++)); write("b255125.txt", n, " ", s25); write("b255126.txt", n, " ", s26); write("b255071.txt", n, " ", i); };
for(n=1,42,write_A255125_and_A255126_and_A255071(n));
  • Scheme
    (define (A255125 n) (if (zero? n) 1 (let loop ((i (- (expt 2 (+ 1 n)) 4)) (s 0)) (cond ((pow2? (+ 2 i)) s) (else (loop (- i (A005811 i)) (+ s (A133872 i))))))))
;; Alternatively:
(define (A255125 n) (add (COMPOSE A059841 A255057) (A255062 n) (A255061 (+ 1 n))))
(define (A255125 n) (add (COMPOSE A059841 A255067) (A255062 n) (A255061 (+ 1 n))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

Formula

a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255057(k)).
a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255067(k)).
a(n) = A255071(n) - A255126(n).

A255331 a(n) = A255329(n) - A255328(n).

Original entry on oeis.org

-1, 0, 0, -4, 1, 0, -7, 0, -3, 1, 0, 3, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, -5, 0, 4, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0, -5, 0, 4, 0, -12, 0, 0, -5, 0, 4, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0, 7, 1, -10, 15, 0, 0, 1, -11, 2, 0, 0, -5, 0, 4, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Feb 21 2015

Keywords

Comments

a(n) = How many more nodes there are in the finite subtrees branching "right" (to the "larger side") than in the finite subtrees branching "left" (to the "smaller side") from the node n in the infinite trunk of number-of-runs beanstalk (A255056).
The edge-relation between nodes is given by A236840(child) = parent. Odd numbers are leaves, as there are no such k that A236840(k) were odd.
If A255058(n) = 1, then a(n) = 0, but also in some other cases.

Examples

			The only finite subtree starting from the node number 0 (which is 0) is the leaf 1, and it branches to the "left" (meaning that it is less than 2, which is the next node in the infinite trunk), thus the difference between the nodes in finite branches to the right vs. the nodes in finite branches to the left is -1 and a(0) = -1.
The only finite subtrees starting from the node number 1 in the infinite trunk (which is 2), are the leaves 3 and 5, of which the other one is on the "left" side and the other one on the "right" side (i.e. less than 4 and more than 4, which is the next node in the infinite trunk), thus a(1) = 1-1 = 0.
The node 11 in the infinite trunk is A255056(11) = 30. Apart from 32, which is the next node (node 12) in the infinite trunk, it has one leaf-child 31 at the "left side" (less than 32), and one leaf-child 33 (more than 32) at the "right side", and also at that side a subtree of three nodes 34 <- 38 <- 43, thus a(11) = (3+1) - 1 = 3.

Links

Crossrefs

Partial sums: A255332.
Cf. A236840, A255058, A255328, A255329, A255330.

Programs

Formula

a(n) = A255329(n) - A255328(n).

A255333 Partial sums of A255330.

Original entry on oeis.org

1, 3, 3, 7, 8, 8, 15, 15, 18, 19, 19, 24, 26, 32, 32, 38, 38, 41, 42, 42, 47, 49, 61, 61, 63, 68, 68, 72, 74, 80, 80, 86, 86, 89, 90, 90, 95, 97, 109, 109, 111, 118, 119, 131, 135, 135, 137, 142, 142, 146, 148, 160, 160, 162, 167, 167, 171, 173, 179, 179, 185, 185, 188, 189, 189, 194
Offset: 0

Views

Author

Antti Karttunen, Feb 21 2015

Keywords

Links

Crossrefs

Cf. A000079, A000225, A255061, A255062, A255330, A255332.
Analogous sequences: A218785, A230408.

Formula

a(0) = 1; for n >= 1: a(n) = a(n-1) + A255330(n).
Other identities:
a(A255061(n)-1) = A000225(n) - A255062(n) for all n >= 2.
Equally: a(A255061(n)-1) + A255062(n) + 1 = A000079(n) = 2^n for all n >= 2.

A262895 Partial sums of A262894.

Original entry on oeis.org

6, 6, 47, 47, 47, 52, 52, 68, 68, 70, 70, 69, 70, 92, 96, 96, 96, 93, 93, 92, 79, 79, 184, 186, 185, 184, 182, 183, 201, 208, 208, 208, 208, 209, 208, 205, 205, 205, 200, 200, 196, 209, 209, 202, 202, 202, 195, 201, 202, 202, 202, 202, 243, 243, 243, 243, 333, 332, 332, 337, 337, 335, 335, 336, 335, 335, 347, 346, 346, 349, 410, 410, 410, 410, 410, 410, 410, 410, 527
Offset: 0

Views

Author

Antti Karttunen, Oct 04 2015

Keywords

Comments

First negative term occurs as a(2778) = -206.

Links

Crossrefs

Cf. A262893, A262894.
Cf. also A255332.

Formula

a(0) = A262894(0); for n >= 1, a(n) = a(n-1) + A262894(n).
Showing 1-5 of 5 results.

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