A036284 -id:A036284 - cp's OEIS frontend

A136380 Quotient obtained when A036284(n) is considered as a GF(2)[X]-polynomial and it is divided by (x^3 + 1) ^ A000225(n-1).

24, 160, 11968, 49657088, 837028380268032, 237269922100748727235760269312, 18811253173629696438994877569412700111469395859003555753984, 118178826602781220665226658680265194908312590801831513776333330179329649495708436476846379030238467286212637486694400

Offset: 1

Antti Karttunen, Dec 29 2007

A. Karttunen, Table of n, a(n) for n = 1..11
A. Karttunen, C program for computing this sequence

a(n) = 4*A136382(n) = 2*A048724(A136384(n)). A136381 shows the same sequence in octal base. Cf. A036284.

A036285 Sequence A036284 written in octal.

Original entry on oeis.org

6, 30, 2640, 25664300, 2503026643656400, 25315703023056664357316564563000, 2527515731736303020414563050266643413731571505656462723056364000, 25246137277231573150136315064303020122720413505630571426364566664342075141145731573675057360256564674414627536305631724050230000

Offset: 0

Antti Karttunen, Nov 01 1998

A. Karttunen, Table of n, a(n) for n = 0..9
A. Karttunen, C program for computing this sequence

Cf. A136381, A136383, A136385.

a(n) = A007094(A036284(n)).

Entry revised Dec 29 2007

A136378 Number of distinct irreducible polynomials dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

Original entry on oeis.org

2, 2, 3, 4, 5, 5, 5, 7, 7, 10, 15, 10, 10, 12, 15, 11

Offset: 0

Antti Karttunen, Dec 29 2007

nonn

a(n) = A091221(A036284(n)).

A136379 Number of irreducible polynomials (counted with multiplicity) dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

Original entry on oeis.org

2, 4, 9, 17, 25, 42, 76, 143, 273, 533, 1052, 2072, 4122, 8221, 16417, 32799

Offset: 0

Antti Karttunen, Dec 29 2007

nonn

a(n) = A091222(A036284(n)).

A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

Original entry on oeis.org

0, 1, 3, 14, 57, 229, 916, 7761, 29567, 117474, 469113, 3973641, 15138352, 60146777, 240187355, 2070207870, 7733090689, 30791909229, 260408711716, 991495872825, 3942106110215, 15739612088946, 133333733918417

Offset: 0

Antti Karttunen, Jan 28 1999

			When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
and one starts collecting their bits from column-0 to SW-direction (from the least to the most significant end), one gets 000... (0), ...00001 (1), ...00011 (3), ...001110 (14), etc. (See A102370 for similar transformation done on nonnegative integers).

Same sequence in octal: A037098. Cf. also: A102370, A000045, A037094-A037095, A036284.

a(n) := Sum(bit_n(A000045(n+i), i)*(2^i), i=0..inf) [ bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); ]

In practice, n can be used as an upper limit instead of infinity.

Entry revised Dec 29 2007

A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

Original entry on oeis.org

1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360

Offset: 0

Antti Karttunen, Jan 29 1999

This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C program.

Conjecture: For n >= 3, each term a(n), when considered as a GF(2)[X] polynomial, is divisible by the GF(2)[X] polynomial (x + 1) ^ A055010(n-1). If this holds, then for n >= 3, a(n) = A048720(A136386(n), A048723(3,A055010(n-1))).

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
it can be seen that the bits in the n-th column from the right can be arranged in periods of 2^n: 1, 2, 4, 8, ... This sequence is formed from those bits: 1, is binary for 1, thus a(0) = 1. 01, reversed is 10, which is binary for 2, thus a(1) = 2, 0000 is binary for 0, thus a(2)=0, 000110011, reversed is 11001100 = A007088(204), thus a(3) = 204.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Antti Karttunen, Table of n, a(n) for n = 0..11
Antti Karttunen, C program for computing this sequence.
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).

a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.
Other identities and observations:
For n >= 2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).

Entry revised by Antti Karttunen, Dec 29 2007

Name changed and the example corrected by Antti Karttunen, Dec 05 2015

A136382 a(n) = A136380(n)/4.

Original entry on oeis.org

6, 40, 2992, 12414272, 209257095067008, 59317480525187181808940067328, 4702813293407424109748719392353175027867348964750888938496, 29544706650695305166306664670066298727078147700457878444083332544832412373927109119211594757559616821553159371673600

Offset: 1

Antti Karttunen, Dec 29 2007

Note that each term a(n) fits into A007283(n-1) bits.

A. Karttunen, Table of n, a(n) for n = 1..11
A. Karttunen, C program for computing this sequence

a(n) = A048724(A136384(n))/2. A136383 shows the same sequence in octal base. Cf. A036284.

A136384 Quotient obtained when A136380(n)/2 is considered as a GF(2)[X]-polynomial and it is divided by (x + 1).

Original entry on oeis.org

4, 48, 3360, 14043520, 233515838757120, 65982595605873500894008888320, 5233741023536997251047595348728205456443682897303843358720, 32837130684987081672210288030183520098814938795984162933658101468543499651419210151303128996446334767341864627691520

Offset: 1

Antti Karttunen, Dec 29 2007

Note that each term a(n) fits into A007283(n-1) bits.

A. Karttunen, Table of n, a(n) for n = 1..11
A. Karttunen, C program for computing this sequence

A136385 shows the same sequence in octal base. Cf. A036284, A136380, A136382.

A136386 Quotient obtained when A037097(n) is considered as a GF(2)[X]-polynomial and it is divided by (x + 1) ^ A000225(n-1) (= A051179(n-2)).

Original entry on oeis.org

4, 8, 352, 3728, 7269662752, 761166466256046848, 390022035611646394530728097023856870592, 91600670557117582933643002658167825054614175029432880501373395030525438396928, 13417853484388319477475698658536993288839029124735549539652836318808118017743106800015257954250357092148394821846783842030516713870361254572407216621548672

Offset: 3

Antti Karttunen, Dec 29 2007

A. Karttunen, Table of n, a(n) for n = 3..13
A. Karttunen, C program for computing this sequence

A136387 shows the same sequence in binary base. Cf. A037096, A037097, A136380, A136382, A136384.

A037097 Periodic vertical binary vectors of powers of 3, starting from bit-column 2 (halved).

Original entry on oeis.org

0, 12, 120, 57120, 93321840, 10431955353116229600, 8557304989566294213168677685339060480, 102743047168201563425402150421568484707810385382513037790885688657488312400960

Offset: 2

Antti Karttunen, Jan 29 1999

Conjecture: For n>=3, each term a(n), when considered as a GF(2)[X]-polynomial, is divisible by GF(2)[X] -polynomial (x + 1) ^ A000225(n-1) (= A051179(n-2)). If this holds, then for n>=3, a(n) = A048720bi(A136386(n),A048723bi(3,A000225(n-1))) = A048720bi(A136386(n),A051179(n-2)).

			When powers of 3 are written in binary (see A004656), under each other as:
000000000001 (1)
000000000011 (3)
000000001001 (9)
000000011011 (27)
000001010001 (81)
000011110011 (243)
001011011001 (729)
100010001011 (2187)
it can be seen that, starting from the column 2 from the right, the bits in the n-th column can be arranged in periods of 2^(n-1): 4, 8, ... This sequence is formed from those bits: 0011, reversed is 11100, which is binary for 12, thus a(3) = 12, 00011110, reversed is 011110000, which is binary for 120, thus a(4) = 120.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

A. Karttunen, Table of n, a(n) for n = 2..12
A. Karttunen, C program for computing this sequence
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

a(n) = floor(A037096(n)/(2^(2^(n-1)))). See also A036284, A136386.

a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n-1))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

a(n) = Sum_{k=0..A000225(n-1)} ([A000244(k)/(2^n)] mod 2) * 2^k, where [] stands for floor function.

Entry revised Dec 29 2007

cp's OEIS Frontend

A136380 Quotient obtained when A036284(n) is considered as a GF(2)[X]-polynomial and it is divided by (x^3 + 1) ^ A000225(n-1).

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A036285 Sequence A036284 written in octal.

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A136378 Number of distinct irreducible polynomials dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

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A136379 Number of irreducible polynomials (counted with multiplicity) dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

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A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

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A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

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A136382 a(n) = A136380(n)/4.

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A136384 Quotient obtained when A136380(n)/2 is considered as a GF(2)[X]-polynomial and it is divided by (x + 1).

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A136386 Quotient obtained when A037097(n) is considered as a GF(2)[X]-polynomial and it is divided by (x + 1) ^ A000225(n-1) (= A051179(n-2)).

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A037097 Periodic vertical binary vectors of powers of 3, starting from bit-column 2 (halved).

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