A116525 -id:A116525

The OEIS is supported by the many generous donors to the OEIS Foundation.

Search: a116525 -id:a116525

Displaying 1-4 of 4 results found.

page 1

A360189

Number T(n,k) of nonnegative integers <= n having binary weight k; triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows.

+10
15

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 3, 3, 1, 3, 3, 1, 1, 4, 3, 1, 1, 4, 4, 1, 1, 4, 5, 1, 1, 4, 5, 2, 1, 4, 6, 2, 1, 4, 6, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 1, 5, 6, 4, 1, 1, 5, 7, 4, 1, 1, 5, 8, 4, 1, 1, 5, 8, 5, 1, 1, 5, 9, 5, 1, 1, 5, 9, 6, 1, 1, 5, 9, 7, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	COMMENTS	`T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.`
	LINKS	`Alois P. Heinz, Rows n = 0..2^12-1, flattened` `Wikipedia, Iverson bracket`
	FORMULA	`T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.` `T(2^n-1,k) = A007318(n,k) = binomial(n,k).` `T(n,floor(log_2(n+1))) = A090996(n+1).` `Sum_{k>=0} T(n,k) = n+1.` `Sum_{k>=0} k * T(n,k) = A000788(n).` `Sum_{k>=0} k^2 * T(n,k) = A231500(n).` `Sum_{k>=0} k^3 * T(n,k) = A231501(n).` `Sum_{k>=0} k^4 * T(n,k) = A231502(n).` `Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).` `Sum_{k>=0} 3^k * T(n,k) = A130665(n).` `Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).` `Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).` `Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).` `Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).` `Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).` `Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).` `Sum_{k>=0} n^k * T(n,k) = A361257(n).`
	EXAMPLE	`T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.` `T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.` `Triangle T(n,k) begins:` `1;` `1, 1;` `1, 2;` `1, 2, 1;` `1, 3, 1;` `1, 3, 2;` `1, 3, 3;` `1, 3, 3, 1;` `1, 4, 3, 1;` `1, 4, 4, 1;` `1, 4, 5, 1;` `1, 4, 5, 2;` `1, 4, 6, 2;` `1, 4, 6, 3;` `1, 4, 6, 4;` `1, 4, 6, 4, 1;` `...`
	MAPLE	b:= proc(n) option remember; `if`(n<0, 0, `b(n-1)+x^add(i, i=Bits[Split](n)))` `end:` `T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):` `seq(T(n), n=0..23);`
	CROSSREFS	`Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).` `Last elements of rows give A090996(n+1).` `Cf. A000027, A000120, A000225, A000788, A006046, A007318, A116520, A116522, A116525, A116526, A130665, A130667, A161342, A231500, A231501, A231502, A361257.`
	KEYWORD	`nonn,look,tabf,base`
	AUTHOR	`Alois P. Heinz, Mar 04 2023`
	STATUS	`approved`

A161342

Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

+10
9

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`First differences are in A161343. - Omar E. Pol, May 03 2015` `From Gary W. Adamson, Aug 30 2016: (Start)` `Let M =` `1, 0, 0, 0, 0, ...` `8, 0, 0, 0, 0, ...` `7, 1, 0, 0, 0, ...` `0, 8, 0, 0, 0, ...` `0, 7, 1, 0, 0, ...` `0, 0, 8, 0, 0, ...` `0, 0, 7, 1, 0, ...` `...` `Then M^k converges to a single nonzero column giving the sequence.` `The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)`
	LINKS	`Alois P. Heinz, Table of n, a(n) for n = 0..16383` `N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS` `Index entries for sequences related to cellular automata`
	FORMULA	`From Nathaniel Johnston, Nov 13 2010: (Start)` `a(n) = Sum_{k=0..n-1} 7^A000120(k).` `a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.` `a(2^n) = 2^(3n).` `(End)` `a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023`
	MAPLE	b:= proc(n) option remember; `if`(n<0, 0, `b(n-1)+x^add(i, i=Bits[Split](n)))` `end:` `a:= n-> subs(x=7, b(n-1)):` `seq(a(n), n=0..44); # Alois P. Heinz, Mar 06 2023`
	MATHEMATICA	`A161342list[nmax_]:=Join[{0}, Accumulate[7^DigitCount[Range[0, nmax-1], 2, 1]]]; A161342list[100] (* Paolo Xausa, Aug 05 2023 *)`
	CROSSREFS	`Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.`
	KEYWORD	`nonn`
	AUTHOR	`Omar E. Pol, Jun 14 2009`
	EXTENSIONS	`More terms from Nathaniel Johnston, Nov 13 2010`
	STATUS	`approved`

A116526

a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.

+10
5

0, 1, 9, 17, 81, 89, 153, 217, 729, 737, 801, 865, 1377, 1441, 1953, 2465, 6561, 6569, 6633, 6697, 7209, 7273, 7785, 8297, 12393, 12457, 12969, 13481, 17577, 18089, 22185, 26281, 59049, 59057, 59121, 59185, 59697, 59761, 60273, 60785, 64881, 64945, 65457, 65969 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	COMMENTS	`A 9-divide version of A084230.` `The interest this one has is in the prime form of even odd 2^n+1, 2^n.` `From Gary W. Adamson, Aug 30 2016: (Start)` `Let M =` `1, 0, 0, 0, 0, ...` `9, 0, 0, 0, 0, ...` `8, 1, 0, 0, 0, ...` `0, 9, 0, 0, 0, ...` `0, 8, 1, 0, 0, ...` `0, 0, 9, 0, 0, ...` `0, 0, 8, 1, 0, ...` `...` `Then M^k converges to a single nonzero column giving the sequence.` `The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)`
	LINKS	`Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)` `H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.` `Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.` `Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant`
	FORMULA	`a(n) = Sum_{k=0..n-1} 8^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019` `a(n) = Sum_{k=0..floor(log_2(n))} 8^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023`
	MAPLE	`a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9a(n/2) else 8a((n-1)/2)+a((n+1)/2) fi end: seq(a(n), n=0..45);`
	MATHEMATICA	`b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9b[n/2]; b[n_?OddQ] := b[n] = 8b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]`
	CROSSREFS	`Cf. A000120, A006046, A077465, A130665, A116520, A130667, A116522, A161342, A116525, A360189.`
	KEYWORD	`nonn`
	AUTHOR	`Roger L. Bagula, Mar 15 2006`
	EXTENSIONS	`Edited by N. J. A. Sloane, Apr 16 2006`
	STATUS	`approved`

A256141

Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, in which row n lists the partial sums of the n-th row of the square array of A256140.

+10
4

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 9, 5, 1, 1, 6, 9, 16, 11, 6, 1, 1, 7, 11, 25, 19, 15, 7, 1, 1, 8, 13, 36, 29, 28, 19, 8, 1, 1, 9, 15, 49, 41, 45, 37, 27, 9, 1, 1, 10, 17, 64, 55, 66, 61, 64, 29, 10, 1, 1, 11, 19, 81, 71, 91, 91, 125, 67, 33, 11, 1, 1, 12, 21, 100, 89, 120, 127, 216, 129, 76, 37, 12, 1 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	COMMENTS	`Questions:` `Is also A130667 a row of this square array?` `Is also A116522 a row of this square array?` `Is also A116526 a row of this square array?` `Is also A116525 a row of this square array?` `Is also A116524 a row of this square array?`
	LINKS	`Table of n, a(n) for n=0..90.` `Index entries for sequences related to cellular automata`
	EXAMPLE	`The corner of the square array with the first 15 terms of the first 12 rows looks like this:` `--------------------------------------------------------------------------` `A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1` `A000027: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15` `A006046: 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, 57, 65` `A130665: 1, 4, 7, 16, 19, 28, 37, 64, 67, 76, 85, 112, 121, 148, 175` `A116520: 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369` `A130667? 1, 6,11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671` `A116522? 1, 7,13, 49, 55, 91,127, 343, 349, 385, 421, 637, 673, 889,1105` `A161342: 1, 8,15, 64, 71,120,169, 512, 519, 568, 617, 960,1009,1352,1695` `A116526? 1, 9,17, 81, 89,153,217, 729, 737, 801, 865,1377,1441,1953,2465` `.......: 1,10,19,100,109,190,271,1000,1009,1090,1171,1900,1981,2710,3439` `A116525? 1,11,21,121,131,231,331,1331,1341,1441,1541,2541,2641,3641,4641` `.......: 1,12,23,144,155,276,397,1728,1739,1860,1981,3312,3422,4764,6095`
	CROSSREFS	`Cf. A000120, A255740, A255741, A256140.` `First five rows are A000012, A000027, A006046, A130665, A116520. Row 7 is A161342.` `First eight columns are A000012, A000027, A005408, A000290, A028387, A000384, A003215, A000578. Column 9 is A081437. Column 11 is A015237. Columns 13-15 are A005915, A005917, A000583.`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Omar E. Pol, Mar 16 2015`
	STATUS	`approved`

page 1

Search completed in 0.011 seconds

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)