A138524 -id:A138524 - cp's OEIS frontend

A138525 a(n) = Sum_{k=0..n} (2*k)!.

1, 3, 27, 747, 41067, 3669867, 482671467, 87660962667, 21010450850667, 6423384156578667, 2439325392333218667, 1126440053169940898667, 621574841786409380258667, 403913035968392044964258667

Offset: 0

Leroy Quet, Mar 23 2008

Last digit is 7 for n > 2. Therefore there is no square in this sequence except 1. - Altug Alkan, Dec 01 2015

G. C. Greubel, Table of n, a(n) for n = 0..224

Cf. A138523, A138524.

Partial sums of A010050.

A138525:=n->add((2*k)!, k=0..n): seq(A138525(n), n=0..15); # Wesley Ivan Hurt, Dec 01 2015

Table[Sum[(2i)!, {i, 0, n}], {n, 0, 15}] (* Stefan Steinerberger, Mar 25 2008 *)

a(n) = sum(k=0, n, (2*k)!); \\ Altug Alkan, Dec 01 2015

More terms from Stefan Steinerberger and Robert G. Wilson v, Mar 25 2008

A138523 a(n) = Sum_{k=1..n} (2k-1)!.

Original entry on oeis.org

1, 7, 127, 5167, 368047, 40284847, 6267305647, 1313941673647, 357001369769647, 122002101778601647, 51212944273488041647, 25903229683158464681647, 15537113273014144448681647, 10904406563691366305216681647, 8852666400303393320848832681647, 8231691320578226211046411712681647

Offset: 1

Leroy Quet, Mar 23 2008

a(n) is divisible by 107 for n >= 53. - Robert Israel, Dec 01 2015
Last digit is 7 for n > 1. Therefore there is no square in this sequence except 1. - Altug Alkan, Dec 01 2015
a(n) is the rank of [2,1, 4,3, ..., 2n,2n-1] within the permutations of [1, 2, ... 2n-1, 2n] in lexicographic order. See A375302 for the ranking function. - Hugo Pfoertner, Aug 25 2024

G. C. Greubel, Table of n, a(n) for n = 1..225

Cf. A007489, A138524, A138525, A375302.

a:=proc(n) options operator, arrow: sum(factorial(2*k-1), k=1..n) end proc: seq(a(n), n=1..14); # Emeric Deutsch, Mar 31 2008

Table[Sum[(2i - 1)!, {i, n}], {n, 15}] (* Stefan Steinerberger, Mar 25 2008 *)

a(n) = sum(k=1, n, (2*k-1)!); \\ Michel Marcus, Oct 28 2015

Recurrence: a(1) = 1, a(2) = 7, a(n) = (4*n^2-6*n+3)*a(n-1) - 2*(n-1)*(2*n-1)*a(n-2). - Vladimir Reshetnikov, Oct 28 2015

More terms from Stefan Steinerberger, Emeric Deutsch and Robert G. Wilson v, Mar 25 2008

A351895 Numbers with an equal number of odd and even digits in their factorial-base representation.

Original entry on oeis.org

2, 5, 25, 26, 29, 30, 34, 37, 38, 41, 42, 46, 51, 55, 56, 59, 63, 67, 68, 71, 73, 74, 77, 78, 82, 85, 86, 89, 90, 94, 99, 103, 104, 107, 111, 115, 116, 119, 723, 727, 728, 731, 735, 739, 740, 743, 745, 746, 749, 750, 754, 757, 758, 761, 762, 766, 771, 775, 776

Offset: 1

Amiram Eldar, Feb 24 2022

			5 is a term since its factorial-base representation, 21, has one odd digit, 1, and one even digit, 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A351893, A351894.
A138524 is a subsequence.
Similar sequences: A031443 (binary), A227870 (decimal).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]

A351897 Numbers that in factorial-base representation have digits with an alternating parity.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 14, 19, 23, 26, 38, 55, 59, 67, 71, 74, 86, 103, 107, 115, 119, 127, 131, 139, 143, 175, 179, 187, 191, 223, 227, 235, 239, 266, 278, 314, 326, 367, 371, 379, 383, 415, 419, 427, 431, 463, 467, 475, 479, 506, 518, 554, 566, 607, 611, 619, 623

Offset: 1

Amiram Eldar, Feb 24 2022

			7 is a term since its factorial-base representation is 101 and the parities of its digits are odd, even, odd.

Amiram Eldar, Table of n, a(n) for n = 1..20477 (terms below 10!)

Cf. A007623.
Similar sequences: A000975 (binary), A030141 (decimal), A033068 (ternary), A179970 (quaternary).
Subsequences: A033312, A138524, A341900.

max = 7; q[n_] := AllTrue[Differences@ Mod[IntegerDigits[n, MixedRadix[Range[max, 2, -1]]], 2], # != 0 &]; Select[Range[0, 1000], q]

cp's OEIS Frontend

A138525 a(n) = Sum_{k=0..n} (2*k)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A138523 a(n) = Sum_{k=1..n} (2k-1)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A351895 Numbers with an equal number of odd and even digits in their factorial-base representation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A351897 Numbers that in factorial-base representation have digits with an alternating parity.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica