login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A236106 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the twice odd numbers (A016825) interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2. 16
2, 6, 10, 2, 14, 0, 18, 6, 22, 0, 2, 26, 10, 0, 30, 0, 0, 34, 14, 6, 38, 0, 0, 2, 42, 18, 0, 0, 46, 0, 10, 0, 50, 22, 0, 0, 54, 0, 0, 6, 58, 26, 14, 0, 2, 62, 0, 0, 0, 0, 66, 30, 0, 0, 0, 70, 0, 18, 10, 0, 74, 34, 0, 0, 0, 78, 0, 0, 0, 6, 82, 38, 22, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Gives an identity for the twice sigma function (A074400), the sum of the even divisors of 2n.

Alternating sum of row n equals A074400(n), i.e., sum_{k=1..A003056(n))} (-1)^(k-1)*T(n,k) = 2*A000203(n) = A074400(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

The number of positive terms in row n is A001227(n).

For more information see A196020.

LINKS

Table of n, a(n) for n=1..76.

FORMULA

T(n,k) = 2*A196020(n,k).

EXAMPLE

Triangle begins:

2;

6;

10,  2;

14,  0;

18,  6;

22,  0,  2;

26, 10,  0;

30,  0,  0;

34, 14,  6;

38,  0,  0,  2;

42, 18,  0,  0;

46,  0, 10,  0;

50, 22,  0,  0;

54,  0,  0,  6;

58, 26, 14,  0,  2;

62,  0,  0,  0,  0;

66, 30,  0,  0,  0;

70,  0, 18, 10,  0;

74, 34,  0,  0,  0;

78,  0,  0,  0,  6;

82, 38, 22,  0,  0,  2;

86,  0,  0, 14,  0,  0;

90, 42,  0,  0,  0,  0;

94,  0, 26,  0,  0,  0;

...

For n = 9 the divisors of 2*9 = 18 are 1, 2, 3, 6, 9, 18, therefore the sum of the even divisors of 18 is 2 + 6 + 18 = 26. On the other hand the 9th row of triangle is 34, 14, 6, therefore the alternating row sum is 34 - 14 + 6 = 26, equaling the sum of the even divisors of 18.

If n is even then the alternating sum of the n-th row of triangle is simpler than the sum of the even divisors of 2n. Example: for n = 12 the sum of the even divisors of 2*12 = 24 is 2 + 4 + 6 + 8 + 12 + 24 = 56, and the alternating sum of the 12th row of triangle is 46 - 0 + 10 - 0 = 56.

CROSSREFS

Cf. A000203, A000217, A001227, A003056, A016825, A074400, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A236104, A236112.

Sequence in context: A240761 A117541 A244060 * A095105 A220338 A052194

Adjacent sequences:  A236103 A236104 A236105 * A236107 A236108 A236109

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Jan 23 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 24 03:51 EDT 2019. Contains 326260 sequences. (Running on oeis4.)