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A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092. 4
3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

LINKS

Table of n, a(n) for n=1..57.

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

FORMULA

a(n) = A024916(n) + A002378(n).

a(n) = A153485(n) + A045943(n).

a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020

EXAMPLE

For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.

For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

PROG

(PARI) a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

CROSSREFS

Partial sums of A224880.

Cf. A002378, A024916, A045943, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Sequence in context: A089693 A194116 A208951 * A246520 A338631 A272764

Adjacent sequences:  A299689 A299690 A299691 * A299693 A299694 A299695

KEYWORD

nonn

AUTHOR

Omar E. Pol, Mar 06 2018

STATUS

approved

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Last modified March 4 14:10 EST 2021. Contains 341794 sequences. (Running on oeis4.)